Here's an Emily Dickinson poem that's been widely studied by scholars. I'm still not sure what to make of it. I can, however, connect it to what I've experienced in my life. My personal comments follow.
This poem has kept me coming back for well over a year. Here are a few thoughts about the poem, which reads more like a small story or long riddle than a philosophical or political point of view.
This poem is at least indirectly about Emily. It's about her life as a prolific poet, a well-known figure in her setting, and lover of the outdoors. And the reality that she is a woman. My first comment, then, is that she's contemplating her life as she has experienced it. A loaded gun standing there in the corner–waiting, as something she doesn't fully own.
The action begins only after the owner appears, identifies himself and carries her away. Not as a person, but as a weapon that will benefit him. It strikes me as sad that the adventure is in the forests, valleys and mountains she loves to roam. We know this from other poems. Yet now her function isn't to talk to the animals, the trees or the birds, but to do her owner's bidding. Shoot to kill, on demand. Beginning with a Doe about which we know nothing more.
Emily comments on her new-found 'half-life' (my term, not hers). Her Master depends on her to do his bidding. Not some of the time, but spectacularly, all the time. She finds comfort in this new-found power to guard her Master's head, as well as in the reputation and safety she now enjoys as the rifle/voice of the Master.
It's a messy situation. We don't know where Emily stands with all this. In the last stanza she struggles with an unresolved question about power. If her Master dies, what will happen to her? Perhaps she fears she'll be picked up by someone else and used as his obedient, powerful speaker/killer. Surely she didn't enjoy killing that Doe.
The poem reminds me of times when so-called Owners used me, beginning with my father. In these situations they used my voice or my words without my permission, to distort truth or amplify their own power. I often wished I could die or disappear.
Yes, it does convey 'that sense of contradiction.' Thanks, Fran!
Thank you for deciphering this poem for me. I was a little bewildered.
It's a bewildering poem! I don't know what Emily would think of my comments. I do, however, know what it feels like to be used for someone else's less than life-giving purposes. I also believe many girls and women live their lives through the men they serve–often at great cost to themselves and others. It's a tough poem no matter how you read it. Maybe Emily was bewildered, too? Thanks, Candice.


Facebook, Twitter, YouTube---some see the Internet as a place for do-it-yourself marketing. But without a coherent strategy, your tweets and video uploads are just more clutter in the digital atmosphere. From web design and SEO, to web ads and blog posts, and to e-newsletters and e-blasts, LDL can help you make the most of the power of online channels.


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In competency based training, the applicant seeking assessment must agree to be assessed once they consider themselves ready.
A competent result is considered to be a 90% pass mark for each topic.
If there is an insufficient score for a competent result, the assessor will counsel the trainee depending on what the final score is. In cases where 80% or greater was achieved, the assessment will be reviewed with the applicant given an opportunity to verbally clarify incorrect responses provided. If in the opinion of the assessor, the responses are satisfactory, the result can be marked up and recorded on the file copy of the paper assessment signature sheet. For results below 80%, another re-test must be attempted by the applicant after determining what additional training is required, including whether the training program is suitable for the applicants learning style.
To proceed with re-accreditation, open the 'Curriculum Tab' where you will the various topics covered. Lessons contain an introduction to the topic, and a link to a revision workbook that can be used to refresh your knowledge on the topic. Under the Assessments panel, are the actual 'quizes' you will need to complete and pass.
The assessment is closed book because in the field, you will not be able to refer to manuals and therefore an underlying knowledge base is essential to safety carry out your duties.
There is no time limit to complete the lessons however the assessments are given generous time limits.
By continuing on with the lessons I am indicating I have read and understood the above instructions and agree that I believe I am ready to complete this assessment.


text:
Q: How to add a Git submodule on a subdirectory that is excluded in .gitignore? Let's say I have the following structure:
my-project
├── modules
│   ├── my-first-module
│   ├── my-second-module
│   └── my-fourth-module

I included the modules directory in my .gitignore file and made an exception for my-third-module:
/my-project/modules/*
!/my-project/modules/my-third-module/

Then when I execute git submodule add git@git.my-app/my-third-module.git from my-project/modules/ path, I get:
The following path is ignored by one of your .gitignore files:
my-project/modules/my-third-module
Use -f if you really want to add it.

It seems the exclusion of /my-project/modules/my-third-module/ in .gitignore is not taken into account.
So, I tried to create an empty my-third-module directory but in this case the git submodule add command complains:
'my-third-module' already exists and is not a valid git repo

Apparently git submodule add requires that the directory doesn't exists, so it's not the way to go.
Why the exclusion of my-third-module is not taken into account by Git?

A: As mentioned in this answer:

Unless --force is specified, submodule add checks if the destination path is ignored by calling git add --dry-run --ignore-missing

So, I executed git add --dry-run --ignore-missing my-third-module and I got:
The following paths are ignored by one of your .gitignore files:
my-project/modules/my-third-module
Use -f if you really want to add them.

That made me suspect that there was probably an issue with excluding in .gitignore a directory that doesn't exist.
After I created manually my-third-module, I executed again the git add command and now it's OK.
That means Git needs the directory to be created to be able to exclude it. But since it's created it's well excluded.
The command git submodule add simply doesn't take charge of this automatically.
Therefore it's safe using -f in this case to add the submodule:
git submodule add -f git@git.my-app/my-third-module.git

The my-third-module directory will be well excluded from being ignored by Git after its creation.

redpajama_set_name: RedPajamaStackExchange


text:
DOWN WITH INTEREST
With inflation easing, Raghuram Rajan cuts interest rates to the lowest in over five years
Reuters/Paco Chuquiure
Mapping the road ahead.
By Madhura Karnik
April 5, 2016 This article is more than 2 years old.
With India's inflation in a comfortable zone and the fiscal math under control, Reserve Bank of India (RBI) governor Raghuram Rajan finally cut key interest rates on Tuesday (April 5).
In his first monetary policy review in the new financial year, Rajan cut the repo rate (pdf)—the rate at which the central bank lends to commercial banks—by 25 basis points to 6.5%.
The repo rate is now at the lowest in over five years.
A decrease in repo rate means the cost of funding for banks will ease, in turn boosting liquidity in the market. Rajan also increased the reverse repo rate—the rate at which the banks lend to the central bank—by 25 basis points to 6%.
In a press briefing in Mumbai, Rajan said that this was done to help banks transmit the benefit of the rate cut to the final consumers and to increase the availability of cash in the system. The central bank has cut interest rates by 150 basis points since the monetary easing cycle began.
Most analysts and experts were expecting a rate cut since finance minister Arun Jaitley presented the budget on Feb. 29.
"It (rate cut) is sensible given the uncertainty in the economy," Rajan said. "Borrowing rates are coming down significantly in the economy and we will see more transmission in the next few months too," he added.
The central bank's policy statement although indicated that it will continue to monitor inflation and monsoon to decide the future trajectory of interest rates.
India's retail inflation cooled down to 5.18% in February, well under the RBI's target. Fiscal deficit is now at 3.69% of the GDP and the government intends to bring this down to 3.5% in the 2017 fiscal, indicating that the gap between the government's expenses and revenue is narrowing.

