"Balancing the Partner's Contribution" - Analyzing the Risks and Benifits in the case of an Indian Joint Venture.
A Joint Venture can be defined an agreement between two or more firms, leading to the establishment of a separate legal entity. The firms may be held responsible for their own individual tasks for which they have special expertise, knowledge or past achievements. For instance, a new firm might be responsible for supplying inputs for a subset of the firms, conducting research and development activities or downstream activities such as marketing and distribution of the products. The participants in a Joint Venture agree to share capital, human resources, risks, profits, technology that may be needed for the formation of the separate entity under shared control. Fuji-Xerox is an example of a successful joint venture.
The Post World War II diversification strategies of large corporations and the existence of globalization have led to a rapid proliferation of strategic alliances. The presence of an increasing number of alliances between firms is especially felt in the case of developing countries. Countries like India and China now provide the best examples for alliance strategies that result in a Joint Venture, or in Business Process Outsourcing, Mergers etc. as these countries stand on the cusp of explosive growth.
Thus, I take the opportunity to analyze the potential risks and benefits for the partners of a joint venture and evaluate how they can deal with the delicate issue of balancing their individual contributions into the alliance with their returns from it. I will analyse the case of an Indian Joint Venture involved in textile manufacturing, where I have previous work experience for almost 3 years. This Joint Venture consists of 3 main participant firms who shere the same general cultural background (that is, they all belong to the same nation, namely, India) but who might have differences in terms of their individual objectives and expectations with regard to the alliance. This research will, therefore, mainly reflect on whether all partners benefit equally from their participation in the common venture and highlight a number of potential limitations and factors that might determine the success or failure of the Joint Venture.
In analysing the case study, I will particularly emphasize the limitations as they may relate directly to the the failure of a Joint Venture. It has been estimated that one fourth of all joint ventures last less than 3 years and the average life span for such cooperative alliances is about 7 years. Several studies have reported failure rate as high as 80 percent, failure that usually leads to their dissolution or acquisition by one of the partners (Gulati et all, 1994). Many researchers have mainly stressed the drawbacks which relate to International Joint Ventures, and highlighted factors such as Cultural Differences, Language barriers, Political Uncertainty and Instability etc. In order to differentiate my study from the mainstream, and to provide an original contribution I have chosen to highlight the difficulties and limitations that are further involved and exist in a national Joint Venture. In my research I would like to highlight the key topics of trust amongst the partners, their motives and their processes of partner selection as the literature on Joint Ventures indicate their relevance for the maintenance of a sound balance between partners' contributions and returns. Moreover, the shifts in partners' bargaining power which might emerge during the life of the Joint Venture and may cause insecurity, thereby undermining its probability of success, is another critical aspect that I would like to incorporate in my study.
According to Fitzpatrick (1998), in order to contribute to understanding complexity, researchers have to choose from a range of possible methodologies and cast aside entrenched theoretical & ideological positions. For my research I would like to adopt a fundamentally qualitative research method. Quantitative research methods are, in fact, useful for answering questions like how many, how often and how much. Moreover, qualitative research methods are useful in understand peoples' behaviours and their attitudes. Qualitative research methods would provide a more in depth analysis and make the research more meaningful by assisting me in better understanding the motives of the joint venture under study and the reasons of its failure.
As previously stated, I will be analysing the case of the alliance strategy of a textile manufacturing company in India called Avis Industries (P) Ltd.* This organisation has textile manufacturing units in the cities of Mumbai and Surat and markets its product throughout India. The company entered into a Joint Venture with one of its main buyers and formed a separate entity called Avis Impex (P) Ltd. This particular Joint Venture lasted for little more than a year after which the partner who was initially the buyer opted out and started a competing parallel firm called the Lotus creations (P) Ltd.* in the field of manufacturing. Lotus creation (P) Ltd. had gained all its knowledge about the production process, the suppliers and the buyers in the industry through the participation in the Joint Venture, and therefore this case appears to be particularly suited to the study of the problems and risks associated with the need to maintain a balance between partners in a Joint Venture in terms of their respective contributions and returns.
The research will involve a mix of primary and secondary sources. My main source of primary data will my own experience and contact with the firm constituting the case study. Since I was working in Avis Industries (P) Ltd., I will use my detailed knowledge of that company and past experience as a participant observer in the firm and its involvement with the Joint Venture. Also, data will be collected through interviews with the directors and executives of the firm. This will enable me to understand the complex circumstances under which the manufacturing firm entered into the Joint Venture and also the circumstances leading to the venture being dissolved in such a short span of time.
Secondary sources for my research would be equally important. Sources such as books, journals, past researches and dissertations, financial and economic times, electronic database, company reports and agreements or contracts (if available), and various search engines and meta search engines will be useful for me to outline the key framework within which my research will be conducted. It will also be useful in successfully analyzing and interpreting the primary data in the most appropriate manner.
The reason alliances are formed at all is that most participants expect to get better payoffs than they would without the alliances (Gulati et all,1994). The main aim of the research is somewhat similar to the above statement i.e, to determine whether all partners benefit equally by participating in a Joint Venture.
b) Analyzing the case of Avis Impex (P) Ltd.
c) To understand the keys issues such as partner selection, trust and motives behind entering into a joint venture.
d) Limitations involved in the joint Venture.


text:
Sony would be surprised by publisher DRM for PS4 games
Written by David Hing
Tags: #drm #ps4
Companies: #e3 #sony
Sony considers it unlikely that third-party publishers will want to bring DRM limitations to the PS4 titles.
Sony senior vice-president of World Wide Studios America, Scott Rohde, said that it would be 'surprising' if publishers wanted to bring game-access limitations similar to those described for the XBox One to the PS4 following the overwhelmingly positive reception Sony got at E3 for refusing to follow Microsoft's lead.
'Technically, they could do something, but it's the standard we're setting that we believe is the right standard and I believe that's the way it's going to work,' Rhode told Polygon.
He added that all the publishers that could have the option to implement DRM systems for their games were present for the Sony press conference at E3 and all heard the cheers of attendees when it was made clear that the PS4 would not have the same game-sharing restrictions as the Xbox One.
Rohde went on to dismiss the fact that Sony filed a patent relating to the limitation of second-hand sales of games last year was simply due to the fact that Sony is a large company that is always looking at different ways that the business could work.
Sony is considered by many industry pundits as the 'winner' of E3 this year after it revealed its next generation console would not have any of the features that the Xbox One has to fight used game sales and would not require a semi-persistent internet connection.
Microsoft subsequently attempted to show Sony up as having merely hidden DRM tactics, but this was later revealed to be Sony saying that implementing DRM is still an option for individual publishers, in much the same way that online passes have worked during the current console generation.
However, in a series of tweets yesterday, veteran developer Cliff Bleszinski accused Sony of 'playing on the internet outrage for free PR' in relation to its stance on used games, claiming that current expectations for triple-A titles are not financially sustainable with used and rental games as a possibility.
US Copyright Office adds 'right to repair' exemptions
They don't cover consoles, though.
October 26, 2018 | 11:05
CD Projekt launches FCK DRM campaign
Promotes GOG.com, naturally.
Chipworks publishes PS4 chip die analysis
Suggests there's room for future growth.

redpajama_set_name: RedPajamaCommonCrawl


A crucial theological question is central to spirituality. Our concern here is not with something that we feel or experience as "beyond" or "transcendent," but with the issue of who is speaking and acting, who is calling, and who is addressing us. Of course, impersonal entities neither speak nor act, and if we have little or no accurate information about the referent of our spirituality, we should have some serious questions. A real and genuine spiritual connection, through the redemptive work of the crucified One to the personal God who is actually there, results in release from sin, a changed heart, and a transformed mind—life amidst the divine community. From Living Spirituality: Illuminating the Path.


IPG works intensively with approximately 300 education accounts and sends hundreds of personalized catalogs and samples to customers who require them. IPG attends Ed Expo, the largest national school supply trade association meeting, and the Educational Book and Media Association meeting, during which the IPG education specialist builds new and fosters current relationships with educational wholesalers from around the country.
Success with the education accounts requires a highly disciplined approach. The buyers are experts in their fields and will not wade through long catalogs hoping to find a title or two that is relevant to their interests. IPG's experienced education rep presents to these buyers only titles that they will want to see.
Before joining IPG, Karen worked as a buyer for Scholastic Book Fairs for eight years and at Books Are Fun for three years. She is a former elementary school teacher and assistant principal. Her specialty is the educational supply market, a diverse group of accounts that includes teacher supply stores, home school mail order catalogs, preschools, teaching facilities for senior citizens, trade schools, special needs and gifted education accounts, as well as several book fair accounts.


text:
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redpajama_set_name: RedPajamaWikipedia