redpajama_set_name: RedPajamaCommonCrawl


text:
Ranch Brokers
Lifestyle & News
The Call of the Range
by JHREA | Jan 31, 2020 | Lifestyle
For generations the West has held a certain captivation over the hearts and minds of the public. From its depiction on the silver screen to the often gritty day-to-day operations, the ranching lifestyle seems to beckon adventurers to try their luck – and their determination – against the land. From early dude ranches to the resurgence of sustainable agriculture, ranching has found a perennial place in American culture.
But what does it truly take to build and sustain a ranching operation in the Mountain West? Unlike any other type of property, purchasing a ranch requires special planning and consideration, whether it be a commercial business or a legacy estate. Like any enterprise, and perhaps more so than most, ranching requires careful foresight, vision and the right resources to succeed.
Matching Ambition with Foresight
There's an old joke about a young rancher who wins the lottery. Asked what he is plans to do with his newfound wealth, he replies, "Well, I guess I'll just keep ranching until it runs out."
While humorous, the adage reflects the realities facing most commercial livestock operations. Ranching is a labor- and capital-intensive venture, with often tight margins. Returns are subject to market forces, which can cause prices to fluctuate wildly from year to year. Compounded by equal variability among inputs, like feed and growing conditions, and mounting competition from foreign suppliers and government regulations, the appeal of ranching can quickly lose its luster without the proper groundwork at the outset.
As any profitable rancher will say, preparation is key to long-term success. That requires determining clear management roles and responsibilities, staffing and operational plans. Small ranches often are family owned and operated, but even then, it often takes additional labor to facilitate day-to-day operations.
For cattle ranches, it's important to establish the type of operation—whether it will support yearlings, which are typically raised from calves until market ready, or cow-calf herds, in which the calves are raised to market but heifers are kept year-round. The latter generally requires more infrastructure, pasture and labor, since livestock is maintained continually.
Both methods require a certain carrying capacity, or the land and resources needed to support the herd. Those considerations are subject to environmental factors, like climate and access to water. Areas with harsh winters, for example, demand seasonal feed programs, which require special attention to ensure livestock is properly nourished.
Beyond operations, ranching requires business foresight to navigate the ups and downs of the market. Rancher regularly rank livestock prices as one of the hardest aspects of the business. It's not unusual for rates to swing from year to year depending on a number of factors, including market saturation, feed costs and even federal and state policies. Boom years can be followed by years of depressed prices, which have been the downfall of many ranches unable to weather the trough.
Finally, ranch buyers should consider succession planning. Estate taxes are frequently identified as one of the largest barriers to a ranch's long-term operation, and rightfully so. Many family operations have been subdivided or sold entirely simply to cover transfer costs. But less obvious challenges, like divergent visions or disinterest among future generations, have turned many ranches into untenable obligations.
'Whiskey's for Drinking, Water's for Fighting'
There's an old saying among ranchers that "whiskey's for drinking, water's for fighting." It underscores the importance of access to water, which in many places can determine the productivity of the land. But that's just one important piece of the equation.
The Mountain West boasts some of the most diverse ecology in the country. But the region's rugged beauty can often belie the fertility of the land. Most established ranches support hay or grain crops, which in turn provide feed for livestock. But sustaining the productivity of the land takes careful management.
Recently, many ranches have returned to practices of regenerative agriculture, first developed by early pioneers, which include rotational grazing and "range composting." These systems help maintain nutrient levels and foster new growth, together which can prevent erosion and help mitigate droughts.
Determining the quality of the land has been aided in recent years by new technologies and better productivity reports, which can identify soil conditions, yield loads and even deterioration risks. Ranch buyers are wise to fully vet the quality and potential of a property to ensure it meets their needs.
Value Is in the Buy
Unlike any other type of property, purchasing a ranch requires a keen understanding of the industry and the many factors that determine the value of a deal—both agricultural and economic. The countless considerations – location, productivity and tax liabilities, to name only a few – can be overwhelming for the unfamiliar buyer. In that regard, the right broker can make all the difference.
As a profile of Western Ranch's Live Jackson Hole team in the Wall Street Journal notes: "Routinely spending years with clients before they buy, ranch brokers must be equal parts tour guide, park ranger, financial adviser and agriculture expert, adept at representing both lifelong cattle ranchers and urban billionaires, and discussing heli-skiing in the same breath as complex water and mineral rights."
The dedicated team of professionals at Western Ranches represents the most ranch and farm properties in the Mountain West, with more listing volume in Wyoming, Montana and Idaho than our nearest competitors combined. Our success owes to the commitment of our agents. They know the industry, its challenges and its opportunities because they live in it, with experience and backgrounds as diverse as the region itself.
"A good ranch broker does everything necessary to learn about a client's needs and has the understanding to match those to the exactly right property," explains Steve Duerr, who has facilitated numerous high-profile sales. "It can be complicated, especially without the right knowledge. We are in the business of empowering both buyers and sellers to ensure their success."
"There is no other life like ranching," says Deidre Griffin. "It's a passion—and one that we are fortunate to share."
To learn more about what makes ranching in the Mountain West so extraordinary, contact our team.
Lifestyle & News Home
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text:
Tropical Storm Humberto gets closer to the area devastated by a hurricane in the Bahamas
Posted 7:47 am, September 14, 2019, by CNN Wire Service
GRAND BAHAMA — Humberto is now a tropical storm barreling its way through the Atlantic and threatening the same northwestern Bahamas islands ravaged by Hurricane Dorian nearly two weeks ago.
Early Saturday, the storm's center was 70 miles east of Great Abaco island, the National Hurricane Center said. It's expected to move "near or over" the northwestern Bahamas on Saturday, bringing tropical-storm-force winds and heavy rainfall, the center said.
The Bahamas will likely see up to 4 inches of rain, with some isolated areas getting up to 6 inches.
Humberto is not expected to produce significant storm surge in the northwestern Bahamas, the center said.
Regardless of wind strength, "there will be rain … over (an) area that certainly doesn't need any rain," CNN meteorologist Chad Myers said.
A tropical storm warning was in effect for the northwestern Bahamas, excluding Andros island, the center said.
Humberto's center does not appear destined for the US Southeast coast, contrary to earlier forecasts. However, its outer bands still could drop 1 to 2 inches of rain on parts of the Florida and Georgia coasts, the National Hurricane Center said.
Early Saturday, Humberto's maximum sustained winds were near 40 mph. Winds of tropical-storm strength extended 90 miles from its center.
TRACK THE STORM
It's expected to become a hurricane after leaving the Bahamas
After it clears the Bahamas, Humberto is expected to become a hurricane in the Atlantic by Sunday night, the hurricane center said.
But the current forecast track predicts Humberto turning away from the US coast.
"Since the forecast track has shifted farther to the east, the chance of heavy rainfall affecting the southeastern United States has diminished," the hurricane center said.
Swells generated by the tropical storm are expected to increase and affect the coasts of central Florida to South Carolina through early next week. These swells could cause life-threatening surf and rip current conditions, the center said.
The storm comes at the peak of the Atlantic hurricane season — which is usually in the weeks surrounding September 10, when weather conditions favor storms forming quickly.
Bahamas grappling with devastation
Meanwhile, hundreds are still missing in the aftermath of the powerful Category 5 hurricane that smashed into the Abaco Islands and Grand Bahama this month.
The death toll stands at 50 but is expected to rise as search and rescue crews sift through flattened neighborhoods.
"We are a nation in mourning," Prime Minister Hubert Minnis said in a statement. "The grief is unbearable following the devastating impact of Hurricane Dorian, which has left behind death, destruction and despair on Grand Bahama and Abaco, our second and third most populous islands."
About 3,900 evacuees have been processed through south Florida by air and sea so far, officials said.
The number includes US citizens, legal residents, Bahamians and people from other countries who evacuated the islands after the storm hit.
Topics: Tropical Storm Humberto
A tropical cyclone could form in the Gulf of Mexico this week
Hurricane Lorenzo to bring 70-foot waves to Azores
National Hurricane Center: Tropical Storm Nestor forms in Gulf of Mexico
Winter storm is on the way, here's a timeline
A triple threat storm could bring tornadoes, snow and flooding to much of the US this weekend
Storms sweep southern US, Midwest as death toll rises to 11
Snow and freezing rain will hit much of the country this week
3 to 6 inches of snow expected, blustery winds to cause drifting
'Freak storm' damaged Port Milwaukee, South Shore Yacht Club, Pleasant Prairie home
Wintry weather threatens to snarl US holiday travel
Dangerous travel conditions as another big snow is set to hit up north
Residents digging out of record-breaking snowfall in Canada
Wicked mix of storms threatens Thanksgiving travel in US