text:
News Sports Opinion Obituaries USA TODAY eNewspaper Legals
Scottoline finds humor in the truth
Best-selling author to headline Palm Springs Speaks series on Monday night
L.J. Gambone For the Daily Press
In looking for ways to unleash her creativity over and over again, prolific humor writer/crime novelist Lisa Scottoline said she has found that the truth never fails to set her free.
She admits that looking at oneself through an honest lens isn't always easy but she believes that in order for her writing to resonate with her readers, and make them laugh, it must come from an uncensored and authentic place.
"I'm always aiming to write something that rings true or feels real for people," said Scottoline, who is set to appear at 7:30 p.m. Monday in the Palm Spring Speaks lecture series at the Camelot Theatres in Palm Springs.
"I always say that if whatever I'm writing doesn't make me cringe, it won't make the reader laugh. For instance, when you find your first chin hair and realize you're turning into an Amish man.
"Noone wants to talk about such an embarrassing thing, which is why it's so funny when someone does actually bring it up."
Scottoline, a native of Chester County, PA, said this will be her second time speaking in Palm Springs and she is ecstatic about her upcoming lecture. She will discuss her creative process and also how she finds inspiration for her stories in her everyday experiences.
"I love the people from the Palm Springs community," said Scottoline, a former attorney. "They love to read, which is a great thing because books are a wonderful way to feel connected to others. There is so much division among people these days and we kind of forget about the bonds we have, as well as our need to connect with our tribe."
Scottoline's work has been resonating with readers since 1994 when her first crime novel, "Everywhere That Mary Went," was published by HarperCollins. It became a bestseller and was nominated for an Edgar Award by the Mystery Writers of America.
Scottoline went on to write an additional 29 novels, all of which have appeared on bestseller lists, including the New York Times, Wall Street Journal, USA Today, Los Angeles Times, Entertainment Weekly and Publisher's Weekly.
She and her daughter Francesca Serritella also co-write a Sunday humor column for the Philadelphia Inquirer titled "Chick Wit." Those columns have been turned into a series of memoirs, including the New York Times bestseller, "I Need A Lifeguard Everywhere But The Pool."
"My novels revolve around strong female characters," said Scottoline. "And they focus on family and relationships because in the end, that's the stuff that really matters."
Scottoline's upcoming appearance in Palm Springs will conclude the Palm Springs Speaks inaugural series, which is presented by the Palm Springs Cultural Center and Friends of the Palm Springs Library. Speakers that preceded her were television personality Dan Savage, former Vice President Al Gore and self-help author Deepak Chopra M.D.
Tickets range from $30 to $95 and can be purchased online at palmspringsspeaks.org or at the Camelot Theatres' box office, 2300 E. Baristo Road in Palm Springs. For box office hours call 760-325-6565.
For more information, visit www.palmspringsspeaks.org. Visit Scottoline at www.scottoline.com
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Home brain homework help A history of erin brockovich ellis an american legal clerk
A history of erin brockovich ellis an american legal clerk
After earning an Assoicates degree in Applied Arts, trying out several jobs, and divorcing twice, Erin found herself alone with three children to provide for. After being seriously injured in a traffic accident in Reno, Erin moved back to Southern California with her children.
He gained prominence for his portrayal of the taxi dispatcher Louie De Palma in Taxi which won him a Golden Globe, confidential and for his voiceover in such films as Space Jam, Hercules and The Lorax. The production company is known for such as Pulp Fiction, Garden State.
He directs, produces and appears in graphic, short, horror films for his Internet venture The Blood Factory, deVitos short stature is the result of multiple epiphyseal dysplasia, a rare genetic disorder that affects bone growth in those afflicted.
DeVito was raised as a Catholic and attended Oratory Preparatory School and he went to the boarding school when he was 14 after he persuaded his father to send him there to keep him out of trouble.
After leaving the school, he trained at the American Academy of Dramatic Arts. The production began performances at the American Airlines Theatre on February 16, and is set to open on March 16 for a limited run through May 7.
DeVito has become a film and television producer. Roberts was the actress in the world throughout most of the s. Her older brother, Eric Roberts, from whom she was estranged untilsister Lisa Roberts Gillan, Robertss parents, one-time actors and playwrights, met while performing theatrical productions for the armed forces.
They later co-founded the Atlanta Actors and Writers Workshop in Atlanta and they ran a childrens acting school in Decatur, Georgia, while they were expecting Julia.
The children of Martin Luther King, Jr. Robertss hospital bill when Julia was born. Inher mother married Michael Motes, who was abusive and often unemployed, the couple had a daughter, Nancy Motes, who died at 37 on February 9, of an apparent drug overdose.
The marriage ended inwith Betty Lou divorcing Motes on cruelty grounds, Robertss own father died of cancer when she was ten. Roberts wanted to be a veterinarian as a child and she also played the clarinet in her school band. Once there, she signed with the Click Modeling Agency and enrolled in acting classes, Roberts made her first big screen appearance in the film Satisfaction, alongside Liam Neeson and Justine Bateman, as a band member looking for a summer gig.
She had previously performed a role opposite her brother, Eric, in Blood Red, filmed in Her first television appearance was as a rape victim in the initial season of the series Crime Story with Dennis Farina, in the episode titled The Survivor.
Her first critical success with moviegoers was her performance in the independent film Mystic Pizza inthat same year, she had a role in the fourth-season finale of Miami Vice 3. Born in Cupertino, California, he moved to the UK at age 13, several years later, he began his acting career by performing in school plays, before moving to Sydney, Australia for his high school senior year.
He left high school without graduating, but earned a diploma through an education course. For much of the mids, he lived in New York City as a struggling, as an undergraduate at BYU, Eckhart met director and writer Neil LaBute, who cast him in several of his own original plays.
Eckhart was born on March 12, in Cupertino, California, the son of Mary Martha Eckhart, a writer, artist, and poet, and James Conrad Eckhart and he is the youngest of three brothers.
Eckharts family relocated to England infollowing his fathers job in information technology, the family resided in Ripley, Surrey, then moved onto Walton-on-Thames, later moving to Cobham.
He eventually earned his diploma through an education course. His first television roles were in commercials, in he appeared as an extra on the television drama series Beverly Hills, Eckhart followed this part with roles in documentary re-enactments, made-for-television movies. Cochrane, one story has Laemmle watching a box office for hours, counting patrons and calculating the days takings.
Within weeks of his Chicago trip, Laemmle gave up dry goods to buy the first several nickelodeons, for Laemmle and other such entrepreneurs, the creation in of the Edison-backed Motion Picture Trust meant that exhibitors were expected to pay fees for Trust-produced films they showed.
Soon, Laemmle and other disgruntled nickelodeon owners decided to avoid paying Edison by producing their own pictures, in JuneLaemmle started the Yankee Film Company with partners Abe Stern and Julius Stern. Laemmle broke with Edisons custom of refusing to give billing and screen credits to performers, by naming the movie stars, he attracted many of the leading players of the time, contributing to the creation of the star system.
Laemmle, who emerged as president in Julywas the figure in the partnership with Dintenfass, Baumann, Kessel, Powers, Swanson, Horsley.
Erin Brockovich Latest News
Eventually all would be out by Laemmle. Following the westward trend of the industry, by the end of the company was focusing its efforts in the Hollywood area.
On March 15, Laemmle opened the worlds largest motion picture production facility, Universal City Studios, studio management became the third facet of Universals operations, with the studio incorporated as a distinct subsidiary organization.
Unlike other movie moguls, Laemmle opened his studio to tourists, Universal became the largest studio in Hollywood, and remained so for a decade. However, it sought an audience mostly in towns, producing mostly inexpensive melodramas, westerns.
In its early years Universal released three brands of feature films — Red Feather, low-budget programmers, Bluebird, more ambitious productions, and Jewel, their prestige motion pictures.Erin Brockovich, legal clerk and environmental activist, speaks as part of Saint Louis University's Great Issues Committee speaker series.
November 7, A reception is . Erin Brockovich is an American legal clerk and environmental activist, who, despite her lack of formal education in the law, was instrumental in building a case against the Pacific Gas and Electrical Company of California in Famous Paralegals biographies and history.
Greta Goodwin. Greta Hall Goodwin was a Democratic member of the Kansas Senate, representing the 32nd District. Erin Brockovich, a twice-divorced, unemployed mother of two young children and an infant, consults lawyer Ed Masry of the Van Nuys, California firm Masry and Vititoe, regarding claims for injuries she suffered in an automobile accident that was not her fault.
"Erin Brockovich-Ellis (born June 22, ) is an American legal clerk and environmental activist who, despite the lack of a formal law school education, or any legal education, was instrumental in constructing a case against the Pacific Gas and Electric Company (PG) of California in ".
Masry & Vititoe, the law firm for which Brockovich was a legal clerk, received $ million of that settlement, and Brockovich herself was given a bonus of $ million. Brockovich's work in bringing litigation against Pacific Gas and Electric was the focus of the feature film, Erin Brockovich, starring Julia Roberts in the title role.
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Erin Brockovich (VHS, ) & Titanic - 2 Romantic VHS | eBay


(linear, defined below) to run in a second window.
produces an output report right below the form.
input data form for that function  in Window #2.
the user decides to exit from the system.
the output reports from those functions.
Below are simplified masterForm and linear function defintions.


Please refer to the reference manual of i.MX6ULL(https://www.nxp.com/docs/en/reference-manual/IMX6ULLRM.pdf). In the page 3126, there's a description about MCLK Select as follows.
information for the meaning of each option.
00 Master Clock (MCLK) 1 option selected.
01 Master Clock (MCLK) 1 option selected.
10 Master Clock (MCLK) 2 option selected.
11 Master Clock (MCLK) 3 option selected.
Would someone please explain the meaning of each option (1,2,3)?, or please let me know where the chip specific information is?
I got it. Thank you very much for your quick and clear answer!


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Q: Can I use a variable address in an ISBLANK() statement I'm trying to add conditional formatting into a vba code and am having trouble with my ISBLANK statement.
Is it possible to use a variable address to define the cell I would like to check?
Here is what I have so far.
`Sub Employee_Training_Record()
Dim lastrow As Long
Dim lastcol As Long

If Cells(3, 2).Value = "" Then
MsgBox "No Employees in List"
GoTo gohere1:
ElseIf Cells(2, 3).Value = "" Then
MsgBox "No Training in List"
GoTo gohere1:
End If

lastrow = Cells(30, 2).End(xlUp).Row
lastcol = Cells(2, 30).End(xlToLeft).Column

employee = InputBox("Enter Employee Name")
Do While employee = ""
employee = InputBox("Enter Employee Name")
Loop
For i = 3 To lastrow
    a = 0
    If Cells(i, 2).Value = employee Then
    GoTo gohere:
    Else
    a = 1
    End If
Next
If a = 1 Then
MsgBox "Employee Not Found"
GoTo gohere1:
End If
gohere:

trning = InputBox("Enter Training Type")
Do While trning = ""
trning = InputBox("Enter Training Type")
Loop
For i = 3 To lastcol
    b = 0
    If Cells(2, i).Value = trning Then
    GoTo gohere5:
    Else
    b = 1
    End If
Next
If b = 1 Then
MsgBox "Training Not Found"
GoTo gohere1:
End If
gohere5:

exprnc = InputBox("Enter Expiration Date")
For i = 3 To lastrow
    If Cells(i, 2).Value = employee Then
    GoTo gohere3:
    End If
Next
gohere3:
For j = 1 To lastcol
    If Cells(2, j).Value = trning Then
    j = j + 1
    GoTo gohere4:
    End If
Next
gohere4:

Cells(i, j).Value = exprnc

    Cells(i, j).FormatConditions.Add Type:=xlCellValue, Operator:=xlLess, _
        Formula1:="=NOW()+60"
    Cells(i, j).FormatConditions(Selection.FormatConditions.Count).SetFirstPriority
    With Cells(i, j).FormatConditions(1).Interior
        .PatternColorIndex = xlAutomatic
        .Color = 49407
        .TintAndShade = 0
    End With
    Cells(i, j).FormatConditions(1).StopIfTrue = False

    Cells(i, j).FormatConditions.Add Type:=xlCellValue, Operator:=xlLess, _
        Formula1:="=NOW()"
    Cells(i, j).FormatConditions(Selection.FormatConditions.Count).SetFirstPriority
    With Selection.FormatConditions(1).Interior
        .PatternColorIndex = xlAutomatic
        .Color = 255
        .TintAndShade = 0
    End With
    Cells(i, j).FormatConditions(1).StopIfTrue = False

    addrss = Cells(i, j).Address

    Cells(i, j).FormatConditions.Add Type:=xlExpression, Formula1:="=ISBLANK(addrss)=True"
    Cells(i, j).FormatConditions(Selection.FormatConditions.Count).SetFirstPriority
    Cells(i, j).FormatConditions(1).StopIfTrue = True

    MsgBox "Cell Address" & Cells(i, j).Address()

gohere1:
End Sub`

redpajama_set_name: RedPajamaStackExchange


Welcome, www.howardbm-ltd.co.uk which is at the forefront of women and men's fashion house. We bring a lot of experience, you can fashion shopping - his style, dresses, jackets, tops, shoes and fashion. The price of the brightest stars in the discovery of a unique style that is now set to the menu.


• Rs. 9500 To 15000 Fixed Per Month.
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➽ More Details: Please Read Official Advertisement.
➽ How to Apply: Interested & eligible candidates can attend the walk-in-interview with a detailed bio-data in prescribed format.
Place: Please Read Official Advertisement.
➤ How to Apply –Interested candidates may apply online through the official website of SSC Online from21-July-2018to20-August-2018.
DOWNLOAD GUJARAT ROJGAR SAMACHAR DATE: 11/07/2018.
Industrial Training Institute (ITI) Chotila has published Advertisement for visual Requirement 2018. Other details like age limit, educational qualification, selection process, application fee and how to apply are given below.
Directorate of Employment and Training (DET), Government of Gujarat conducts the admission process for the various privately and publically owned ITI colleges of the state of Gujarat.
All the candidates, who have completed their secondary or senior secondary education from any recognized education board of State or Central government, can apply online by visiting the official website of DET, Gujarat. Aspirants should submit the duly filled application form on or before the due date. Here, candidates can find all the details about the Gujarat ITI Admission 2018 Application Form, Eligibility Criteria, Admission Process and the Important Dates.


ARS Researchers studied potato tubers from the Klamath Basin in Oregon infected with the causal bacteria of zebra chip disease. They confirmed the presence of the pathogen but did not identify the sample as one of the six known varieties (haplotypes).
The UK's approach to weed management investment needs to be overhauled, if the needs of industry are to be met. That was the conclusion of the first ever major cross-sector review of weed management, commissioned by AHDB and BBRO.


If you want to contact us or book a riding tour with us fill the form bellow or send us an email to info@sloveniahorseriding.com or call us by phone +386 41 377 030.
For further information feel free to give us a call. We will answer all your questions with a smile! We are open all year around.
Email Field is required. Incorrect email.
Country and Phone number Field is required.
Name and address of accomodation - Do you need a pick up? Field is required.
Your age: _______ Height: _______ Weight: _______ Field is required.
Please read the following Terms & Conditions carefully. Note that the following is at the discretion of LIFEIME EXPERIENCE and rates/itineraries are subject to change. Please return your signed contract below to sLOVEnia HORSE RIDING reservations office by email:info@sloveniahorseriding.com.
Important: Please ensure that you are familiar with our riding policy. We reserve the right to prevent any client from riding if we feel they do not fulfill the minimum competence level as indicated.
When the booking is done there is no money refund in case of bad weather, no-show, not appropriate equipment or other failure to comply with our rules.
Cancellation On confirmation of a booking an invoice will be issued and your booking will then be subject to the below cancellation policy. All cancellations must be in writing.
Confirmation – 61 Days prior to date of arrival: 30% of total safari cost is forfeited. 60 – 31 Days prior to date of arrival: 50% of total safari cost is forfeited. 30 – 0 Days prior to date of arrival: 100% of total safari cost is forfeited.
Should a client, for any reason, fail to join a safari or join it after departure or leave it prior to its completion, no refund can be made.
Please quote your invoice number on the payment and instruct your bank that the beneficiary (Lifetime Experience) should receive the full invoice amount, with no charges deducted.
Account Name: Lifetime Experience, Account number: SI56 6100 0002 0836 529, Bank Name: Delavska hranilnica d.d. Ljubljana: SWIFT Address: HDELSI22.
Be prepared to spend up to six hours a day in the saddle. You will need to feel confident about keeping up with the group, capable of riding at all paces, rising to the trot and controlling your horse at the canter. Riders need to be over 12 years old, be riding fit, and weigh no more than 95 kgs (210 lbs or 15 stone). Riders who are over the weight limit would need to make special arrangements for this with our reservations team and need to be advanced riders. We take a maximum of eight people in each riding group to ensure the best possible standards of safety and enjoyment. We are using English or Western style trail saddles.
If you are not riding regularly we strongly recommend that you get in practice before joining us on safaris. It will be at the discretion of the company whether less experienced riders will be permitted to ride, taking into consideration their safety and enjoyment and that of the group as a whole. We reserve the right to prevent any client from riding if we feel they do not fulfill the minimum competence level detailed above.
Many insurance companies will only insure you when wearing a hard hat or safety helmet whilst in the saddle. It is your responsibility to provide your own properly fitted hard hat or safety helmet. We cannot accept any liability whatsoever for head injuries sustained as a result of ignoring this advice. You must be insured to ride with us - please ensure your policy covers you for horse riding and includes medical evacuation cover.
You must be fully aware and understand that there are risks involved when around horses, in riding generally and in visiting and staying in remote wilderness areas due to the presence of wild and potentially dangerous animals. For these reasons you will be required to sign an indemnity form before you ride on our safaris confirming that you do so entirely at your own risk and that you have adequate insurance for all eventualities.


text:
← Durham Academy: "I Don't Say" Anything About The First Amendment
NC Common Core #ASRC June Meeting Update →
Speech By Presidential Hopeful Sen. Ted Cruz Fires Up The Crowd At The NC GOP Convention
Posted on June 15, 2015	by A.P. Dillon
Applause and cheers greeted 2016 presidential candidate Senator Ted Cruz as he took the stage at the North Carolina Republican Convention.
Opting to move around instead of speaking from behind a podium, Cruz sounded Reaganesque as he spoke on three main points of economic growth, defending our Constitutional rights and restoring America's position in the world.
Cruz's speech was fluid, fast-moving, had the crowd laughing and on their feet applauding multiple times. The speech lasted just over 40 minutes — and he did it all without a teleprompter.
We're seeing tremendous enthusiasm here in NC, and it's a privilege to join y'all at #NCGOPConvention today! #ncpol pic.twitter.com/9LGMdGGYUJ
— Ted Cruz (@tedcruz) June 6, 2015
Senator Cruz warmed up the crowd by thanking them for retiring Senator Kay Hagan and electing Senator Thom Tillis in her place. The Senator then transitioned into a 'central issue' of his campaign, which is "reigniting the promise of America." and the fundamental idea that our kids will 'have it better' than we did.
On economic growth, Cruz pressed the need for tax reform by scrapping the current tax code and adopting a flat-tax. Cruz added that regulation reform was also needed and cracked a joke, that was captured by TWC News, about firing the 90,000 IRS workers and putting them on the border.
"There are about 90,000 employees at the IRS. We need to padlock that building and take everyone of them and put them on our Southern border," he said.
While some of Sen. Cruz's speech could be considered tongue in cheek, some delegates say they appreciated the honesty.
TWC News did not include the punchline of the joke. "If the first thing you see is 90,000 IRS agents," said Cruz, "you'd turn around and go home too,". The audience rippled with applause and laughter.
Obamacare, which the Senator has long opposed, was also made mention of. "We should repeal every word of Obamacare.", the Senator said. Cruz noted Obamacare as being a jobs killer and the recent propose rate hikes by insurance companies, including the 27% hike proposed North Carolina's Blue Cross Blue Shield.
"I will always, always, always stand and fight for religious freedom!", exclaimed Cruz while speaking about defending the Constitution. Senator Cruz also defended the Second Amendment and the 10th, slamming the Obama administration on the latter.
Unlike Governor Walker, Senator Cruz's comments on education almost immediately went to Common Core. Cruz cited the need for the states to 'repeal every word of Common Core'; stressing the need to get rid of the Dept. of Education and return to state and local control of education.
On Cruz's final leg of the speech, he turned towards the failed foreign policy of the Obama administration, condemning the administration's negotiations with Iran. Cruz said we should not be allowing Iran to obtain nuclear weapons.
Senator Cruz criticized President Obama's broken relationship with Israel and for not meeting with Prime Minister Netanyahu. Cruz said, "Imagine a President standing unapologetically next to the Prime Minister of Israel." This statement received the biggest and loudest standing ovation of the afternoon.
Scott Walker Brought the Audience at the North Carolina GOP Convention To Its Feet
On Common Core, I had Sen. Ted Cruz "at Hello"
This entry was posted in ELECTIONS, EXCLUSIVE, LL1885 and tagged Election 2016, NCGOP, NCGOP Convention, Ted Cruz. Bookmark the permalink.
2 Responses to Speech By Presidential Hopeful Sen. Ted Cruz Fires Up The Crowd At The NC GOP Convention
billbe says:
To bk. Senator Cruz and a few other Repubs have the skill and moral leadership to force the Repub RoINO's like Bohner to change Big O's no health care to an improved private plan. It CAN happen if the people make it clear that it MUST be changed. Are YOU going to do anything to help change the direction away from DemacRAT Progressive Statism, or are you just a low life troll trying to stur the pot?
brackenkaren says:
Ted Cruz makes great speeches but like most great speech makers that is about all he will do. If he is elected I can guarantee 2 things. He will NEVER repeal Obamacare and he will NEVER shut down the US Dept. of Education. And when he doesn't do that what will the American people do about it? Absolutely nothing. As a matter of fact they will probably give him a second term in office. This is the problem we fall in love with a candidate on his charm and rhetoric but never look deep into his/her background, And even many that do dismiss it because "he is such a nice guy" and has told us everything we want to hear. Sorry Ted Cruz is more of the same. And the fact he is not a natural born citizen also makes him not eligible to run. But again we ignore the facts because "he is such a nice guy."


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Electra Partners announces the refinancing of Park Resorts
Electra Partners is pleased to announce that it has led the refinancing of Park Resorts' debt facilities. Following completion of the refinancing, funds managed or advised by Electra Partners (including Electra Private Equity PLC and third party investors) will become the majority shareholders in the company with a 54% stake. Existing shareholder GI Partners will become a minority shareholder and management will take a significant stake. The refinancing is signed and is expected to complete within two weeks.
Park Resorts is a leading operator of holiday parks offering caravan holidays and holiday homes at its 39 sites across England, Scotland and Wales.
Upon completion of the refinancing, funds managed or advised by Electra Partners will hold senior debt and PIK notes with a par value of £145 million.
The refinancing of the existing debt facilities will reduce the company's senior debt from £353 million to £316 million and extend its maturity to December 2017. The refinancing will also allow £60 million of capital to be made available for investment to improve facilities in existing holiday parks.
Along with the refinancing, Park Resorts has also announced the appointment of David Boden as CEO. David has over 30 years' experience in the leisure industry having held senior Board level positions at Rank Group Plc and Hippodrome Casinos Ltd. David replaces founder David Vaughan who, following 12 years as CEO, becomes Chairman.
Alex Fortescue, Chief Investment Partner at Electra Partners, said:
"This deal showcases the flexibility of the capital we manage for Electra Private Equity PLC. The initial investment was in Park Resorts' debt, and upon completion our clients will take a controlling equity position through a transaction which gives the company the solid financial foundation it requires to continue its growth strategy."
David Vaughan, Chairman of Park Resorts said:
"I'm thrilled to have worked successfully with Electra Partners to create the right platform to grow and develop the business. There are many exciting opportunities to improve park facilities and to add to the business and we are looking forward to working to deliver these over the coming years."
Alex Fortescue and Sarah Williams are responsible for the investment in Park Resorts and both represent Electra Partners on the Board of the company.
2013 has been a busy year for Electra Partners, announcing five investments and two realisations, the most recent of these being the US$1.35 billion sale, and subsequent US$85 million (£57 million) reinvestment in animal tagging business, Allflex.
Electra Partners refers to Electra Partners LLP and/or EQM Capital LLP acting on behalf of Electra Private Equity PLC and other third party investors.
For Alex Fortescue or Sarah Williams:
Charlotte McMullen, M:Communications +44 (0)20 7920 2349
Nicholas Board and Andrew Kenny, Electra Partners +44 (0)20 7306 3902
About Electra Partners LLP
Electra Partners is an independent private equity fund manager with over 30 years' experience in the mid-market buyout sector. During the last 25 years it has invested in excess of £3 billion in over 150 deals. As at 31 March 2013, the firm had funds under management of over £1.6 billion. Electra Partners' flexible investment approach allows it to invest broadly across the private equity market with particular focus in three areas: buyouts & growth capital, secondaries and debt.
The firm's major client is Electra Private Equity PLC, a private equity investment trust which has been listed on the London Stock Exchange since 1976.
For further information please visit www.electrapartners.com.
Electra Partners LLP is authorised and regulated by the Financial Conduct Authority.
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Vegan YouTuber Rawvana has apologised to fans after they discovered she was no longer vegan when she was caught eating fish.
The YouTube star – real name Yovana Mendoza Ayres – uploaded a 33-minute video apologising and explaining to her fans.
'I just want to say that I'm really really sorry about the way the news was communicated to you,' she began the video.
The 28-year-old recalled adopting a 'raw vegan' diet six years ago, and it helping her to overcome addictions to alcohol and smoking.
'I felt on top of the world,' she recalled.
However, she revealed that two months ago, she began to incorporate eggs and fish into her diet for health reasons.
She went on to explain that her period had stopped after completing a water fast, she was diagnosed as pre-menopausal, and she was borderline anemic.
However, people were simply not having it with many pointing out that she was doing raw vegan challenge videos during the period when she was re-incorporating eggs and fish into her diet.
'No, this isn't an "I'm human I've made a mistake" thing,' one viewer commented.
'This video screams I got caught, so I have to make this,' another added.
'Fish are friends, not food,' one more quipped.
'You addressed everything but the fact you continued to PRETEND to be a vegan for PROFIT!' an angry viewer fumed.
Yovana has stayed away from social media since the controversy.
Metro.co.uk has contacted reps for Yovana for comment.


IPM works in all environments where food is produced and wherever people live, work, go to school, and recreate. In these environments, pests, including traditional and many invasive and emerging species bring adverse effects, either through lower quality food, loss of biodiversity, unhealthy conditions, or reduced integrity of the built environment. A safe and nutritious food supply that is accessible by local residents and reduces harmful and excessive inputs is increasingly important for sustaining communities. Using IPM methods, growers can protect human health as well as that of the environment.
One large project that our Center is integrally involved in is the biology, ecology, and management of the brown marmorated stink bug (BMSB). The brown marmorated stink bug is a voracious pest that was unintentionally introduced to the northeastern United States in the 1990s. It has quickly become a major agricultural and nuisance pest and continues to spread throughout the United States. It feeds on a long list of crops with damages estimated at $37 million to the mid-Atlantic apple industry alone. In partnership with scientists and researchers across the country, our Center is helping to communicate their work in developing IPM solutions to help control this pest and reduce its impacts. Our Center created and now maintains the website for this national project at www.stopbmsb.org.
Another effort that our Center is leading is the improvement of pest control in affordable housing through the project "Promoting IPM in Affordable Housing," which teaches managers, maintenance staff, and residents how to use IPM. Efforts are centered on eliminating obstacles to using IPM in affordable housing and refining an implementation process for housing providers. Indoor pests, including molds, common insects such as cockroaches and bed bugs, and rodents can contribute to health issues. In addition to providing hands-on training, our Center created a website, www.stoppests.org, to share IPM solutions and resources to help people take control of their pest management.
Our Center will continue to broaden its efforts in rural and urban settings. Partnerships will continue to be developed with building managers, grower groups, retailers supplying resources for pest management, and government organizations. Networks and bridges will be built between rural and urban communities, placing a priority on communication and embracing technological advances that help propel project results into field implementation. The participation of regional and local partners together will be encouraged in workshops and presentations on local and regional issues that may include invasive and emerging species, pesticide resistance, and other pest threats to sustaining rural and urban communities.
Why Don't We Know How Best to Fight Ticks and Lyme Disease?
We asked participants at the New England Vegetable & Fruit Conference the following question: What are the two biggest challenges you face implementing sustainable / integrated pest management (IPM) solutions?