Once, Sage Markandey was doing mediation in the Bay of Bengal. It is a condition for the soul seated on the post of Indra that if during his tenure of 72 Chaukadi Yugas, someone on earth performs meditation or a religious sacrificial ceremony required for attaining the post of Indra, and if there is no hindrance in his action, then that worshipper is given the post of Indra, and that position is taken away from the current Indra. Therefore, as far as is possible, Indra does not let any worshipper's meditation or sacrificial ceremony get accomplished during his tenure. He causes hindrance in that worshipper's religious practice; no matter what he has to do for it.
Then Sage Markandey asked, "What will you do when Indra will die?" Urvashi replied, "I will be a wife to 14 Indras." The purport is that there are 1008 Chaturyug in one day of Shri Brahma ji; during this time, 14 Indras die after finishing their tenure of 72 Chaturyug each. The soul of the queen of Indra had done so many virtuous deeds in some human life that as a result of them, becoming the main queen of 14 Indras, she will enjoy the luxuries of heaven and the joy of having a husband.
It has been stated in the above-mentioned speech no. 3 that those who acquire the position of Indra, Kuber, Ish, Brahma ji, Varun and Dharmrai and those who have acquired the position of demi-god after going to Vishnu ji's Lok (place), also remain in the cycle of birth and death.
There are 33 crore posts of demi-gods in Heaven. Like, there are 540 seats of Members of Parliament in the Parliament of India. Individuals keep changing. From amongst those Members of Parliament only, the Prime Minister and other central ministers etc are appointed.
Similarly, amongst those 33 crore demi-gods only, someone acquires the post of Kuber, that is, the God of wealth just like a finance minister. The meaning of (Ish ki Padvi) 'position of Ish' is position of god, who are in total three in number according to (Lokved) folklore: - 1. Shri Brahma ji, 2. Shri Vishnu ji, and 3. Shri Shiv ji.
Varun is the god of water. Dharmrai is the chief justice, who grants the results of deeds to all the living beings. He is also known as Dharmraj. All these attain their positions by doing worship of Kaal Brahm. After the exhaustion of virtues, all the demi-gods are relieved from the position of god, and are put in the lives of 84 lakh types of animals and birds etc. Then new Brahma, new Vishnu and new Shiv sit on these positions.
All these aforesaid gods take birth and die. They are not immortal. Their state will become clear to you in this book's chapter "Creation of Nature" that how much god they are. Whose sons are these and who is their mother?
Other evidence: - In Shri Devi Puran (Sachitra Mota Type Keval Hindi, Published from Gita Press Gorakhpur), third skand, on page 123, it is written that - You have a pure form. This entire world is originating from you. I (Vishnu), Brahma and Shankar are existing by your grace. We have (Aavirbhaav) birth and (Tirobhaav) death. You are Goddess Prakriti.
God Shankar said that – O Goddess! If Brahma, who took birth after Vishnu, has taken birth from you, then am I Shankar, who performs Tamoguni leela (wrathful acts), not your son, that is, you are my mother too. We can only perform tasks as per rules, that is, we can only grant whatever is written in one's destiny. Neither can we increase it, nor decrease it.
Readers! It has become clear from this excerpt of Shri Devi Puran that Shri Brahma, Shri Vishnu and Shri Shankar ji are mortal. Their mother's name is Goddess Durga. You will gain more information from Chapter "Creation of Nature", which is written at the end in this book. These are the chief gods. Others gods are lower than these. All these take birth and die. They are not (Avinashi Ram) Immortal God.
In Shrimadbhagavat Gita Chapter 10 Verse 2, it is written that the Giver of the knowledge of Gita, Kaal Brahm, has said that – "Neither the gods know about my origin nor the Maharishis (great sages) because I am the creator of all these, that is, all of these have originated from me.
In Gita Chapter 4 Verse 5, the Giver of the knowledge of Gita has said that – O Arjun! You and I have had several births. You do not know. I know. It is also clear in Gita Chapter 2 Verse 12 that – You, I and all the kings and the army have also taken birth in the past and will be born in the future too.
In Gita Chapter 14 Verse 3, the Giver of the knowledge of Gita has said that – O Arjun! My Prakriti, that is, Durga conceives and I place the seed in her womb, by which all the living beings are born.
In Gita Chapter 14 Verse 4, the Giver of the knowledge of Gita has said that – O Arjun! Whichever embodied beings are born in all the different types of species of life-forms, (Mahat) Prakriti is their mother who conceives them and (Aham Brahm) I, Brahm, am the seed-laying father.
In Gita Chapter 14 Verse 5, the Giver of the knowledge of Gita has clarified that – O Arjun! – Satvagun Shri Vishnu, Rajgun Shri Brahma and Tamogun Shri Shiv, these three gods i.e. the three gunas, born of Prakriti i.e. Goddess Durga, bind the eternal soul to the body based on one's deeds.
It has been clarified from the above-mentioned evidence that the aforesaid gods are mortal, and they are children of Kaal Brahm.
Even after acquiring the position of Indra, Kuber, Ish i.e., Brahma, Vishnu, Shiv, and the position of Varun, Dharmrai, and after going to the Lok of Shri Vishnu Naath, one remains in the cycle of birth and death.
Readers! Sage Shri Chunak, Sage Shri Durvasa and Seer Muni did Brahm Sadhna. The Giver of the knowledge of Gita has described their way of worship as Anuttam i.e. bad in Gita Chapter 7 Verse 18. Sage Markandey performed sadhna of the same category. He was doing it considering it to be the best, and was also advising Indra to do Brahm Sadhna.
Meaning of Speech no. 1: - They guide others, but they themselves do not know about the path of bhakti.
Meaning of Speech no. 2: - Due to the absence of a Tatvdarshi Saint, the Moti Mukta i.e. the salvation-like pearl, the mantra of salvation, is not visible. This entire world is blind, that is, it is devoid of the eyes of spiritual knowledge. One who has cataract, which is an eye disease, his eyes appear to be healthy, but that person cannot see anything. Giving this example, it has been explained that those who speak fluent Sanskrit seem to be highly learned, but they do not know the profound mysteries of the holy scriptures. They are afflicted with the cataract of ignorance.
Meaning of Speech no. 3: - Those who read Vedas and learn them by heart, because of not understanding the profound mysteries of the Vedas, themselves worship against the Vedas and make others worship against them too. There is no mention of worship of a stone idol in the Vedas. Those so-called scholars of Vedas themselves worship stone idols and make others do that. They have forgotten the Creator mentioned in the Vedas.
In Shrimadbhagavat Gita Chapter 4 Verses 25 to 29, this only has been described that, whichever worship a worshipper is doing, he considers it to be the best. All the worshippers consider their respective way of worship to be the destroyer of sins.
In Gita Chapter 4 Verse 32, it has been clarified that Sachidanand Ghan Brahm has Himself described in detail the knowledge of yagyas, that is, religious rituals in the speech uttered from His lotus-mouth. It is Tatvgyan. Knowing it, a worshipper becomes free from all the sins.
In Gita Chapter 4 Verse 34, it has been clarified that Tatvdarshi Saints know that Tatvgyan. By prostrating before them, and politely asking question, those knowledgeable Mahatmas, who are well-versed with the Tatvgyan, will instruct you in Tatvgyan (true spiritual knowledge).
Because of not getting that Tatvgyan, which has been mentioned as "Nirgun Rasa" in the above speech, all the worshippers remained in the cycle of birth and death.
Sage Markandey was doing Brahm-worship. In Shrimadbhagavat Gita Chapter 8 Verse 16, it is said that those worshippers, who have gone to Brahm Lok, also repeatedly remain in the cycle of birth and death.
Translation: - O human being! You have taken birth and died infinite times. Perform true worship and die whilst you are alive. The meaning of dying whilst alive is that a worshipper realises that every object of this world is temporary. This body is also not permanent. Even the problem of birth and death is horrible. There is nothing but misery in this world. If even after acquiring a human life, one did not attain salvation, then one only lived a life of an animal. Like, in Gita Chapter 7 Verse 29, it is stated that a worshipper who is engaged in religious practice only to gain freedom from the suffering of old age and the misery of death, knows 'Tat Brahm', the entire spirituality and all the actions.
In this way, after gaining Tatvgyan, a person does not have any desire for unnecessary things. He does not consume tobacco, alcohol and meat. Dancing and singing seems to be an act of fools. Whatever one gets to eat, one feels contented in it.
To get one's welfare done, a worshipper contemplates that if I will not come to satsang, I will not be able to obtain a sight of Guru ji. Because of not listening to the satsang, the mind will again start getting engrossed in vices. Leaving all his tasks, that worshipper sets forth to listen to the satsang. He ponders that every day we hear and see that leaving behind young children, a father departs from the world; he dies. Big capitalists die in mishaps. They leave behind all the wealth that they had accumulated in their entire lives. They do not return to look after that wealth. Before death, they did not feel like leaving work even for a single day, and now everything is left permanently.


text:
\section{Introduction}

In probability theory it is customary to investigate two broad families of problems, (i) the convergence of sums of large numbers of random variables, and (ii) the estimation of likelihoods of deviations of those sums from their large number limits \cite{Feller}.
The probability spaces of the random variables in question are usually fixed
and one investigates how fast do properties of a sample drawn from the statistical population \textit{\textbf{converge}} towards properties of that statistical population (to be called population for brevity, later on).

A good example of this is one of the most fundamental theorems of probability, the \textit{\textbf{Law of Large Numbers}} (LLN). The weak version can be stated (see~\cite{Grinstead} for example) as :
\begin{quotation}
Let $X_1, X_2, \ldots, X_N$ be an independent trials process, with finite expected value $\mu = E[X_i]$ and variance $\sigma^2 = Var(X_i)$. Let $S_N = \sum_{i=1}^{N} X_i$. Then $\forall \mbox{ } \epsilon > 0$,
\begin{equation}
\lim_{N \to \infty} P\left[ \left| \frac{S_N}{N} - \mu \right| \geq \epsilon \right] = 0 \label{eq:LLN}
\end{equation}
\end{quotation}
Noting that the quantity $\frac{S_N}{N}$ is nothing more than the sample mean, $\overline{X}_N$, the Law of Large Numbers can be stated in words as ``for independent trials of a sample of a distribution with finite population mea
, the sample mean should approach the population mean, as the number of trials (i.e. size of the sample) gets very large''.

In this paper we pose questions in a different way. We consider a sequence of populations from discrete, finite probability spaces whose size (to be termed number of outcomes in what follows) increases. We draw a sample of a fixed size from each of these populations, and analyze under what conditions, and how quickly, do the properties of the samples \textit{\textbf{diverge}} from properties of the populations.
In effect we are increasing monotonically the number of outcomes available to each trial, while keeping the number of trials, of sample size ($N$) constant.
In other words we formulate the \textbf{\textit{\textbf{``Reverse'' of the Law of Large Numbers}}}.

We illustrate this using a thought experiment based on a real life situation - betting on horses.
Each horse is given a unique label.
Here the random variable in question is the label of the winning horse, the number of outcomes $M$ is the number of horses in the race, and the sample size $N$ stands for the number of repetitions of the race.
If the number of horses in the race is fixed ($M=\mbox{const}$) then, if the number of race repetitions becomes large ($N\rightarrow \infty$), the average label of the winning horse converges towards that of the expected value of the label of the winning horse. We have made the idealized assumption that the probability distribution itself remains fixed thus ignoring aging effects, weather changes, different tracks etc.
Now imagine that, as time progresses, new horses are added to the race indefinitely ($M\rightarrow \infty$) and that each time a new horse has been added, the same number of races ($N=\mbox{const}$) is held.
In this situation we intuitively perceive that the average label of the winning horse becomes less likely to be linked to the expected value of the label of the winning horse.
In other words, the variance of the average label of the winner will increase at a rate that depends on the number of horses that participate in the race and on the probability distribution that a given horse wins.

Another thought experiment to illustrate this idea is to think of an unbiased $M$-sided ``die''. Here we ask what value appears face up when we throw the die. If we think of the standard die, with $M=6$, and consider $N = 1000$ throws, we would expect the average throw to be pretty close to the expected value of, in this case, $3.5$. However, if we increase $M$ to larger and larger values, but keep $N$ fixed, we would no longer expect to always get sample means close to the population mean (expected value).

In the above examples, we looked at one particular statistic - the sample mean. The Law of Large Numbers, in equation (\ref{eq:LLN}), tells us that the sample mean approaches the population mean as the number of trials increases. This is normally proven using Chebyshev's inequality (assuming that the variance, $\sigma^2$, is finite), which for the sample mean, is given by:
\begin{equation}
P\left[ \left| \overline{X}_N - \mu \right| \geq \epsilon \right] \leq \frac{\sigma^2}{N \epsilon^2} \label{eq:Cheby}
\end{equation}
Here we used the fact that the variance of the sample mean is given by: $\sigma^2_{\overline{X}_N} = \frac{\sigma^2}{N}$.
Chebyshev's inequality offers a sense of ``how large'' $N$ must be to see the desired convergence described by $\epsilon$. For fixed $N$ and $\epsilon$, the variance of the random variable, $X$, controls the bound on the probability of the sample mean being close to the population mean\footnote{Of course, this probability can never be greater than $1$ so bounds exceeding $1$ are ``loose''.}.

In this paper, we look at probability distributions of finite discrete random variables. We consider what happens when additional outcomes are deemed possible, but the distribution still retains the same basic form. We then have, in essence, a sequence consisting of probability distributions that are all similar, but with higher terms in the sequence corresponding to those distributions with a higher number of outcomes ($M$). We also have corresponding sequences for the expected values, and variances of these distributions. We are interested in those such sequences of variances which diverge as the number of outcomes becomes large. Then, from equation (\ref{eq:Cheby}), and for fixed $N$ and $\epsilon$, it is clear that sample means for the corresponding random variables would not be likely to be close to the expected values of those random variables.

Therefore, we seek conditions for these sequences of variances to diverge as the number of outcomes becomes very large. We also attempt to determine classes of (sequences of) distributions that correspond to particular rates of divergence of the variance. In the course of this work we will use the terms probability measure and probability distribution interchangeably. We will call the sequences of distributions described above, as simply distributions (that are functions of $M$), and the sequences of variances as variances (that are functions of $M$). We will attempt to formulate our considerations and results in the axiomatic language of Kolmogorov's theory of probability~\cite{Kolmogorov}.

\section{Theoretical formulation}

\subsection{The Probability Distributions}

Both the weak and the strong LLNs are proven without imposing any assumptions on the sample space $\Omega$ and on the probability measure ${\mathbb P}$.
Instead one only assumes the existence of the first moment, in the case of the weak law, and both the first and the second moments in the case of the strong law \cite{Varadhan}.

However, exploring the other extreme, namely the limit of the number of outcomes becoming very large ($M\rightarrow \infty$) subject to the sample size $N$ being fixed, does in fact require the knowledge of both the sample space and the probability measure.
Hence, we have to formulate assumptions about $\Omega$ and ${\mathbb P}$.

Consider a finite discrete random variable $X$ with $(M+1)$ possible outcomes $\Omega = \lbrace m_j \rbrace_{j=0}^{M}$, with associated probabilities $\lbrace \bar{\mathbb P}_j \rbrace_{j=0}^{M}$ with $M \in \mathbb N \geq 1$.\footnote{There are $(M+1)$ possible outcomes as we index from $0$ to $M$. For compactness we will continue to use $M$ as the number of outcomes. Indeed, as $M$ becomes very large, there is little difference between $M$ and $(M+1)$.} For our purposes it is necessary to define the probability distribution as an explicit function of the outcomes, $m_j$. Thus, we use a polynomial representation given by:
\begin{equation}
\bar{{\mathbb P}}_j := \frac{{\mathbb P}_j}{\mathfrak N} =
\frac{1}{\mathfrak N}\sum\limits_{n=0}^{M} \tilde{a}_n {m_j}^n \quad \mbox{for $j=0,\dots,M$}
\label{eq:ProbSpace_gen}
\end{equation}
Here the numbers $\left({\mathbb P}_j\right)_{j=0}^M$ are like ``unnormalized'' probabilities and the normalization factor, ${\mathfrak N}$, is given by:
\begin{equation}
{\mathfrak N} := \sum\limits_{j=0}^M {\mathbb P}_j  \label{eq:norm}
\end{equation}
The coefficients, $\left\{\tilde{a}_n\right\}_{n=0}^M$, are real numbers which can be computed by inverting the linear relationship in (\ref{eq:ProbSpace_gen}) and thus inverting the matrix $\underline{\underline{J}}_{j, n} = {m_j}^n$.
As $\underline{\underline{J}}$ is the VanderMonde matrix~\cite{Vandermonde} this can always be done; thus, all finite discrete probability distributions can be represented in this form.

We hope to look at random variables with more arbitrary outcomes in future work but for this paper, we look at a specific set of outcomes, $m_j = j \mbox{, }j \in \lbrace 0, 1, \ldots M \rbrace$. In this case, equation (\ref{eq:ProbSpace_gen}) becomes:
\begin{equation}
\bar{{\mathbb P}}_j := \frac{{\mathbb P}_j}{\mathfrak N} =
\frac{1}{\mathfrak N}\sum\limits_{n=0}^{M} \tilde{a}_n j^n \quad \mbox{for $j=0,\dots,M$}
\label{eq:ProbSpace}
\end{equation}
We then use the fact that ${\mathbb P}_0 = \tilde{a}_0$, and call the quantities $\left(\frac{{\mathbb P}_j - {\mathbb P}_0}{j}\right)_{j=1}^M$ ``reduced probabilities''. We can write the other coefficients $\lbrace \tilde{a}_n \rbrace_{n=1}^M $ in terms of the reduced probabilities:
\begin{equation}
\tilde{a}_n := \sum\limits_{j=1}^M {\mathfrak A}_{j,n} \frac{{\mathbb P}_j -{\mathbb P}_0}{j}
\label{eq:InverseRel}
\end{equation}
for $n=1,\dots,M$.
Here $\left({\mathfrak A}_{j,n}\right)_{j=1,n=1}^{M, M}$ is the inverse of
$\underline{\underline{J}} := \left(j^{n-1}\right)_{j=1,n=1}^{M,M}$.
Again, since $\underline{\underline{J}}$ is a Vandermonde matrix it always invertible.

We note that any quantities that describe the sample (the sample mean, the variance of the sample mean, for example) depend explicitly
on the coefficients in (\ref{eq:InverseRel}). We will see that, depending on those coefficients, we will obtain different
large-$M$ behaviour of those quantities and, in particular, of the variance of the sample mean.
Some of these coefficients may be zero. For our analysis it is useful to use $s > 0$ for the order of the polynomial in (\ref{eq:ProbSpace}) such that $\tilde{a}_s \ne 0$ and $\tilde{a}_j = 0$ for all $s<j\le M$. We will see that the variance of the sample mean scales differently with the number of outcomes depending on $s$. Both $s$ and the coefficients $\tilde{a}_n$ may vary as $M$ varies.

Recall that the variance of the sample mean depends on the variance of the underlying random variable. To look at the behaviour of the variance for different $M$, we need to find a closed form expression
for the inverse Vandermonde matrix.
Furthermore we believe that the closed form expression for the inverse will be useful for other mathematical problems, like polynomial least square fitting, Lagrange interpolation polynomials \cite{HoffmanKunze}, and reconstruction of a statistical distribution from the moments of the distribution \cite{VonMises}.

The expression for the inverse was, to the best of our knowledge, previously unknown.
We give it below.  The inverse
$\left({\mathfrak A}_{j,n}\right)_{n=1,j=1}^{M,M}$ reads:
\begin{eqnarray}
\lefteqn{{\mathfrak A}_{j,n} = \frac{(-1)^{j+n}}{(n-1)!(M-n)!}
\cdot
\left(\sum\limits_{p=0}^{M-j} P_{j+p}(M) (-n)^p\right)}
\label{eq:InverseVandermonde} \\
\!\!&&=\!\!
(-1)^{j+n} C^M_n
\sum\limits_{p=0}^{j-1}
\left(
\sum\limits_{1\le q_1 < \cdots < q_p \le n-1} \prod\limits_{l=1}^{p} \frac{1}{q_l}
\right)
\left(
\sum\limits_{n+1\le q_{p+1} < \cdots < q_{j-1} \le M} \prod\limits_{l=p+1}^{j-1} \frac{1}{q_l}
\right)
\label{eq:InverseVandermonde1}
\end{eqnarray}
Here the polynomials, $P_i(j) $, satisfy the following recursion relations:
\begin{equation}
P_{M-j}(M)-P_{M-j}(M-1) = -\sum\limits_{p=1}^{j} P_{M-j+p}(M) (-M)^p
\label{eq:PolynomialsRecurs}
\end{equation}
for $j=1,\dots,M-1$ with $P_M(M) = 1$.
The lowest ten polynomials ($j=1,\dots,10$) are listed in (\ref{eq:Pol1})-(\ref{eq:Pol10}) in Appendix C.
Therein attached is also a piece of Mathematica code that tests the validity of those expressions.
The proof of (\ref{eq:InverseVandermonde}) is given in Appendix B.
Note that rows with low ($1$, $2$, $3$) and with high ($M$, $M-1$, $M-2$) indices  have a particularly
simple form; the complexity of the expression rises when the row index tends towards $M/2$.
Setting $j=1,2,3$ in (\ref{eq:InverseVandermonde1}) and $j=M,M-1,M-2,M-3$ in (\ref{eq:InverseVandermonde}) we obtain:
\begin{eqnarray}
\lefteqn{{\mathfrak A}_{1,n} = (-1)^{1+n} C^M_n}  \\
&&\!\!\!\!
{\mathfrak A}_{2,n} =
(-1)^1 {\mathfrak A}_{1,n} \left(\sum\limits_{q=1}^{n-1} \frac{1}{q} (1_{1\le q \le n-1} + 1_{n+1\le q \le M})\right) \\
&&\!\!\!\!
{\mathfrak A}_{3,n} = (-1)^2 {\mathfrak A}_{1,n}
\left( \sum\limits \frac{1}{q_1 q_2} 1_{n+1\le q_1<q_2\le M} \right. \nonumber \\
&&
\left. + \sum\limits \frac{1}{q_1 q_2} 1_{1\le q_1 \le n-1} 1_{n+1 \le q_2 \le M} 1_{1\le q_1 < q_2 \le n-1}\right) \\
&&\vdots \nonumber \\
&&\!\!\!\!
\frac{{\mathfrak A}_{M-3,n}}{{\mathfrak A}_{M,n}(-1)^{-3}} =
  \left( \frac{(M-2)(M-1)M^2(M+1)^2}{48}\right. \nonumber \\
&&
 \left. -\frac{n}{24}(M-1)M(M+1)(2+3 M) + \frac{n^2}{2} M(M+1) -n^3\right)\\
&&\!\!\!\!
\frac{{\mathfrak A}_{M-2,n}}{{\mathfrak A}_{M,n}(-1)^{-2}} =
\left(\frac{(M-1)M(M+1)(2+3 M)}{24} - \frac{n}{2} M(M+1)  + n^2\right) \\
&&\!\!\!\!
\frac{{\mathfrak A}_{M-1,n}}{{\mathfrak A}_{M,n}(-1)^{-1}} =
  \left(\frac{M(M+1)}{2}  - n  \right) \\
&&\!\!\!\!
{\mathfrak A}_{M,n} = \frac{(-1)^{M+n}}{(n-1)!(M-n)!}
\end{eqnarray}