Of the recreational choices, one may visit Six Flags Over Georgia, with more than 200 acres of entertainment and just a few minutes from downtown Atlanta. Also, the downtown area offers a wide range of entertainment possibilities including Underground Atlanta, the World of Coke, and the Civic Center where concerts, shows and other attractions occur.
For the sports-minded, Atlanta has excellent professional teams, headlined by the two year back-to-back National League Champions, and the 1995 World Champions, The Atlanta Braves. Atlanta also hosts the Atlanta Falcons and the Atlanta Hawks. In addition, the city offers several artistic attractions, such as the High Museum of Art, the famous Fox Theatre, and the Atlanta Symphony Orchestra. Downtown Atlanta features many historic sites including the Martin Luther King, Jr. Center, the Carter Presidential Center, and the gold-domed State Capitol.
Just minutes from the College, one may visit historic Stone Mountain, regarded as the largest solid granite stone in the world. Carved in its side are figures of Jefferson Davis, Robert E. Lee, and Stonewall Jackson. Boat rides, a carillon, and many other amusements draw millions of people to the park each year. Atlanta truly offers a great and exciting "home away from home" while attending Gupton-Jones College of Funeral Service!


text:
\section{Introduction}

One of the most enduring challenges in statistical function estimation
is to devise procedures that adapt to the locally variable complexity
of the unknown function. For example, if one observes a random sample
$X_{1},...,X_{n}$ with density $f_{0}:\mathbb{R}\to\mathbb{R}$,
then $f_{0}$ may exhibit \textit{spatially inhomogeneous smoothness}:
The density could be infinitely-differentiable on most of its support
except for a few points $x_{m}$ where it behaves locally like $|x-x_{m}|^{\alpha_{m}}$
for some distinct numbers $\alpha_{m}$. The location of the irregular
points $x_{m}$ will usually not be known, and neither the corresponding
degree of smoothness $\alpha_{m}$. Moreover $f_{0}$ could possess
a so-called \textit{multifractal} behavior, changing its H\"{o}lder exponents
continuously on its domain of definition -- in fact, as shown in Jaffard
\cite{jaff}, `typical' functions in the Besov spaces usually considered
in nonparametric statistics are always multifractal. Donoho and Johnstone
\cite{dj} and Donoho, Johnstone, Kerkyacharian, and Picard \cite{djkp1},
\cite{djkp2} have suggested that methods based on wavelet shrinkage
can, to a certain extent, adapt to spatially inhomogeneous complexity
of the unknown function $f_{0}$. Moreover, Lepski, Mammen, and Spokoiny
\cite{lms} showed that this is not intrinsic to wavelet methods,
and that similar spatial adaptation results can be proved for kernel methods based on locally variable bandwidth choices.

There are several ways in which one can measure spatial adaptivity
of an estimator. A minimal requirement may be to devise a rule $\hat{f}_{n}(x)$
that estimates $f_{0}(x)$ in an optimal way at every point $x$,
and the methods suggested in \cite{dj} and \cite{lms} meet this
requirement. These procedures depend on the point $x$, and the natural
question arises as to how a given procedure performs globally as an
estimator for $f_{0}$. To address this question, Donoho et~al.~\cite{djkp2}
and Lepski et al.~\cite{lms} considered global $L^{r}$-loss, $r<\infty$,
and argued that taking $L^{r}$-loss over Besov-bodies $B(s,p,q)$
where smoothness is measured in $L^{p}$, $r>p$, gives a way to assess
the spatial performance of an estimator. A probably more transparent
approach to the spatial adaptation problem is to consider sup-norm
loss for estimators with locally variable bandwidths: one aims to
find an estimator $\hat{f}_{n}(x)$ that is locally optimal for estimating
$f_{0}(x)$, and simultaneously so for all $x$. This approach was
not considered in the literature so far -- the results \cite{gn0},
\cite{gn}, \cite{gn1}, \cite{golep} address the spatially homogeneous
setting only.

A first contribution of this article is to show that a dyadic histogram
estimator with variable bin size spatially adapts to possibly inhomogeneous
local H\"{o}lder smoothness of $f_{0}$, in global sup-norm loss. More
precisely, for $K(x,y)$ the Haar wavelet projection kernel, we shall
construct \[
\hat{f}_{n}(x)=\frac{2^{\hat{j}_{n}(x)}}{n}\sum_{i=1}^{n}K(2^{\hat{j}_{n}(x)}x,2^{\hat{j}_{n}(x)}X_{i}),\]
 where $\hat{j}_{n}(x)$ is a variable resolution level that depends
both on $x$ and the sample, and show that the random variable \[
\sup_{x}\frac{1}{r(n,x,f_{0})}\left|\hat{f}_{n}(x)-f_{0}(x)\right|\]
 is stochastically bounded, where $r(n,x,f_{0})$ is the optimal risk of an 'oracle estimator' for $f_{0}$ at the point $x$. We show moreover that this rate equals the pointwise minimax rate of adaptive estimation for $f_0(x)$ at every $x$, and that spatial adaptation occurs uniformly in $x$ except near discontinuities of the H\"{o}lder exponent function $t(f,x)$, see after Theorem \ref{holdc} for a detailed discussion.

While this result shows that spatial adaptation is indeed possible
in a strong theoretical way, a drawback shared by most results in
the literature on adaptive estimation remains: The theoretical findings
give no indication whatsoever as to how to choose the numerical constants
in the thresholds that feature in shrinkage- or Lepski-test-based
methods. It has become a common practice that thresholding constants
are chosen according to simulation results where simulations are drawn
as if the true underlying signal is very simple (say, uniform or piecewise
constant). This practise has not had any general theoretical corroboration
until recently Spokoiny and Vial \cite{sv} gave, in a simple Gaussian
regression model, a certain justification based on the idea of `propagation'.
The results in \cite{sv} are heavily tied to the simplicity of the
model used, in particular to the strong Gaussianity assumption employed,
and to the fact that pointwise loss is considered. In the present
paper we show how the ideas of \cite{sv} generalise, subject to some
nontrivial modifications, to nonparametric density estimation. A key
idea in the proofs in \cite{sv}, translated into the density estimation
context, is to replace the sampling distribution by a locally constant
product measure. The 'transportation cost' of this replacement is
easy to control in the Gaussian setting of \cite{sv}, but in the
density estimation case the fluctuations of the likelihood ratios
between the unknown sampling distribution and relevant locally constant
product measures do not obey a Gaussian regime, but turn out to be
of Poisson type, so that the 'Gaussian intuitions' of \cite{sv} could
be entirely misleading. We show however that the main information
theoretic idea of \cite{sv} remains sound in this Poissonian setting
as well: We use a Lepski-type procedure to construct $\hat{j}_{n}(x)$,
and we show that if we compute sharp thresholds for this procedure
as if the true density $f_{0}$ belonged to a family $\mathcal F$
of locally constant densities, then the resulting estimator is spatially adaptive in sup-norm loss. In contrast to the results in \cite{sv},
the rates of convergence we obtain for the risk of the final estimator
are \textit{exact} rate-adaptive.

While the techniques and results of this paper generalise in principle to more complex estimation problems that involve in particular adaptation to higher degrees of smoothness, we prefer to stay within the simpler setting of Haar wavelets, which allows for a clean exposition of the main ideas.

\section{Uniform spatial adaptation using propagation methods}

We will use the symbol $\|g\|_{T}$ to denote the supremum $\sup_{t\in T}|g(t)|$
of a function $g$ over some set $T$, but we will still use the symbol
$\|g\|_{\infty}$ to denote $\sup_{x\in\mathbb{R}}|g(x)|$ if no confusion
can arise.

For any $j\in\mathbb{N}$, we define a dyadic partition of $(0,1]$
into $2^{j}$-many disjoint subintervals by setting $I_{j,k}=(k2^{-j},(k+1)2^{-j}]$,
$k=0,\ldots,2^{j}-1$; and for $0<x\le1$ we denote by $I_{j,k(x)}$
the unique interval containing $x$. For $j\in\mathbb{N}$, $k=1,\ldots,2^{j}-1$,
let $V_{j,k}$ be the space of all bounded density functions on $\mathbb{R}$
that are constant on $I_{j,k}$. Via the local projections\begin{align*}
K_{j,x}(f)(z): & =\begin{cases}
2^{j}\int_{I_{j,k(x)}}f(y)dy & \textrm{if }z\in I_{j,k(x)},\\
f(z) & \textrm{otherwise},\end{cases}\end{align*}
 we map any bounded density $f$ onto $V_{j,k(x)}$. (Note that $K_{j,x}(f)$
is indeed a density since $K_{j,x}(f)$ and $f$ assign the same probability
to the interval $I_{j,k(x)}$.) For $f\in V_{j,k}$ and $j'\ge j$
we clearly have $K_{j',x}(f)=f$.

\subsection{Estimation procedure\label{Section: Estimation-Procedure}}

Let $X,X_{1},...,X_{n}$ be i.i.d.~with bounded density $f_{0}:\mathbb{R}\to[0,\infty)$, $n>1$. We wish to construct a single estimator which estimates $f_{0}(x)$
in an optimal way, uniformly so for points $x$ in the interval $(a,b]$.
We shall take without loss of generality $(a,b]=(0,1]$, and we shall
assume throughout that $f_{0}$ is bounded away from zero on $(0,1]$.
Let $K(x,y)=\sum_{k}\phi(x-k)\phi(y-k)$ be the projection kernel
based on the Haar wavelet $\phi=1_{(0,1]}$. We shall write $K_{j}(x,y)=2^{j}K(2^{j}x,2^{j}y)$,
and the associated linear density estimator is the dyadic histogram
estimator given by\[
f_{n}(j,x):=\frac{1}{n}\sum_{i=1}^{n}K_{j}(x,X_{i}).\]
 We make the important observation that $E_{f}f_{n}(j,x)=2^{j}P_{f}(I_{j,k(x)})$,
which directly follows from the identity $K_{j}(x,y)=2^{j}1_{I_{j,k(x)}}(y)$.
If $f$ is constant on $I_{j,k(x)}$ this in particular implies $E_{f}f_{n}(j,x)=f(x)$.
(In other words: for any locally (at $x$) constant density $f$ the
bias of $f_{n}(j,x)$ equals zero if the resolution level is chosen
fine enough.)

We finally note that the estimator $f_{n}(j,x)$ by construction only
depends on data points falling into $I_{j,k(x)}$. This amounts to
$n2^{-j}$ being the `effective' sample size for estimating $f_{0}$
at $x$.

\subsection{Local choice of the resolution level}

We fix $j_{\max}:=j_{\max,n}\in\mathbb{N}$ satisfying $2^{-j_{\max}} \ge (\log n)^{2}/n$ for some $d > 0$. For thresholds $\zeta_{n}$ to be specified below, and for $J\in\mathbb{N}$, $J\le j_{\max}$ and $0<x\le1$, we define
\begin{eqnarray}
\lefteqn{\hat{j}_{n}(J,x)=\min\biggl\{ j\in\mathbb{N},\, J\le j\le j_{\max}:}\nonumber \\
 &  & \sqrt{n2^{-j'}}\left|f_{n}(j',x)-f_{n}(j,x)\right|\le\zeta_{n} \sqrt{f_{n}(j,x)} \textrm{ for all }j',\, j<j'\le j_{\max}\biggr\}\label{lepskiH}\end{eqnarray}
 as well as \begin{equation}
\hat{j}_{n}(x)=\hat{j}_{n}(0,x).\label{lepski}\end{equation}
 (If the condition in \eqref{lepskiH} is not met for any $j$, $J\le j\le j_{\max}$,
we set $\hat{j}_{n}(J,x)=j_{\max}$.) Given the locally variable resolution
level $\hat{j}_{n}$, we define the family of nonlinear estimators
\begin{equation}
\hat{f}_{n}(J,x):=f_{n}(\hat{j}_{n}(J,x),x),~~~~~\hat{f}_{n}(x):=f_{n}(\hat{j}_{n}(x),x),~~~~~x\in[0,1].\label{estimators}\end{equation}
 These are estimators for $f_{0}(x)$ based on a locally variable
resolution level depending on $x$, and they are density-analogues
of the estimators introduced in \cite{lms} in the context of the
Gaussian white noise model. Note that by construction $\hat{j}_{n}(x)$
is a step function in $x$. Introducing the parameter $J$ will be
useful in what follows -- effectively, $\hat{f}_{n}(J,x)$ is a nonlinear
estimator based on a search over the resolution levels $j\ge J$ that
stops at $j_{\max}$.