\noindent{\bf Note:} The inversion of the Vandermonde matrix has applications in
control theory \cite{Tou,deBrule,Reis}, in signal processing problems \cite{Neagoe}
and in systems theory \cite{Reis,Kan,Goeknar,Wertz}.
In order to solve the problem one typically makes use of the Lagrange interpolation formula
and finds the rows of the inverse matrix
by computing the coefficients of the Lagrange interpolation polynomials related to the matrix elements \cite{Tou}.
An alternative approach was proposed in \cite{Kaufman} where the inverse matrix is expressed
as a product of two matrices
one of which is diagonal and the elements of the other one are given through certain recursion relations.
In this way the total number of operations needed to invert is reduced from $O(M^3)$ to $O(M^2)$.
Finally in \cite{Neagoe} one expresses the elements of the inverse through totally symmetric polynomials
of the elements of the original matrix.

Our method of computing the inverse (presented in Appendix B) is a way of actually
solving the recursion relations from \cite{Kaufman} or summing up
the totally symmetric polynomials analytically if the elements of the original matrix are certain real or complex
powers of a constant or of an arithmetic progression.
Indeed, even though the elements of the original matrix are first powers of an arithmetic progression in our work i.e.\ $x_j=j$,  the manipulations
(\ref{eq:MatrixInverse})-(\ref{eq:MatrixInverse1a}) can also be done analytically in the generic case of arbitrary powers, with little effort.
Thus, we believe, that our method is superior to the methods known in the literature
and can be applied to produce more efficient numerical algorithms of use in the areas described above.

Now we fix $N \ge 1$, we draw a sample of size $N$ from the population described in (\ref{eq:ProbSpace}) and we conjecture that the quantities that describe the sample (estimators of population parameters) diverge from those that describe the population, if the number of outcomes $M$ becomes very large.
To be specific, we analyze the variance of the sample mean, and we show that, except for very ``unusual distributions'', it increases monotonically with $M$, if $M$ is big enough, which implies that, for fixed $N$, the sample mean diverges from the population mean when $M \to \infty$.
In most computations below we will use asymptotic limits ($M\to \infty$) rather than exact results.
However, deriving exact results is not an essential difficulty; it amounts to performing more work, which is not necessary for our purposes.

\subsection{The variance of the sample mean}

The variance of the sample mean reads:
\begin{equation}
\sigma^2_{\overline{X}_N} := \mbox{var}\left[ \frac{\sum\limits_{i=1}^N X_i}{N} \right] = \frac{\sigma^2}{N} := \frac{\left<\left(X - \left<X\right>\right)^2\right>}{N} = \frac{\left<X^2\right> - \left<X\right>^2}{N}
\label{eq:SampleVardef}
\end{equation}
It depends on the variance, $\sigma^2$, of the random variable. Using equation (\ref{eq:ProbSpace}) and Faulhaber's formula (\ref{eq:Faulhaber}), this variance can be written as:
\begin{eqnarray}
\sigma^2 &=&
\frac{1}{{\mathfrak N}^2}
\left[
{\mathfrak N} \left(\sum\limits_{j=0}^M j^2 {\mathbb P}_j\right)
-
\left(\sum\limits_{j=0}^M j {\mathbb P}_j\right)^2
\right]
=
\frac{\sum\limits_{n=0}^{2{M}+4} \beta^{(1)}_n
(M+1)^n
}{
\sum\limits_{n=0}^{2{M}+2} \beta^{(2)}_n
(M+1)^n
}
\label{eq:MyVariance}
\end{eqnarray}
where
\begin{eqnarray}
\beta^{(1)}_n &:=&
\sum\limits_{n=n_1+n_2+4-(k_1+k_2)} 1_{n_1\le M}1_{n_2\le M}
\tilde{a}_{n_1} \tilde{a}_{n_2} B_{k_1} B_{k_2} \nonumber\\
&&
\left(
\frac{C^{n_2+3}_{k_2}  1_{k_2\le n_2+3}}{n_2+3} \frac{C^{n_1+1}_{k_1}1_{k_1\le n_1+1}}{n_1+1}
-
\frac{C^{n_2+2}_{k_2}  1_{k_2\le n_2+2}}{n_2+2} \frac{C^{n_1+2}_{k_1}1_{k_1\le n_1+2}}{n_1+2}
\right)\label{eq:CoeffsRatioFct0}\\
\beta^{(2)}_n &:=&
\sum\limits_{n=n_1+n_2+2-(k_1+k_2)} 1_{n_1\le M}1_{n_2\le M}
\tilde{a}_{n_1} \tilde{a}_{n_2} B_{k_1} B_{k_2} \nonumber \\
&& \times \left( \frac{C^{n_1+1}_{k_1}1_{k_1\le n_1+1}}{n_1+1}\right) \left( \frac{C^{n_2+1}_{k_2} 1_{k_2\le n_2+1}}{n_2+1} \right)
\label{eq:CoeffsRatioFct}
\end{eqnarray}
where $n_1, n_2, k_1, k_2 \in \mathbb{N}$. The numbers, $B_{k_i}$, are the Bernoulli numbers.

Thus both the variance $\sigma^2$ and the variance of the sample mean are rational functions of the number of outcomes $M$.
Since we are only interested in the large-$M$ behaviour of the variance of the sample mean, we do not need to simplify expressions (\ref{eq:CoeffsRatioFct0}) and (\ref{eq:CoeffsRatioFct}) for the coefficients but instead we only need to work out leading order terms in both the numerator and the denominator in (\ref{eq:MyVariance}).
Thus, we assume that the order of the polynomial in (\ref{eq:ProbSpace}) is $s>0$, meaning
that the coefficent $\tilde{a}_s$ is non-zero and the last (M-s) coefficients, $\lbrace \tilde{a}_n \rbrace_{n=s+1}^{M}$ in the expansion (\ref{eq:ProbSpace}), are zero.
Then the coefficients for the highest order terms in the numerator read:
\begin{eqnarray}
\beta^{(1)}_{2 s+4} &=&  \frac{(\tilde{a}_s)^2}{(s+1)(s+3)(s+2)^2} \\
\beta^{(1)}_{2 s+3} &=&  \frac{2 \tilde{a}_{s-1} \tilde{a}_s}{s(s+1)(s+2)(s+3)} -  \frac{(\tilde{a}_s)^2}{(s+1)(s+2)(s+3)} \\
\beta^{(1)}_{2 s+2} &=&   \frac{-\tilde{a}_s \tilde{a}_{s-2}}{s(s+1)^2 (s+2)} +
\frac{\tilde{a}_{s-1} \tilde{a}_{s-1}}{s(s+1)^2 (s+2)} +  \frac{3 \tilde{a}_s \tilde{a}_{s-2}}{(s-1)s(s+2)(s+3)}
\\\nonumber &&
+  \frac{\tilde{a}_s \tilde{a}_{s-1} 2(3+2s)}{s(s+1)(s+2)(s+3)}
+  \frac{(\tilde{a}_s)^2 (2s+3)}{(s+1)(s+2)(s+3)} \\
\beta^{(2)}_{2 s+2} &=&  \frac{(\tilde{a}_s)^2}{(s+1)^2} \\
\beta^{(2)}_{2 s+1} &=&  \frac{2\tilde{a}_{s-1} \tilde{a}_s}{s(s+1)} -  \frac{(\tilde{a}_s)^2}{(s+1)}
\end{eqnarray}

Dividing the polynomials in (\ref{eq:CoeffsRatioFct0}) and (\ref{eq:CoeffsRatioFct}) by one another we obtain the following expression for $\sigma^2$, for large $M$:
\begin{eqnarray}
 \sigma^2 =
(M+1)^2 \frac{(s+1)}{(s+3)(s+2)^2}
 +
(M+1) \left(2 \frac{\tilde{a}_{s-1}}{\tilde{a}_s} - s\right) \frac{(s+1)}{s (s+2)^2(s+3)}
+O(1)
\label{eq:Variance}
\end{eqnarray}
Subbing this into equation (\ref{eq:SampleVardef}) will also give an expression for the variance of the sample mean as a function of $M$.

\subsection{The Large $M$ behaviour of the Variance}\label{ss:Large_M}

We now wish to examine whether $\sigma^2$ converges or diverges for large $M$. Equation (\ref{eq:Variance}) can be rewritten as:
\begin{equation}
\sigma^2 =
\frac{(s+1)}{(s+3)(s+2)^2} \left[M^2 + 2M \left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) + M + 2\left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) \right]
\label{eq:Variance1}
\end{equation}
As noted previously, it is possible for both the order, $s$, and the coefficients, $\tilde{a}_n$, to vary with $M$. Looking at the term in the square brackets, we notice that the first term ($M^2$) will always dominate the third term ($M$) and the second term ($2M \left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right)$) will always dominate the fourth term ($2\left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right)$). Therefore we want to consider the large $M$ behaviour due to the first and second terms. The exact behaviour of the second term will depend on the distribution concerned, but we can identify four broad cases depending on the behaviour of the ratio $\left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right)\left(\frac{1}{s}\right)$ in the large $M$ limit:
\begin{enumerate}
\item
$\displaystyle\lim_{M \to \infty} \left|\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) < \infty$
\item
$\displaystyle\lim_{M \to \infty} \left|\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) = \infty$ and  $2M\left| \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) \ll M^2$
\item
$\displaystyle\lim_{M \to \infty} \left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) = \infty$ and $2M\left( \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) \gtrsim  M^2$
\item
$\displaystyle\lim_{M \to \infty}\left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) = -\infty$ and $2M\left( \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) \lesssim -M^2$
\end{enumerate}
The first case corresponds to when the limit of the ratio is finite. The second case is when the ratio grows in absolute value with $M$ but at a rate slower than $M$. The third case is when the ratio grows with $M$ at a rate faster than $M$. Finally, the fourth case is when the ratio grows in absolute value with $M$ but is negative. We now consider each case separately.

\subsubsection{Case 1}

In this case, we do not need to worry about the coefficients, and how they depend on $M$ to understand the large $M$ behaviour of the variance. It is clear that only the first term from equation (\ref{eq:Variance1}) matters as $M \to \infty$. Thus we have:
\begin{equation}
\sigma^2 \approx \frac{(s+1)}{(s+3)(s+2)^2} M^2 \label{eq:case1}
\end{equation}

Now, let us consider three examples. Firstly, $s$ does not depend on $M$; secondly,
$s \simeq \sqrt{M}$, i.e.\ $\lim\limits_{M\rightarrow \infty} s/\sqrt{M} = \gamma$ where $0 < \gamma < \infty$; and finally,  $s \simeq M$, i.e.\ $\lim\limits_{M\rightarrow \infty} s/M = \tilde{\gamma}$ where $0 < \tilde{\gamma} < \infty$.
From (\ref{eq:case1}) we see that in all three instances, both the variance and the variance of the sample mean (for given $N$)
diverge as the second, the first and the zeroth power of the number of outcomes respectively. That is, we have:
\begin{eqnarray}
\lim_{M\rightarrow \infty} \frac{\sigma^2}{M^2} &=&  \frac{(s+1)}{(s+3)(s+2)^2} < \infty \label{eq:AccDiv}\\
\lim_{M\rightarrow \infty} \frac{\sigma^2}{M^1} &=&  \frac{1}{\gamma^2} \label{eq:Div}\\
\lim_{M\rightarrow \infty} \frac{\sigma^2}{M^0} &=&  \frac{1}{\gamma_1^2}\label{eq:DecDiv}
\end{eqnarray}

We now go on to provide three explicit examples of such distributions. In all three examples, for fixed sample size $N$, the sample mean, or the estimator of the population mean, diverges from the population mean, yet at different rates.
We describe these examples as \textit{\textbf{Accelerating Divergence}}, \textit{\textbf{Divergence}}, and \textit{\textbf{Decelerating Divergence}}, respectively.

\paragraph{Accelerating divergence:}
We generated a sequence of non-zero real parameters $\left(\tilde{a}_n\right)_{n=1}^s$ and computed the normalized probabilities
$\left(\bar{\mathbb P}_j\right)_{j=0}^M$ from (\ref{eq:ProbSpace}).
We run over $s=2,\dots,6$ and for each value of $s$ we plot both the normalized probabilities, for $M=100$, as a function of $j$, and the variance of the distribution as a function of the number of outcomes $M$, in the double logarithmic scale (see Figure \ref{fig:ProbVar}).
It is readily seen that the variance behaves asymptotically as the second power of the number of outcomes, and, in addition the asymptotic behaviour is attained quite quickly.

\noindent{\bf Note:}
We reiterate that this type of divergence is obtained for every distribution whose unnormalized probability distribution can be represented by a polynomial of order $s$ that does not depend on $M$, when $M$ is large.

\begin{figure}[hbt]
\hbox{
\psfig{figure=Probability100.ps,width=0.49\textwidth}
\psfig{figure=Variance100.ps,width=0.49\textwidth}
}
\caption{The probability distributions in equation (\ref{eq:ProbSpace}) for different
values of the order of the polynomial $s=2,\dots,6$ (left)
and the variance of that distribution as a function of the number of outcomes $M=100000,\dots,200000$ (right).
A second order polynomial was fitted by least square regression to each of the data sets on the right. (This was done to confirm the relation in equation (\ref{eq:case1})).
The parameters of that fit are
(-3.2831 (-3.2834), 1.998, 0.000087), (-3.6240 (-3.6243), 1.998, 0.000097), (-3.9195 (-3.9199), 1.998, 0.000138), (-4.17872 (-4.1795), 1.9959, 0.00023), (-4.4096 (-4.4102), 1.99677, 0.00018) for $s=2,\dots,6$ respectively.
The theoretical values of the first parameter (the intercept) are given in brackets and read
$\log((s+1)/((s+3)(s+2)^2)$.
The theoretical values of the second and the third parameter are two and zero respectively.
\label{fig:ProbVar}
}
\end{figure}

\paragraph{Divergence:}
Here we repeat the procedure from the previous point with one difference, namely that the number of parameters depends on the number of outcomes like equation (\ref{eq:Div}).
For the sake of simplicity let us take $\tilde{a}_n = \delta_{n,\gamma \sqrt{M}}$ where $\gamma>0$.
Then, asymptotically, the unnormalized probabilities ${\mathbb P}_j$, and the norm ${\mathfrak N}$, read ${\mathbb P}_j = j^{\gamma \sqrt{M}}$, ${\mathfrak N} = \frac{M^{\gamma \sqrt{M} + 1}}{(\gamma\sqrt{M}+1)}$ respectively.
The probability distribution is plotted in the first graph of Figure~\ref{fig:ProbVar1}.
The first and second moments, and the variance read:
\begin{eqnarray}
\left<j\right> &=& \sum\limits_{j=0}^M j j^{\gamma \sqrt{M}} \left(\frac{\gamma \sqrt{M} + 1}{M^{\gamma \sqrt{M}+1}}\right)
=
M \left( \frac{\gamma \sqrt{M}+1}{\gamma \sqrt{M}+2} \right) \label{eq:FirstMoment}\\
\left<j^2\right> &=& M^2 \left( \frac{\gamma \sqrt{M}+1}{\gamma \sqrt{M}+3} \right) \label{eq:2ndMoment}\\
\sigma^2 &=& \left<j^2\right> - \left<j\right>^2 = M^2 \left( \frac{\gamma \sqrt{M}+1}{(\gamma \sqrt{M}+3)(\gamma \sqrt{M}+2)^2} \right)
\mathop{=}_{M\rightarrow \infty} \frac{1}{\gamma^2} M
\end{eqnarray}
In Figure \ref{fig:ProbVar1} we plot the variance of the distribution as a function of the number of outcomes $M$.
Now the variance diverges asymptotically as the first power of the number of outcomes. Here, however,
the asymptotic behaviour is attained much slower than in the previous case.

\noindent{\bf Note:} We emphasize that this type of divergence is characteristic for every distribution whose probability function can be represented by a polynomial of order $s$  that behaves like the square root of the number of outcomes $M$, for large $M$.
Indeed, setting $s=s_0\sqrt{M}$ in (\ref{eq:Variance1}) we get:
\begin{equation}
\sigma^2 \mathop{=}_{M\rightarrow\infty} M \frac{1}{{s_0}^2} + O(1)
\end{equation}

\begin{figure}[hbt]
\hbox{
\psfig{figure=ProbabilityLin100.ps,width=0.49\textwidth}
\psfig{figure=VarianceLin100.ps,width=0.49\textwidth}}
\caption{The probability distribution ${\mathbb P}_j = j^{\sqrt{M}}$ (left) and the variance of that distribution as a function of the number of outcomes $M=100000,500000$ (right)
Here, the parameters of the linear regression second order polynomial fit are $(-0.3\pm 0.0027,1.04\pm 0.0004,0.002\pm 0.000017)$ compared to the theoretical parameters $(0.0,1.0,0.0)$,
\label{fig:ProbVar1}
}
\end{figure}

\paragraph{Decelerating Divergence:}
Finally, we consider an exponential distribution
\begin{equation}
\bar{{\mathbb P}}_j = \exp(-j\alpha) \left( \frac{1- \exp(-\alpha)}{1 - \exp(-(M+1)\alpha)}\right)
\label{eq:ExpDistr}
\end{equation}
for $j=0,\ldots,M$ and $\alpha > 0$.
For large $M$,
the distribution can be well approximated by its Taylor expansion truncated at the $M$\textsuperscript{th} order
and as such, it would correspond to the polynomial form in equation (\ref{eq:ProbSpace}).
Then $\tilde{a}_n = (-\alpha )^n/n!$
and, in the large $M$ limit, the variance takes the following form:
\begin{equation}
\sigma^2 = \frac{1}{\alpha^2}
\left( \frac{1 - \exp(-\alpha M)(1+\alpha M+\frac{(\alpha M)^2}{2}+\frac{(\alpha M)^3}{6})}{1 - \exp(-\alpha M)(1+\alpha M)}\right)
\mathop{=}_{M\rightarrow \infty}
\frac{1}{\alpha^2}
\label{eq:SampleVarExp}
\end{equation}
Recall that equation (\ref{eq:SampleVarExp}) is only an approximation. In order to check the validity of that
approximation, we plot the approximated variance of the sample mean $\sigma^2_{\overline{X}_N}=\sigma^2/N$ along with the exact result\footnote{We have derived the exact variance of the sample mean using MATHEMATICA. The expression is lengthy and cumbersome and, in our opinion, it does not bring much to quote it here.} in Figure \ref{fig:SampleVariance}.
We can see from the figure that the results do not differ by more than $1\%$ for $M>10$.