\subsection{Threshold choice by propagation}

One of the main challenges for all adaptive procedures is the choice
of the thresholds $\zeta_{n}$ used in the tests defined in \eqref{lepskiH}. Define the standardisation
\[
 \frac{1}{s_n(j,x)} := \begin{cases}
                        \frac{1}{\sqrt{f_n(j,x)}} & \text{if } f_n(j,x) > 0; \\
			0 & \text{otherwise}.
                       \end{cases}
\] We suggest to choose the thresholds in such a way that the following
condition is satisfied:

\begin{condition} \label{prop} Let $\mathcal{F}_{j,k}$ be any triangular
array of subsets of $V_{j,k}$, $j\le j_{\max}$, $k=0,\ldots,2^{j}-1$,
and let $k(m)$ be the unique $k$ such that $I_{j_{\max},m}\subseteq I_{j,k}$.
We say that the thresholds $\zeta_{n}$ satisfy the uniform propagation
condition $\textup{UP}(\alpha,\mathcal{F}_{j,k})$ for some fixed
$\alpha>0$ if for every $n$, every $j\le j_{\max}$, every $m=0,\ldots,2^{j_{\max}}-1$,
and every $f\in\mathcal{F}_{j,k(m)}$ we have that \begin{equation}
E_{f}\left(\sup_{x\in I_{j_{\max},m}}\max_{j\le j'\le j_{\max}}\sqrt{\frac{n2^{-j'}}{\log n}}\left|\frac{\hat{f}_{n}(j',x)-f_{n}(j',x)}{s_n(j',x)}\right|\right)^{2}\le\frac{\alpha}{n2^{2j_{\max}}}.\label{UP}\end{equation}
 \end{condition}
(Note that since $\hat{j}_n(j') \ge j'$ we have that $f_n(j',x) = 0$ implies $\hat{f}_n(j',x) = 0$ for the fully data-driven estimator $\hat{f}_n(j')$, and so the error $|\hat{f}_n(j',x) - f_n(j',x)|$ is then $0$.) An interpretation of this condition can
be given along the following lines: For $0<x\le1$ the class $\mathcal{F}_{j,k(x)}$
contains only densities $f$ that can be exactly reconstructed on
$I_{j,k(x)}$ by $\int K_{j}(x,y)f(y)dy$, so that the bias of the
linear estimator $f_{n}(j',x)$ equals zero locally. In particular, any choice
of the resolution level finer than $j'$ will only increase the variance
without reducing the bias, and we would want $\hat{j}_{n}(j',x)$
to detect that and equal, with large probability, $j'$. This property
of $\hat{j}_{n}$ will then be mirrored in the fact that $\hat{f}_{n}(j',x)-f_{n}(j',x)=0$
for every $j'\ge j$ on an event with large probability, in which
case the l.h.s.~of \eqref{UP} is exactly equal to zero. The quantity
$\alpha/(n2^{2j_{\max}})$ stands for the a priori expected tolerance
for a probabilistic error of $\hat{j}_{n}$ to detect the `correct'
resolution level on each interval $I_{j_{\max},m}$ in this `no-bias'
situation.

The following lemma shows that Condition \ref{prop} is not empty
and that thresholds $\zeta_{n}$ satisfying the uniform propagation
condition exist. It shows furthermore that the thresholds can be taken
to be of order $\sqrt{\log n}$ and independent of $f$, which will
be crucial in understanding the adaptive properties of $\hat{f}_{n}$
below.

\begin{lemma} \label{propver} Let $\mathcal{F}_{j,k}$ equal $V_{j,k}$
intersected with the set\[
\left\{ f:0<\delta\le\inf_{0 < x \le 1}f(x),\,\|f\|_{\infty}\le M\right\} \]
 for some fixed $0<\delta,M<\infty$. Then for every given $\alpha>0$
there exists a numerical constant $\kappa>0$ that depends only on $\alpha$ such
that for any threshold choice \[
\zeta_{n}\ge\kappa\sqrt{\log n}\]
 the uniform propagation condition $\textup{UP}(\alpha,\mathcal{F}_{j,k})$
is at least satisfied for $n$ larger than some index that only depends
on $\delta$ and $M$.
\end{lemma}

While Lemma~\ref{propver} proves the existence of thresholds of the order $\sqrt {\log n}$ under the uniform propagation condition -- a fact that will be seen to imply adaptivity of $\hat f_n$ below -- it does not suggest a practical choice of $\zeta_n$. Instead, this choice can be made by direct evaluation of \eqref{UP}, as follows: Condition \ref{prop} only concerns the local error bounds
over small intervals $I_{j_{\max},m}$ on which the function $f$
is constant, which effectively means that it suffices to check this
condition only for classes of densities which are constant on the
interval of interest. The particular choice of the interval $I_{j_{\max},m}$
is unimportant. Secondly, all quantities in Condition \ref{prop}
depend on known quantities after $f$ is chosen. By construction of
the estimators $f_{n}$ and $\hat{f}_{n}$ the random variable featuring
in \eqref{UP} -- we call it $T$ -- only depends on the number of
data points falling into each of the (uniquely determined) $j'$-fine
intervals containing $I_{j_{\max},m}$. This observation allows for
an easy computation of the l.h.s.~of \eqref{UP} along the following
lines: Fix $0\le p\le1$. Then, for any $f\in\mathcal{F}_{j,k(m)}$
satisfying $2^{-j}f=p$ on $I_{j,k(m)}$, the number $Z$ of observations
falling into the interval $I_{j,k(m)}$ is binomial $B(n,p)$. Conditionally
on $Z=k$, take $k$-many independent random variables that are uniform
on $I_{j,k(m)}$ and count the number of observations $V_{j'}$ in
each of the $j'$-fine intervals. Then compute $f_{n}$, $\hat{f}_{n}$;
and $T$. This shows that $T$ does only depend on $V_{j'}$, $j\le j'\le j_{\max}$,
and that the l.h.s.~of \eqref{UP} is therefore equal to $E[E[T(V_{j},\ldots,V_{j_{\max}})|Z]]$.

The practical choice of $\zeta_{n}$ can then be obtained via a Monte
Carlo simulation of \eqref{UP} by choosing $\zeta_{n}$ as the smallest
threshold for which \eqref{UP} is satisfied in the simulation for
one specific interval $I_{j_{\max},m}$ uniformly over the class of
all densities constant on this interval. Given $j_{\max}$ and $\alpha$, this procedure has to be performed only for one fixed interval $I_{j_{\max},m},$ and then applies for every $m$ simultaneously.

\subsection{Local small bias condition}

\label{lsbc}

The idea behind Condition \ref{prop} is that we take `idealised'
classes of densities $\mathcal{F}$ for which we compute sharp thresholds
$\zeta_{n}$. The danger arises that the true density $f_{0}$ may
be very different from the elements in $\mathcal{F}$, which may lead
to wrong thresholds (and inference). We have to assess the error that
comes from replacing $f_{0}$ by an element from $\mathcal{F}$, in
a neighborhood of a given point $x$. This can be fundamentally quantified
in terms of the log-likelihood ratio between $f_{0}$ and its local
(at $x$) approximand in $\mathcal{F}$. As we shall see, one of the
deeper reasons behind the fact that propagation methods imply adaptation
results is that this error can be related to the usual bias term in
linear estimation.

\begin{condition} \label{sbc} Given real numbers $\Delta_{j,x}$, $0 < x \le 1$, $j \in \mathbb N \cup \{0\}$ satisfying $\Delta_{l',x}\le\Delta_{l,x}$ for every
$l'>l$, we say that $f_{0}$ satisfies the
local small bias condition at $x \in (0,1]$ and with $\Delta_{j,x}\equiv \Delta_{j,x}(f_{0})$ if\[
\textup{Var}_{K_{j,x}(f_{0})}\log\frac{f_{0}}{K_{j,x}(f_{0})}\le\Delta_{j,x}(f_{0})\]
for all $j\in\mathbb{N}$. \end{condition}

The local 'cost' of transporting a product measure $\prod_{i=1}^n f_0(x_i)$ to $\prod_{i=1}^n K_{j,x}(f_{0})(x_i)$ can be quantified by $n$ times the variance featuring in the above condition, and we shall have to restrict ourselves to resolution levels $j$ for which this transportation cost is at most a fixed constant times the logarithm of the sample size $n$. The smallest resolution level for which this is still the case will be defined as $j^*(x)$: More precisely, for some fixed positive constant $\Delta$,
define the local resolution level\begin{equation}
j^{*}(x):=j^{*}(x,n,\Delta,f_{0})=\min\left\{ j\in\mathbb{N}:j\le j_{\max},\, n\Delta_{j,x}(f_{0})\le\Delta\log n\right\} .\label{Definition: Local bandwith}\end{equation}
While this is an information-theoretic definition of $j^*$, a key observation of this subsection is that it has the classical `bias-variance'
tradeoff generically built into it for suitable choices of $\Delta_{j,x}(f_{0})$.

\begin{lemma} \label{conc} Suppose $f_{0}$ is bounded by some finite
number $M>0$ and that
\[
 \inf_{0<x\le1}f_{0}(x)\ge\delta>0.
\]
Then $f_{0}$ satisfies Condition~\ref{sbc} with\[
\Delta_{j,x}(f_{0})=\frac{M}{\delta^{2}}2^{-j}\|f_{0}-K_{j,x}(f_{0})\|_{\infty}^{2}.\]

\end{lemma}
\begin{proof}
First, observe that $K_{j,x}(f_{0})$ is bounded by $M$ and bounded
below by $\delta>0$. Then, using that $K_{j,x}(f_{0})$ coincides
with $f_{0}$ outside of $I_{j,k(x)}$ and the inequality $|\log x-\log y|\le\max(x^{-1},y^{-1})|x-y|$,
we get\begin{eqnarray*}
 &  & \text{Var}_{K_{j,x}(f_{0})}\log\frac{f_{0}}{K_{j,x}(f_{0})}\\
 &  & \le\int\left(\log\frac{f_{0}(y)}{K_{j,x}(f_{0})(y)}\right)^{2}K_{j,x}(f_{0})(y)dy\\
 &  & \le\int\max\left(f_{0}(y)^{-2},K_{j,x}(f_{0})(y)^{-2}\right)(f_{0}(y)-K_{j,x}(f_{0})(y))^{2}K_{j,x}(f_{0})(y)dy\\
 &  & \le\frac{M}{\delta^{2}}\int(f_{0}(y)-K_{j,x}(f_{0})(y))^{2}dy \le\frac{M}{\delta^{2}}2^{-j}\|K_{j,x}(f_{0})-f_{0}\|_{\infty}^{2}.
\end{eqnarray*}
\end{proof}

\medskip

The lemma shows that the quantity $(n/\log n)\Delta_{j,x}(f_{0})$
can be viewed as the square of the `bias divided by the variance'
of linear projection estimators for $f_{0}(x)$. \emph{Hence, to choose
the smallest $j\le j_{\max}$ such that $(n/\log n)\Delta_{j,x}(f_{0})$
is still bounded by a fixed constant $\Delta$ means to locally balance
the `variance' and `bias' term in the nonparametric setting.}