Now the variance behaves asymptotically as the zeroth power of the number of outcomes and this behaviour is attained much faster than in the previous case.

\noindent{\bf Note 1:} Again we stress that this type of divergence is characteristic for every distribution whose probability
function is a polynomial whose order $s$ is proportional to the number of outcomes $M$.
Indeed, setting $s=s_ M$ in (\ref{eq:Variance}) we get:
\begin{equation}
\sigma^2 \mathop{=}_{M\rightarrow\infty}  \frac{1}{{s_0}^2} + O(\frac{1}{M})
\end{equation}

\begin{figure}[thb]
\hbox{
\psfig{figure=ExpVarianceApproxvsExact1.ps,width=0.48\textwidth}
\psfig{figure=ExpVarianceApproxvsExact2.ps,width=0.48\textwidth}
}
\caption{On the left: The approximated variance (short dash) versus the exact variance (long dash) of an exponential distribution as a function of the number of outcomes for different values of $\alpha=1.0,1.5,2.0$ (from top to bottom).
On the right: The difference of the approximated and the exact variance  divided by the exact variance as a function of the number of outcomes.
\label{fig:SampleVariance}}
\end{figure}

From the above discussion it is clear that the condition:
\begin{equation}
\lim_{M \to \infty} \left|\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) < \infty \label{eq:Case1_Constraint}
\end{equation}
is sufficient for the variance to exhibit divergent behaviour in the large $M$ limit.
It is however an open question to specify both \textit{\textbf{necessary and sufficient conditions}} for distributions to exhibit the behaviour specified above.
One way of doing that, is to require that the last $(M-s)$ coefficients $\left(\tilde{a}_n\right)_{n=s+1}^M$ are equal to zero.
This, from (\ref{eq:InverseRel}), yields $(M-s)$ linearly independent linear equations for the unknown probabilities, given by:
\begin{equation}
\tilde{a}_n = \sum\limits_{j=1}^M {\mathfrak A}_{j,n} \frac{{\mathbb P}_j - {\mathbb P}_0}{j} = 0
\label{eq:SysEq}
\end{equation}
for $n=s+1, s+2, \ldots, M$.
The solution to (\ref{eq:SysEq}) is given in (\ref{eq:ReducedProbabilitiesFinal1})
along with (\ref{eq:ModesProbs})
and its proof is in Appendix D.

\paragraph{Summary of Case 1} An $s$-parameter family of unnormalized distributions
is given by:
\begin{equation}
{\mathbb P}_j = {\mathbb P}_0 + \sum\limits_{q=1}^s {\mathfrak V}_{q,j} \frac{j}{q} ({\mathbb P}_q - {\mathbb P}_0)
\label{eq:ReducedProbabilitiesFinal11}
\end{equation}
for $j=s+1,\dots,M$ with the quantities ${\mathfrak V}_{q,j}$ being defined in (\ref{eq:ModesProbs}).
A probability distribution satisfying (\ref{eq:ReducedProbabilitiesFinal11}) and the constraint (\ref{eq:Case1_Constraint}) has a property that its variance $\sigma^2$, along with the variance of the sample mean $\sigma^2_{\overline{X}_N}$ (for a given $N$), exhibits an asymptotic behaviour as in (\ref{eq:Variance1}).
This implies that the large-$M$ behaviour of the variance of the sample mean can range in a continuous fashion from a quadratic divergence $(\sigma^2_{\overline{X}_N} \simeq M^2)$ to asymptotic convergence $(\sigma^2_{\overline{X}_N} \simeq M^0)$, for fixed sample size, $N$.
Furthermore, if we assume the order of the polynomial in (\ref{eq:ProbSpace}), $s$, has a power law dependence on $M$ for large $M$, of the form  $s \simeq M^{\alpha}$, with $0 \le \alpha \le 1$, then, from (\ref{eq:Variance1}), and for fixed $N$ the variance of the sample mean behaves as $\sigma^2_{\overline{X}_N} \simeq M^{2-2\alpha}$.

We did some numerical testing of our results. For particular values of $M=50$ and $s=10$ we have found the solutions (\ref{eq:ModesProbs}) by solving numerically  equations
(\ref{eq:SysEq}).
Subsequently we found the unnormalized probabilities from (\ref{eq:ReducedProbabilitiesFinal11}) by taking arbitrarily ${\mathbb P}_0=0$ and ${\mathbb P}_q=(-1)^q$ for $q=1,\dots,s$.
We plot the results in Figure \ref{fig:ProbDistrsModes}.

In addition, for a particular value of $s$ ($s=10$), and for $M=12,\dots,50$, we computed numerically the variance of all the $s$ different probability distributions in (\ref{eq:ReducedProbabilitiesFinal11}) and we plotted the results, as a function of $M$ in Figure \ref{fig:VarianceNumPlots} as well.
As we can see, the variance displays a quadratic dependence on the number of outcomes and the leading order coefficient fits in well with the result in (\ref{eq:Variance1}).

\begin{figure}[tbh]
\hbox{
\psfig{figure=ProbDistr50_10.ps,width=0.45\textwidth}
\psfig{figure=VariancevsM50_10.ps,width=0.45\textwidth}
}
\caption{On the left: All probability distributions for case 1. \label{fig:ProbDistrsModes}
Here $M=50$ and $s=10$. We plot the quantities $\left({\mathfrak V}_{q,j} \frac{j}{q}\right)_{j=s+1}^M$ for $q=1,\dots,s$ and the distributions are constructed as in (\ref{eq:ReducedProbabilitiesFinal11}).
On the right: The dependence of the variance of the probability distributions shown in the left of Figure \ref{fig:ProbDistrsModes} on the number of outcomes.
Here $s=10$ is fixed and $M=s+2,\dots,50$.
The graphs are overlayed with a polynomial of second order fitted by least square regression.
\label{fig:VarianceNumPlots}.
The leading order coefficients read $0.00603$, $0.00606$, $0.00607$, $0.00609$,  $0.00609$, $0.00608$, $0.00607$, $0.00601$, $0.00593$, and $0.00587$ compared to the theoretical value $0.00587$ (compare equation (\ref{eq:Variance})).}
\end{figure}

\subsubsection{Case 2}

Recalling section \ref{ss:Large_M}, case 2 includes all probability distributions that satisfy the following conditions:
\begin{eqnarray}
\lim_{M \to \infty} \left|\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) = \infty \label{eq:Case2_Constraint1} \\
2M\left| \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right| \left(\frac{1}{s}\right) \ll M^2 \label{eq:Case2_Constraint2}
\end{eqnarray}
For this case, looking at equation (\ref{eq:Variance1}) it is clear that only the first term is important for large $M$. Thus one can use equation (\ref{eq:case1}) in the previous section and follow the discussion of divergence there. Thus case 2 distributions will also exhibit divergence in the variance as $M \to \infty$.  This implies, as before, that, for fixed $N$, the large-$M$ behaviour of the variance of the sample mean can range in a continuous fashion from a quadratic divergence $(\sigma^2_{\overline{X}_N} \simeq M^2)$ to asymptotic convergence $(\sigma^2_{\overline{X}_N} \simeq M^0)$.

\subsubsection{Case 3}

Case 3 distributions are those that satisfy:
\begin{eqnarray}
\lim_{M \to \infty} \left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) = \infty \label{eq:Case3_Constraint1}\\
2M\left( \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) \gtrsim  M^2 \label{eq:Case3_Constraint2}
\end{eqnarray}
Unlike in previous cases, it is clear that now the second term in equation (\ref{eq:Variance1}) will contribute to the variance in the large $M$ limit. However, the variance and the variance of the sample mean (for fixed $N$) will still diverge as $M \to \infty$. But if the inequality in (\ref{eq:Case3_Constraint2}) is strict, then the second term in equation (\ref{eq:Variance1}) now controls the exact behaviour of the divergence and it will be faster divergence than the maximum possible ($M^2$) in the previous two cases. However, unlike in the previous cases, we do not analyze this behaviour any further.

\subsubsection{Case 4}

In the previous three cases, for a given $N$, the variance of the sample mean increases with the number of outcomes, which might imply that this holds for every distribution.
However this is not the case.
One can construct distributions such that the variance of the sample mean decreases with the number of outcomes.

For example, take the (unnormalized) probability distriubtion given by:
\begin{equation}
{\mathbb P}_j  = j^{M^2}\label{eq:Counter_Ex}
\end{equation}
Although this suggests a polynomial of order $M^2$, for each $M$, it can always be decomposed in a unique way into a polynomial of the form in (\ref{eq:ProbSpace}).
In this case the leading order term in the decomposition corresponds to $s=M$.
Asymptotically, we get ${\mathfrak N} = M^{M^2+1}/(M^2+1)$ and the variance reads:
\begin{equation}
\sigma^2 = M^2 \left( \frac{1+M^2}{(3+M^2)(2+M^2)^2} \right)
\label{eq:DecayingDiverg}
\end{equation}
Hence the variance is inversely proportional to the square of the number of outcomes $M$, for large $M$. Thus we have convergent behaviour for the variance.