To be more concrete, let us briefly discuss what this means in the
classical situation where the bias is bounded by local regularity
properties of the unknown density $f_{0}$. Since we are interested
in spatial adaptation, we wish to take locally inhomogeneous smoothness
into account by appealing to local H\"{o}lder conditions: Let $0<t\le1$
and let us say that a function $g:\mathbb{R}\to\mathbb{R}$ is locally
$t$-H\"{o}lder at $x\in\mathbb{R}$ if for some $\eta>0$ \[
\sup_{0<|m|\le\eta}\frac{|g(x+m)-g(x)|}{|m|^{t}}<\infty.\]
 Define further a `local' H\"{o}lder ball of bounded functions \[
\mathcal{C}(t,x,L,\eta):=\left\{ g:\mathbb{R}\to\mathbb{R},\,\max\left(\|g\|_{\infty},\,\sup_{0<|m|\le\eta}\frac{|g(x+m)-g(x)|}{|m|^{t}}\right)\le L\right\} .\]
 Condition~\ref{sbc} then has the following more classical interpretation
in terms of local smoothness properties of $f_{0}$:

\begin{lemma} \label{daub} If $f_{0}\in\mathcal{C}(t,x,L,\eta)$
for some $0<t\le1$, then the local bias $\|f_{0}-K_{j,x}(f_{0})\|_{\infty}$
is bounded by $c2^{-jt}$ for some constant $c\equiv c(t,L,\eta)$. Furthermore,
if
\[
 \inf_{0<x\le1}f_{0}(x)\ge\delta>0,
\]
then Condition \ref{sbc}
is satisfied with \begin{equation}
\Delta_{j,x}(f_{0})=c^{2}\frac{L}{\delta^{2}}2^{-j(2t+1)}.\label{hbd}\end{equation}
 \end{lemma}
\begin{proof}
Let $y\in I_{j,k(x)}$ be arbitrary. Then, using the substitution
$2^{j}z=2^{j}y-u$, \begin{eqnarray*}
|f_{0}(y)-K_{j,x}(f_{0})(y)| & = & \left|2^{j}\int_{I_{j,k(x)}}(f_{0}(y)-f_{0}(z))dz\right|\\
 & \le & \int_{-1}^{1}|f_{0}(y)-f_{0}(x)+f_{0}(x)-f_{0}(y-2^{-j}u)|du\\
 & \le & 2|f_{0}(y)-f_{0}(x)|+\int_{-1}^{1}|f_{0}(x)-f_{0}(y-2^{-j}u)|du\end{eqnarray*}
By definition of $x,y,I_{j,k(x)}$ we have $|y-x|\le2^{-j}$, and
also $|y-2^{-j}u-x|\le2^{-j+1}$ by the triangle inequality, so that
for $2^{-j+1}\le\eta$ the last quantity is bounded by $c_{0}2^{-jt}$
in view of $f_{0}\in\mathcal{C}(t,x,L,\eta)$. If $2^{-j}>\eta/2$,
then the quantity in the last display can still be bounded by $6\|f_{0}\|_{\infty}\le6L$,
so that choosing $c_{1}=6L(2/\eta)^{t}$ establishes the desired bound
for $c=\max(c_{0},c_{1})$. To prove the second claim, apply Lemma~\ref{conc}.
\end{proof}

\medskip

Using the bound from the last lemma to verify Condition \ref{sbc},
we see that, by definition of $j^{*}(x)$ and for $f_{0}\in\mathcal{C}(t,x,L,\eta)$,
\begin{equation}
\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\sim\left(\frac{n}{\log n}\right)^{\frac{t}{2t+1}}\label{locad}\end{equation}
 is the locally (at $x$) optimal adaptive rate of convergence, so
that the local small bias condition constructs a minimax optimal resolution
level $j^{*}(x)$ at every $x\in[0,1]$.

\subsection{Main results}

We now state the main results, starting with the following `oracle'
inequality. Note that the oracle $f_{n}(j^{*}(x),x)$ is not an estimator in itself as it depends on unknown quantities.

\begin{theorem} \label{main} Let $\hat{f}_{n}(\cdot)$ be the density estimator defined in \eqref{estimators}
with thresholds $\zeta_{n}$ that satisfy the uniform propagation
condition $\textup{UP}(\alpha,\mathcal{F}_{j,k})$ for some $\mathcal{F}_{j,k}$. Suppose $f_{0}$
satisfies Condition \ref{sbc} for every $0<x\le1$, and let $j^{*}(x)$
be as in \eqref{Definition: Local bandwith}. Then we have \begin{eqnarray}
 &  & E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|\nonumber \\
 &  & \le\frac{\zeta_{n}}{\sqrt{\log n}}+\sqrt{\frac{\alpha}{n}}n^{\Delta e^{4U}}\label{Oracle bound}\end{eqnarray}
 for any $U$ satisfying \begin{equation}
U\ge\sup_{0<x\le1}\left\Vert \log\frac{f_{0}}{K_{j^{*}(x),x}(f_{0})}\right\Vert _{\infty}.\label{U}\end{equation}
\end{theorem}

If $\zeta_{n}=O(\sqrt{\log n})$ -- as follows under the conditions of Lemma \ref{propver}
 -- and if one chooses $\Delta<1/2$, $U$ as in the remark below, then the r.h.s.~of \eqref{Oracle bound} is $O(1)$ as $n$ tends to infinity. Theorem \ref{main} thus implies that the estimator $\hat{f}_{n}$ with resolution
levels chosen by the propagation approach is close to the linear {}`oracle
estimator' evaluated at the locally optimal resolution level $j^{*}(x)$,
and this uniformly so on $(0,1]$.
\begin{remark}

If $\mathcal{F}_{j,k}$ is as in Lemma~\ref{propver} and $f_{0}$
is bounded by $M$ and bounded below by $\delta$, we may apply Lemma~\ref{conc}
\emph{(}using $2^{-j_{\max}}\ge d(\log n)^{2}/n$\emph{)} to obtain the bound\[
\log\frac{f_{0}}{K_{j^{*}(x),x}(f_{0})}=\log\left(1+\frac{f_{0}-K_{j^{*}(x),x}(f_{0})}{K_{j^{*}(x),x}(f_{0})}\right)\le\log\left(1+\frac{\Delta}{dM\log n}\right),\]
which tends to zero as $n$ tends to infinity.
\end{remark}
Our results then imply the following uniform spatial adaptation result:

\begin{theorem} \label{hold} Assume that $f_{0}$ is bounded by $M$ and satisfies $\inf_{0<x\le1}f_{0}(x)\ge\delta>0$. Let $\hat{f}_{n}(\cdot)$ be the density estimator
from \eqref{estimators} with thresholds $\zeta_{n}=O(\sqrt{\log n})$
that satisfy the uniform propagation condition $\textup{UP}(\alpha,\mathcal{F}_{j,k})$ for $\mathcal{F}_{j,k}$ as in Lemma~\ref{propver}. Let $j^{*}(x)$ be as in \eqref{Definition: Local bandwith} with
$\Delta<1/2$ and with $\Delta_{j,x}$ as in Lemma~\ref{conc}. Then
\begin{equation}
\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\hat{f}_{n}(x)-f_{0}(x)\right|=O_{{\Pr}_{f_0}}(1).\label{Adaptive rate optimality}
\end{equation}
\end{theorem}

Thus the fully data-driven estimator $\hat f_n$ for $f_0$ achieves the locally optimal risk of the 'oracle' based on $j^*(x)$, uniformly at all points in $(0,1]$. If $j^{*}(x)$ -- with $\Delta<1/2$ -- is based on $\Delta_{j,x}$
as in Lemma~\ref{daub}, then \eqref{Adaptive rate optimality} holds
true and the 'oracle' rate is the adaptive locally minimax rate of convergence
at every $0<x\le1$ where $f_{0}$ is locally $t$-H\"{o}lder with
$0<t\le1$, see the discussion in Section~\ref{lsbc} surrounding \eqref{locad}. This means that at any given point $x$ our estimator is rate-adaptive to local H\"{o}lder smoothness (with the usual $\log n$ penalty for adaptation).

One may ask further if spatial adaptation in the minimax sense occurs \textit{uniformly} for every $x \in (0,1]$. A consequence of Theorem \ref{hold} is the following.

\begin{theorem} \label{holdc} Suppose the assumptions of Theorem \ref{hold} are satisfied and that the true density $f_{0}$ lies
in $\mathcal{C}(t(x),x,L(x),\eta(x))$, $0 < x \le 1$, for some $t(\cdot),L(\cdot),\eta(\cdot)$ that
are bounded and uniformly bounded away from zero on $(0,1]$. Let
$j^{*}(x)$ be as in \eqref{Definition: Local bandwith} with $\Delta<1/2$
and with $\Delta_{j,x}$ as in Lemma~\ref{daub}. Then\[
\sup_{0<x\le1}\left(\frac{n}{\log n}\right)^{t(x)/(2t(x)+1)}\left|\hat{f}_{n}(x)-f_{0}(x)\right|=O_{{\Pr}_{f_0}}(1).\]
 \end{theorem}

The assumptions on the functions $t, L, \eta$ need discussion. For densities that locally look like  $|x-x_m|^{\alpha_m}$ we would wish to choose $t(x)$ equal to their pointwise H\"{o}lder exponents $t(x_m)=\alpha_m$ and $t(x)=1$ otherwise, but then $\eta$ is not uniformly bounded away from zero for points $x \to x_m$. However, Theorem \ref{holdc} holds for \textit{any} choice of the functions $t, L, \eta$ for which $f_0$ satisfies $f_0 \in \mathcal{C}(t(x),x,L(x),\eta(x))$, $0 < x \le 1$. In other words, in the above example we can choose $t(x)=\alpha_m$ on the interval $(x_m-\eta_0, x_m+\eta_0)$ and $t(x)=1$ otherwise, where $\eta_0$ is some arbitrary lower bound for $\eta(x)$. This comes at the expense of not being adaptive near $x_m$, i.e., for $x \in (x_m-\eta_0, x_m+\eta_0) \setminus \{x_m\}$, which is sensible as we cannot expect adaptation for points $x$ arbitrarily close to $x_m$ from a finite sample. Inspection of the proofs (particularly the dependence on $\eta$ in Lemma \ref{daub}) shows that, for fixed $n$, the above theorem holds for densities $\mathcal{C}(t(x),x,L(x), \eta_{(n)})$, $0 < x \le 1$, where $\eta_{(n)}$ can be taken of order $n^{-1/3}$, the binwidth corresponding to the maximal smoothness $t=1$ one wants to adapt to in our setting, and this is again reasonable: H\"{o}lder smoothness of $f_0$ in an interval $[x \pm r_n]$ where $r_n = o(n^{-1/3})$ does not allow to control the bias at $x$ with the locally optimal binwidth of order $n^{-1/3}$. By the same arguments multifractal densities $f_0$ which change their H\"{o}lder exponent continuously can be handled by taking $t(x)$ piecewise constant on a partition of $(0,1]$ into bins of size of order $n^{-1/3}$, the estimator achieving the local uniform minimax rate on each bin of the partition.

\section{Proofs}

\subsection{Proof of Theorem \ref{main}}

A first idea is to use a moment bound, localised at any point $x$
of estimation, on the log-likelihood ratio between $f_{0}$ and its
approximand in $V_{j,k}$.

\begin{lemma} \label{Lemma: Second moment of the likelihood}

If, for fixed $0<x\le1$,\begin{equation}
\textup{Var}_{K_{j,x}(f_{0})}\log\frac{f_{0}}{K_{j,x}(f_{0})}\le\frac{D\log n}{n}\label{Second moment bound}\end{equation}
 for some $0<D<\infty$ and every $n\in\mathbb{N}$, then, for every
$n\in\mathbb{N}$, \[
E_{K_{j,x}(f_{0})}\left(\prod_{i=1}^{n}\frac{f_{0}(X_{i})}{K_{j,x}(f_{0})(X_{i})}\right)^{2}\le n^{2De^{4U}}\]
 holds for any $U$ satisfying\[
U\ge\left\Vert \log\frac{f_{0}}{K_{j,x}(f_{0})}\right\Vert _{\infty}.\]

\end{lemma}
\begin{proof}
Since the Kullback-Leibler distance \[
\mathcal{K}(f_{0},K_{j,x}(f_{0}))=-E_{K_{j,x}(f_{0})}\log\frac{f_{0}}{K_{j,x}(f_{0})}\ge0\]
is non-negative, we have\begin{eqnarray*}
E_{K_{j,x}(f_{0})}\left(\prod_{i=1}^{n}\frac{f_{0}(X_{i})}{K_{j,x}(f_{0})(X_{i})}\right)^{2} & \le & \left(E_{K_{j,x}(f_{0})}e^{2\left(\log\frac{f_{0}}{K_{j,x}(f_{0})}-E_{K_{j,x}(f_{0})}\log\frac{f_{0}}{K_{j,x}(f_{0})}\right)}\right)^{n}\end{eqnarray*}
by the i.i.d.~assumption. Using the power series expansion of the exponential function and that the variables in the exponent are centered, one easily
bounds the previous display by\[
\left(1+\frac{2De^{4U}\log n}{n}\right)^{n}\le e^{2De^{4U}\log n}=n^{2De^{4U}}.\]
\end{proof}

\medskip

Here is the proof of Theorem~\ref{main}: We first note that Condition~2
allows us to take $\Delta_{j,x}(f_{0})$ to be constant on the intervals
$I_{j,k}$. Consequently, $j^{*}(\cdot)$ from \eqref{Definition: Local bandwith}
is then constant on every interval $I_{j_{\max},m}$, and we set\[
j_{m}^{*}=\sup_{x\in I_{j_{\max},m}}j^{*}(x).\]
 To prove the theorem, we split \begin{eqnarray*}
 &  & E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|\\
 &  & \le E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|1_{\{\hat{j}_{n}(x)<j^{*}(x)\}}\\
 &  & \quad+E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|1_{\{\hat{j}_{n}(x)\ge j^{*}(x)\}}\\
 &  & =:I+II\end{eqnarray*}
 according to whether $\hat{j}_{n}(x)$ comes to lie below the local
resolution level $j^{*}(x)$ or not. By definition of $\hat j_n(x)$ in \eqref{lepskiH} one immediately has \[
I\le\frac{\zeta_{n}}{\sqrt{\log n}}.\]
 About $II$: Define\begin{equation}
S_{m}=\sup_{x\in I_{j_{\max},m}}\max_{j_{m}^{*}\le j\le j_{\max}}\sqrt{\frac{n2^{-j}}{\log n}}\left|\frac{\hat{f}_{n}(j,x)-f_{n}(j,x)}{s_{n}(j,x)}\right|.\label{S_m}\end{equation}
 Using that on the event $\hat{j}_{n}(x)\ge j^{*}(x)$ we necessarily
have $\hat{f}_{n}(x)=\hat{f}_{n}(j^{*}(x),x)$, we see that\begin{eqnarray}
II & \le & E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(j^{*}(x),x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|\nonumber \\
 & \le & E_{f_{0}}\max_{m}\sup_{x\in I_{j_{\max},m}}\max_{j_{m}^{*}\le j\le j_{\max}}\sqrt{\frac{n2^{-j}}{\log n}}\left|\frac{\hat{f}_{n}(j,x)-f_{n}(j,x)}{s_{n}(j,x)}\right|\nonumber \\
 & \le & 2^{j_{\max}}\max_{m}E_{f_{0}}S_{m}.\label{Bound: II}\end{eqnarray}
 We use the Cauchy-Schwarz inequality to bound\begin{eqnarray*}
 &  & E_{f_{0}}S_{m}\\
 &  & =\int\cdots\int S_{m}(x_{1},\ldots,x_{n})\prod_{i=1}^{n}f_{0}(x_{i})dx_{1}\cdots dx_{n}\\
 &  & =\int\cdots\int S_{m}(x_{1},\ldots,x_{n})\prod_{i=1}^{n}\frac{f_{0}(x_{i})}{K_{j_{m}^{*},x}(f_{0})(x_{i})}\prod_{i=1}^{n}K_{j_{m}^{*},x}(f_{0})(x_{i})dx_{1}\cdots dx_{n}\\
 &  & \le\sqrt{E_{K_{j_{m}^{*},x}(f_{0})}S_{m}^{2}}\,\sqrt{E_{K_{j_{m}^{*},x}(f_{0})}\left(\prod_{i=1}^{n}\frac{f_{0}(X_{i})}{K_{j_{m}^{*},x}(f_{0})(X_{i})}\right)^{2}}\end{eqnarray*}
 by the square-root of the second moment of $S_{m}$ under the `idealised'
density $K_{j_{m}^{*},x}(f_{0})$ times the square-root of the second
moment of the likelihood ratio. (Here, $x$ is any point in $I_{j_{\max},m}$.)
Using Condition~1 and Lemma~\ref{Lemma: Second moment of the likelihood},
we obtain a bound for the last term in \eqref{Bound: II} of order\[
2^{j_{\max}}\max_{m}E_{f_{0}}S_{m}\le\sqrt{\frac{\alpha}{n}}n^{\Delta e^{4U}},\]
which concludes the proof of the theorem.

\subsection{Proof of Theorems \ref{hold} and \ref{holdc}}

We first prove Theorem \ref{hold}: Clearly,
\begin{align*}
   \sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\hat{f}_{n}(x)-f_{0}(x)\right| &\le\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|\frac{\hat{f}_{n}(x)-f_{n}(j^{*}(x),x)}{s_{n}(j^{*}(x),x)}\right|\sqrt{f_{n}(j^{*}(x),x)} \\
 & ~~~~~+\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|f_{n}(j^{*}(x),x)-f_{0}(x)\right|.
 \end{align*}
 The first factor of the first summand is bounded in probability in
view of Theorem \ref{main} and of Lemma \ref{propver} and the hypothesis $\zeta_n = O(\sqrt{\log n})$. The second
factor of the first summand is also bounded in probability since \[
\sup_{0<x\le1}\max_{j\le j_{\max}}|f_{n}(j,x)-E_{f_{0}}f_{n}(j,x)|=o_{P_{f_{0}}}(1)\]
 by Proposition \ref{uexp}, using $2^{-j_{\max}}\ge d(\log n)^{2}/n$,
and since $\sup_{x,j}|E_{f_{0}}f_{n}(j,x)|\le\|f_{0}\|_{\infty}<\infty$.
It remains to prove that the second summand is bounded in probability,
and we achieve this by bounding the moment \begin{eqnarray*}
 &  & E_{f_{0}}\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|f_{n}(j^{*}(x),x)-E_{f_{0}}f_{n}(j^{*}(x),x)\right|\\
 &  & \qquad+\sup_{0<x\le1}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}\left|E_{f_{0}}f_{n}(j^{*}(x),x)-f_{0}(x)\right|\\
 &  & \le E_{f_{0}}\sup_{0<x\le1}\max_{j\le j_{\max}}\sqrt{\frac{n2^{-j}}{\log n}}\left|f_{n}(j,x)-E_{f_{0}}f_{n}(j,x)\right|\\
 &  & \qquad+\max_{m}\sup_{x\in I_{j_{\max},m}}\sqrt{\frac{n2^{-j^{*}(x)}}{\log n}}|E_{f_{0}}f_{n}(j^{*}(x),x)-f_{0}(x)|.\end{eqnarray*}
 The first term is bounded by a fixed constant using Proposition \ref{uexp}
below. Recalling the definition of $j_{m}^{*}$ from the beginning
of the proof of Theorem~\ref{main} and choosing $\Delta_{j,x}(f_{0})$
from Lemma~\ref{conc}, the second term is bounded by\begin{eqnarray*}
 &  & \max_{m}\sqrt{\frac{n2^{-j_{m}^{*}}}{\log n}}\sup_{x\in I_{j_{\max},m}}|E_{f_{0}}f_{n}(j_{m}^{*},x)-f_{0}(x)|\\
 &  & \le\max_{m}\sqrt{\frac{n2^{-j_{m}^{*}}}{\log n}}\|K_{j_{m}^{*},x}(f_{0})-f_{0}\|_{\infty}\\
 &  & \le\delta\sqrt{\frac{\Delta}{M}},\end{eqnarray*}
where $x$ is any point in $I_{j_{\max},m}$, and this completes the proof.

We next prove Theorem \ref{holdc}: Using the hypotheses on $t(\cdot),L(\cdot),\eta(\cdot)$,
the proof of Lemma~\ref{daub} shows that $f_{0}$ satisfies Condition~\ref{sbc}
with\begin{equation}
\Delta_{j,x}(f_0)=c'2^{-j(2t(x)+1)},\label{Delta independent of x}\end{equation}
$0<c'<\infty$, where $c'$ does not depend on $x$. Using that $t(\cdot)$
is bounded below by some positive number implies that
$$\Delta_{j_{\max},x}(f_0)=c' 2^{-j_{\max}(2t(x)+1)} \le \frac{\Delta \log n}{n}.$$
holds for $n$ large enough (independent of $x \in (0,1]$), so that $j^{*}(x)$, when based on $\Delta_{j,x}(f_0)$ as in $\eqref{Delta independent of x}$, is asymptotically equivalent
to the minimax optimal locally adaptive rate, uniformly so for all
$x$.

\subsection{Proof of Lemma \ref{propver}}

The proof relies on Propositions \ref{uexp1} and \ref{uexp} which
are given below. Recall first from Section~\ref{Section: Estimation-Procedure}
that for $f\in V_{j,k}$ and $j'\ge j$ we necessarily have $E_{f}f_{n}(j',x)-f(x)=0$
for every $x\in I_{j,k}$, so the bias at $x\in I_{j,k}$ is exactly
zero, a fact we shall use repeatedly below without separate mentioning.
Write \begin{eqnarray}
 &  & \sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\left|\frac{\hat{f}_{n}(j',x)-f_{n}(j',x)}{s_n(j',x)}\right|\notag\label{oldest}\\
 &  & =\sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\sum_{l > j'}\left|\frac{f_{n}(l,x)-f_{n}(j',x)}{s_n(j',x)}\right|1_{\{\hat{j}_{n}(j',x)=l\}}.\end{eqnarray}
 To treat the indicator, observe that\begin{eqnarray*}
 &  & \{\hat{j}_{n}(j',x)=l\} \subseteq\left\{ \sqrt{n2^{-l'}}|f_{n}(l',x)-f_{n}(l-1,x)|>\zeta_{n} \sqrt{f_{n}(l-1,x)} \textrm{ for some }l' \ge l\right\} \\
& & \subseteq \Biggl\{ \sqrt{n2^{-l'}} |f_{n}(l',x) - E_f f_n(l',x) + E_f f_n(l-1,x) - f_{n}(l-1,x)| \\
& & \quad > \zeta_n \frac{\sqrt{f(x)}}{2} \textrm{ for some }l' \ge l \Biggr\}  \cup \left\lbrace \min_{\ell \ge j} \sqrt{f_n(\ell,x)} \le \frac{\sqrt{f(x)}}{2}\right\rbrace
\end{eqnarray*}
 Observe that the first set is a subset of\begin{eqnarray*}
 &  & \left\{ \sqrt{n2^{-l'}}\left| f_{n}(l',x)-E_{f}f_{n}(l',x)\right| \ge \frac{\zeta_{n}\sqrt{f(x)}}{4} \textrm{ for some } l' \ge l\right\} \\
 &  & \quad\cup\left\{ \sqrt{n2^{-(l-1)}}\left| f_n(l-1,x)-E_f f_{n}(l-1,x)\right| > \frac{\zeta_{n}\sqrt{f(x)}}{4} \right\} \\
 &  & \subseteq\left\{ \max_{\ell \ge j} \sqrt{n2^{-\ell}}\|f_{n}(\ell)-E_{f}f_{n}(\ell)\|_{I_{j_{\max},m}} > \frac{\zeta_{n}\sqrt{\|f\|_{I_{j,k(m)}}}}{4} \right\} =:B_{1}
\end{eqnarray*}
and that, using $y \ge \sqrt{\delta y}$ for $y \ge \delta$, the second set is contained in
\begin{eqnarray*}
 & & \left\lbrace \max_{\ell \ge j} |f_n(\ell,x) - E_f f_n(\ell,x)| > \frac{f(x)}{2} \right\rbrace  \\
& & \subseteq \left\lbrace \max_{\ell \ge j} |f_n(\ell,x) - E_f f_n(\ell,x)| > \frac{\sqrt{\delta f(x)}}{2} \right\rbrace \\
& & \subseteq \left\lbrace  \sup_{x \in I_{j_{\max},m},\, \ell \ge j} |f_n(\ell,x) - E_f f_n(\ell,x)| > \frac{\sqrt{\delta \|f\|_{I_{j,k(m)}}}}{2}\right\rbrace := B_2;
\end{eqnarray*}
so that $\{\hat{j}_{n}(j',x)=l\}\subseteq B_{1}\cup B_{2}=:B$, a
set which does not depend on $j',x$ or $l$. Hence, $1_{\{\hat{j}_{n}(j',x)=l\}}\le1_{B}$
uniformly in $j',x,l$, so that the quantity in \eqref{oldest} is
bounded from above by \[
1_{B}\sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\sum_{l > j'}\left|\frac{f_{n}(l,x)-f_{n}(j',x)}{s_n(j',x)}\right|,\]
and therefore the second moment of \eqref{oldest} is bounded, using
the Cauchy-Schwarz inequality, by \[
{\Pr}_f(B)^{1/2}\left\Vert \sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\sum_{l > j'}\left|\frac{f_{n}(l,x)-f_{n}(j',x)}{s_n(j',x)}\right|\right\Vert _{4,{\Pr}_f}^{2}=:I\times II.\]
 We first bound $II$: By the triangle inequality and since the bias
is exactly zero, this term is less than or equal to \begin{eqnarray}
 &  & 2\left\Vert \sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\sum_{l > j'} \left|\frac{f_{n}(l,x)-E_{f}f_{n}(l,x)}{s_n(j',x)}\right|\right\Vert _{4,{\Pr}_f}^{2}\notag\label{nuis}\\
 &  & \quad + 2\left\Vert \sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\sqrt{\frac{n2^{-j'}}{\log n}}\sum_{l > j'}\left|\frac{f_{n}(j',x)-E_{f}f_{n}(j',x)}{s_n(j',x)}\right|\right\Vert _{4,{\Pr}_f}^{2}.
\end{eqnarray}
Define now $S=\{\sup_{x\in I_{j_{\max},m}}\min_{j' \ge j}f_{n}(j',x)\ge\delta/2\}$.
Note that, by definition of $f_{n}(j')$, $f_{n}(j',x)>0$ implies
$f_{n}(j',x)\ge2^{j'}/n$. Then, for every $1\le p<\infty$, \begin{eqnarray*}
 &  & E_{f}\left(\sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\frac{1}{s_{n}(j',x)}\right)^{p}=E_{f}\left(\sup_{x\in I_{j_{\max},m}}\max_{j' \ge j}\frac{1}{s_{n}(j',x)}(1_{S}+1_{S^{c}})\right)^{p}\\
 &  & \le\frac{2^{3p/2-1}}{\delta^{p/2}}\\
 &  & \quad+2^{p-1}n^{p/2}E_{f}1\left\{ \sup_{x\in I_{j_{\max},m}}\min_{j' \ge j}|f_{n}(j',x)-E_{f}f_{n}(j',x)+f(x)|<\frac{\delta}{2}\right\} \\
 &  & \le\frac{2^{3p/2-1}}{\delta^{p/2}}+2^{p-1}n^{p/2}{\Pr}_f\left\{ \sup_{x\in I_{j_{\max},m},\, j' \ge j}|f_{n}(j',x)-E_{f}f_{n}(j',x)|>\frac{\delta}{2}\right\} \\
 &  & \le\frac{2^{3p/2-1}}{\delta^{p/2}}\\
 &  & \quad+2^{p-1}n^{p/2}{\Pr}_f\left\{ \sup_{x\in I_{j_{\max},m},\, j' \ge j}\sqrt{n2^{-j'}}|f_{n}(j',x)-E_{f}f_{n}(j',x)|>\frac{\delta\sqrt{d}\log n}{2}\right\} \\
 &  & \le\frac{2^{3p/2-1}}{\delta^{p/2}}+2^{p-1}n^{p/2}cn^{-\frac{\delta^{2}d}{4c}\log n}\end{eqnarray*} for large $n$ in view of Proposition \ref{uexp1} (using that $2^{-j_{\max}}\ge d(\log n)^{2}/n$),
so that this expectation is bounded uniformly in $n$ by some constant
$c_{1}(p,\delta,M)$.
Using this, the Cauchy-Schwarz inequality and
Proposition \ref{uexp}, the square of the first term in \eqref{nuis}
is less than or equal to \begin{eqnarray*}
& & c_2 2^{2j_{\max}} j_{\max}^4 \\
& & \quad \times E_f \left( \sup_{x\in I_{j_{\max},m}}\max_{l \ge j}\sqrt{\frac{n2^{-l}}{\log n}} \left|f_{n}(l,x)-E_{f}f_{n}(l,x)\right| \sup_{x\in I_{j_{\max},m}}\max_{l \ge j}\frac{1}{s_{n}(j',x)} \right) ^4 \\
 &  & \le c_{2} 2^{2j_{\max}} j_{\max}^4 \left(E_{f}\left(\sup_{x\in I_{j_{\max},m}}\max_{l \ge j}\sqrt{\frac{n2^{-l}}{\log n}|}f_{n}(l,x)-E_{f}f_{n}(l,x)|\right)^{8}\right)^{1/2}\\
 &  & \quad\times\left(E_{f}\left(\sup_{x\in I_{j_{\max},m}}\max_{l \ge j}\frac{1}{s_{n}(l,x)}\right)^{8}\right)^{1/2}\le c_{3} 2^{2j_{\max}} j_{\max}^4;\end{eqnarray*}
 and the same reasoning also implies that the second term in \eqref{nuis}
is less than or equal to some constant, so that we can conclude, using
the lower bound of $2^{-j_{\max}}$, that \begin{equation}
II\le c_{4}n\label{two}\end{equation}
 for some fixed constant $c_{4}$ that depends only on $\delta$ and
$M$.

To bound $I$, we have the following: First, using Proposition \ref{uexp1}
below, we see \begin{eqnarray}
{\Pr}_f(B_{1}) & = & {\Pr}_f \left\{ \max_{\ell \ge j}\sqrt{n2^{-\ell}}\|f_{n}(\ell)-E_{f}f_{n}(\ell)\|_{I_{j_{\max},m}}>\frac{\zeta_{n}\sqrt{\|f\|_{I_{j,k(m)}}}}{4}\right\} \nonumber \\
& \le & Dn^{-\frac{\kappa^{2}\delta}{4D}} \label{infcon}\end{eqnarray}
for large $n$, with $D$ only depending on $M$. Furthermore, using $2^{-j_{\max}}\ge d(\log n)^{2}/n$
and Proposition~\ref{uexp1} below,\begin{eqnarray*}
& & {\Pr}_f(B_2) \\
 &  & \le {\Pr}_f\left\{ \sup_{x\in I_{j_{\max},m},\, \ell \ge j}\sqrt{n2^{-\ell}}|f_{n}(\ell,x)-E_{f}f_{n}(\ell,x)|>\frac{\sqrt{d\delta\|f\|_{I_{j,k(m)}}}\log n}{2}\right\} \\
 &  & \le Dn^{-\frac{d\delta^{2}}{4D}\log n}\end{eqnarray*}
for large $n$. Thus, choosing $\kappa$ large enough but finite depending on the
choice of $\alpha$, we obtain for $n$ large enough \[
I\times II\le c_{4}Dn\left(n^{-\frac{\kappa^{2}\delta}{4D}}+n^{-\frac{d\delta^{2}}{4D}\log n}\right)\le\frac{\alpha}{n2^{2j_{\max}}}.\]
 This completes the proof.

\subsection{Uniform-in-bandwidth bounds for Haar wavelet density estimators and
some consequences}

\label{ube}

The following exponential inequality was used repeatedly in the proofs.

\begin{proposition} \label{uexp1}Let $j_{\max}\in\mathbb{N}$ such
that $2^{-j_{\max}} \ge d(\log n)^2/n$. Let $I=(2^{-j}k,2^{-j}(k + 1)]$ for some $j \le j_{\max}$ and $k\in \mathbb{Z}$, and suppose $f:\mathbb{R} \rightarrow [0,\infty)$ is a density that satisfies $\|f\|_I \le M$ and
\[
 \inf_{x \in I} f(x) \ge \delta > 0.
\]
There exist constants $C_1(d)$, $C_2(d)$ and an index $n(\delta,M)$ such that for all $n \ge n(\delta,M)$ and all $C_3 \ge C_2(d)$, if
\begin{equation} \label{range}
C_{1}(d)\sqrt{\|f\|_{I} \log n} \le u \le C_{3} \|f\|_{I} \sqrt{n2^{-j_{\max}}},
\end{equation}
then
\[
{\Pr}_f \left\{ \sup_{x\in I}\max_{j\le j' \le j_{\max}}\sqrt{n2^{-j'}}|f_{n}(j',x)-E_f f_{n}(j',x)|\ge u\right\} \le De^{-\frac{u^{2}}{D}},
\]
where $D$ only depends on $C_3$ and $M$.
\end{proposition}
\begin{proof}
Writing \begin{eqnarray*}
& &  \sqrt{n2^{-j'}}\left|f_{n}(j',x)-E_{f}f_{n}(j',x)\right| \\
& & = 2 \sqrt{\frac{2^{j_{\max}}}{n}} \frac{\sqrt{2^{j'-j_{\max}}}}{2}\left|\sum_{i=1}^{n}(K(2^{j'}x,2^{j'}X_{i})-E_f K(2^{j'}x,2^{j'}X_i))\right|,
\end{eqnarray*}
 we have to consider the supremum \[
2 \sqrt{\frac{2^{j_{\max}}}{n}}\sup_{h\in\mathcal{H}}\left|\sum_{i=1}^{n}(h(X_{i})-E_f h(X_i))\right|\]
 of the (scaled) empirical processes indexed by the class of functions
\[
\mathcal{H}:=\left\{ \frac{\sqrt{2^{j'-j_{\max}}}}{2}K(2^{j'}x,2^{j'}(\cdot)):x\in I,\, j' \ge j \right\} .\]
 This class has constant envelope $1/2$ since $j' \le j_{\max}$ and
since $\sup_{x,y}|K(x,y)|=1$. Furthermore, noting that $K^{2}(x,y)=K(x,y)$
for every $x,y$, we have for $h\in\mathcal{H}$ that
\begin{eqnarray*}
E_f h^{2}(X) & = & \frac{2^{j'-j_{\max}}}{4}\int K^{2}(2^{j'}x,2^{j'}y)f(y)dy \\
& = & \frac{2^{j'-j_{\max}}}{4} \int_{2^{-j'}k(x)}^{2^{-j'}(k(x) + 1)} f(y) dy \le\frac{2^{-j_{\max}}}{4} \|f\|_{I}.
\end{eqnarray*}
 Note further that $\mathcal{H}$ is a VC-type class of functions
by using Lemma 2 in \cite{gn} and a simple computation on covering
numbers (including an obvious covering of the set $[2^{-j_{\max}},1]\subseteq[0,1]$).
Rewrite\begin{eqnarray*}
 &  & {\Pr}_f \left\{ \sup_{x\in I}\max_{j\le j' \le j_{\max}}\sqrt{n2^{-j'}}|f_{n}(j',x)-E_f f_{n}(j',x)|\ge u\right\} \\
 &  & = {\Pr}_f \left\{ \sup_{h\in\mathcal{H}}\left|\sum_{i=1}^{n}(h(X_{i})-E_f h(X_i))\right|\ge \frac{u\sqrt{n2^{-j_{\max}}}}{2}\right\} \end{eqnarray*}
 and apply expression (21) in \cite{gn1}, with
\[
 \sigma^2 := \frac{2^{-j_{\max}}\|f\|_I}{4} \wedge \frac{1}{4}
\]
and
\[
 \lambda := \begin{cases}
            c_1(d) \sqrt{\frac{\log n}{\|f\|_I \log \frac{n}{\|f\|_I}}} & \text{if } \|f\|_I \le 1, \\
		c_2(d) & \text{otherwise};
           \end{cases}
\]
for appropriate constants $c_1(d)$, $c_2(d)$ that only depend on $d$.
\end{proof}
\begin{proposition} \label{uexp} Let $j_{\max}$, $I$ and $f$ be as in Proposition~\ref{uexp1}. Then there exists a
constant $D(d,\delta,M)$ such that
for every $1\le p<\infty$ we have \begin{equation}
E_f \left(\sup_{x \in I}\max_{j\le j' \le j_{\max}}\sqrt{\frac{n2^{-j'}}{\log n}}\left|f_{n}(j',x)-E_f f_{n}(j',x)\right|\right)^{p}\le D^{p}.\end{equation}
 \end{proposition}
\begin{proof}
The proof follows from considering the same empirical process as in
the proof of Proposition \ref{uexp1}, and using bounds for $p$-th
moments of empirical processes indexed by uniformly bounded VC-classes
of functions, e.g., the bound in the display following (21) in \cite{gn1}, with $\sigma^2$ and $\lambda$ as in the proof of Proposition \ref{uexp1}, together with Proposition~3.1 in \cite{glz}. \end{proof}

\medskip

\noindent \textbf{Acknowledgement.} The authors would like to thank an anonymous referee for critical remarks, particularly on Theorem \ref{holdc}.

redpajama_set_name: RedPajamaArXiv


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