If we now solve for the ratio $\frac{\tilde{a}_{s-1}}{\tilde{a}_s}$  in equation (\ref{eq:Variance1}) by requiring that the variance behaves like $M^{-2}$ for large $M$, then we get:
\begin{equation}
\frac{\tilde{a}_{s-1}}{\tilde{a}_s} = \frac{1}{2}\left[ \frac{\left(-M^5 - 2M^4 + 7M^2 + 16M + 12 \right)}{\left( M(M^2 + 2M + 1 \right)} \right]\label{eq:ratio}
\end{equation}

This equation combined with the fact that $s = M$ confirms that the probability distribution in (\ref{eq:Counter_Ex}) fits under case 4 as its satisfies the conditions given by:
\begin{eqnarray}
\lim_{M \to \infty}\left(\frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) = -\infty \label{eq:Case4_Constraint1} \\
2M\left( \frac{\tilde{a}_{s-1}}{\tilde{a}_s}\right) \left(\frac{1}{s}\right) \lesssim -M^2 \label{eq:Case4_Constraint2}
\end{eqnarray}

It is also possible to show that all distributions of the form ${\mathbb P}_j  = j^{M^x}$ where $x >1$, like that in equation (\ref{eq:Counter_Ex}), will have a variance that converges as $M \to \infty$. These distributions are, however, quite unusual since most probabilities are assigned to very few outcomes. It is still an open question as to how many distributions will have variances that will exhibit convergent behaviour for a large number of outcomes.

\subsection{The Percentiles}

So far we used the variance as a measure of the ``width'' of the distribution.
This may be unsatisfactory because for certain distributions the variance may not carry the relevant information - an example is the case where the variance does not exist. It is thus much more useful to compute percentiles of the distribution of the difference between the sample and the population mean, $Z := (\sum\limits_{i=1}^N X_j)/N - \mu$, and investigate the conditions under which they diverge.
Here $\mu := \left<X\right>$ is the mean of the random variable $X$.
Recall that the $p$-percentile $z_p$ of random variable $Z$ is defined as such a value of argument such that the Cumulative Distribution Function (CDF) equals $p$\%, i.e.\
\begin{equation}
P\left( Z > z_p \right) = \frac{p}{100}
\end{equation}
Here $0< p < 100$.
We denote by ${\mathfrak N} := \sum\limits_{j=0}^M P_j$ the normalization constant and we compute the CDF as follows:
\begin{eqnarray}
P\left(Z > z\right) &=& \sum\limits_{j_1,\dots,j_N=0}^M 1_{\sum\limits_{p=1}^N j_p > (z+\mu) N} \prod\limits_{p=1}^N P_{j_p}
\label{eq:CDFSampleMean}\\
&=&
\frac{1}{{\mathfrak N}^N}
\sum\limits_{n_1,\dots,n_N=0}^{M} \left(\prod\limits_{p=1}^N \frac{a_{n_p}}{n_p!}\right)
\sum\limits_{l=(z+\mu) N}^{N M}
\sum\limits_{\sum\limits_{p=1}^N j_p = l}
\left(\prod\limits_{p=1}^N j_p^{n_p} 1_{j_p \le M}\right)
\label{eq:CDFSampleMean1}\\
&=&
\frac{1}{{\mathfrak N}^N}
\sum\limits_{n_1,\dots,n_N=0}^{M}
\frac{\prod\limits_{p=1}^N a_{n_p}}{(\left|\vec{n}\right|+N)!}
\left(
(NM)^{\left|\vec{n}\right|+N}
-
(z+\mu)^{\left|\vec{n}\right|+N}
\right)
\label{eq:CDFSampleMean2}\\
&=&
\frac{1}{{\mathfrak N}^N}
\frac{1}{(N-1)!}
\sum\limits_{p=0}^{{M} N}
\frac{(N M)^{p+N} - ((z+\mu)N)^{p+N}}{(p+N) p!} a^{\otimes N}_p
\label{eq:CDFSampleMean3} \\
&=&
\frac{1}{{\mathfrak N}^N}
\sum\limits_{l=(z+\mu)N}^{M N}
\frac{l^{N-1}}{(N-1)!}
\sum\limits_{p=0}^{{M} N}
\frac{l^{p}}{p!} a^{\otimes N}_p
\label{eq:CDFSampleMean4}
\end{eqnarray}
In (\ref{eq:CDFSampleMean}) we used the definition of the CDF and
the Law of Conditional Probabilities, and we conditioned on the random variables in the sample, while we used equation (\ref{eq:ProbSpace}) in (\ref{eq:CDFSampleMean1}).
In (\ref{eq:CDFSampleMean2}) we neglected the indicator functions in the last term in parentheses, which we can do for large $M$, and we  summed over the $j$ values using  the identity (\ref{eq:SumovSimplexI}) and we summed over $l$ using  elementary number theoretic identities.
We also defined
$\left|\vec{n}\right| := \sum\limits_{p=1}^N n_p$
as the $L^1$ norm of the $\vec{n}$ vector.
In (\ref{eq:CDFSampleMean3}) we introduced the normalized $N$\textsuperscript{th} auto-convolution $a^{\otimes N}_p$ of the coefficients
$\left(a_n\right)_{n=0}^{M}$ via:
\begin{equation}
a^{\otimes N}_p := \frac{1}{C^{N-1+p}_p} \sum\limits_{\sum\limits_{q=1}^N n_q = p} \prod\limits_{q=1}^N a_{n_q}
\label{eq:NthAutoConv}
\end{equation}

\noindent{\bf Note:} Expression (\ref{eq:CDFSampleMean4}) uses an approximate identity (\ref{eq:SumovSimplexI}) and is therefore only a starting point for analyzing the scaling of the percentiles of the difference between the sample mean and the population mean.
In order to provide further insight into this problem one needs to compute the $N$\textsuperscript{th} autoconvolution
from (\ref{eq:NthAutoConv}), (\ref{eq:InverseRel}) and from the expression (\ref{eq:InverseVandermonde})
 for the inverse Vandermonde matrix. This will make it possible to uniquely classify distributions
according to the asymptotic behaviour of their percentiles. We will accomplish that goal in future work.

\section{Conclusions}

All finite discrete probability distributions may be represented by polynomials in the possible outcomes of their random variables, by inverting the relevant VanderMonde matrix.
We used this fact to examine the convergence of certain sample properties of finite discrete distributions.
In particular, we considered families of probability distributions that had different numbers of outcomes ($M$) but were otherwise similar.
Such families of distributions can be represented as sequences whose terms are indexed by $M$. Their variances can also be described as sequences in $M$.

In this paper we considered the special case of integer outcomes $\lbrace 0, 1, \ldots M \rbrace$.
We calculated an  expression for the variances of such families of distributions using large $M$ approximations, and then examined their behaviour as $M \to \infty$.
We found conditions for \textit{divergence} of the variance with increasing $M$ and gave some examples. We discussed some necessary and sufficient conditions for distributions to exhibit specific types of divergence. We also gave examples of families of distributions that \textit{did not} satisfy these conditions for divergence and whose variances actually \textit{converged} as $M \to \infty$. Finally, we have provided an expression for the Cumulative Distribution Function (CDF) of the difference between the sample mean and the population mean. This expression can be used to analyze the asymptotic behaviour of the percentiles of the difference between the sample mean and the population mean as a function of the number of outcomes $M$.

In obtaining the results, we derived many mathematical identities and they are listed in the appendices. We also derived an expression for the inverse of the Vandermonde matrix used in this paper, which was previously unknown. We believe these formulae could be useful in many different fields.

We hope, in future work, to generalize our results to arbitrary probability distributions, to quantify better what distributions these results apply to, as well as examine the behaviour of the variances of these distributions, for an increasing number of outcomes, relative to the mean.

In conclusion, this work should be relevant to situations where one must sample the distributions of finite discrete random variables. It highlights how adding new outcomes to such random variables (without any change in the overall form of the probability distribution) may still affect how the statistics converge, with increasing sample size, to the population properties. More importantly, we have proven that for certain finite discrete probability distributions  when $N$ is kept \textit{fixed}, the variance of the sample mean actually \textit{diverges} as $M \to \infty$. We termed this result as the \textit{``Law of Many Outcomes''}, or alternatively, the \textit{``Reverse of the Law of Large Numbers''}.

redpajama_set_name: RedPajamaArXiv


text:
Here's what Courtney Love thinks of Miley Cyrus' recent Hole cover
Rachael Dowd Published: December 8, 2020
[Photos via Miley Cyrus/YouTube, Courtney Love/Manfred Werner via Wikimedia Commons]
Last week, Miley Cyrus celebrated the release of her new rock-inspired album Plastic Hearts with an emotional cover of Hole's "Doll Parts."
Now, Courtney Love has shared what she thinks of Cyrus' recent cover performance.
Read more: See My Chemical Romance bring back the 'Revenge' era in a new teaser
2020 has been a big year for Miley Cyrus. On Nov. 27, she finally released her highly-anticipated new album Plastic Hearts. The rock-inspired LP includes collabs with Billy Idol, Dua Lipa, Stevie Nicks and even Joan Jett.
Prior to the album's release, Cyrus unleashed various rock covers throughout 2020. Along with covering the Cranberries' classic "Zombie," she also debuted a rendition of Blondie's "Heart of Glass."
Then, last week, Cyrus gave her first live performance since the start of the coronavirus pandemic on The Howard Stern Show. Along with talking about her new album, she gave a powerful cover of Hole's "Doll Parts."
Read more: This villain's return may have just confirmed that 'Spider-Man 3' crossover
The song originally appears on Hole's 1994 album Live Through This which also includes the hits "Miss World" and "Violet." For the cover, Cyrus truly channeled her inner Courtney Love, belting out the song's iconic lyrics while giving it her own rock 'n' roll spin.
At the time, Cyrus told Howard Stern that Love had not heard the cover. In fact, she had never even performed the song live before.
"No one's ever heard me do it, I'd never heard me do it until yesterday," she said.
Read more: See why fans think new music from twenty one pilots is arriving soon
Now, Courtney Love has shared her thoughts on Cyrus' recent Hole cover. On Monday, she shared Cyrus' performance on Instagram and called her cover of "Doll Parts" a "sweet version."
"@mileycyrus Doing a sweet version of " doll parts " on @sternshow (hi Howard)
A post shared by Courtney Love Cobain (@courtneylove)
In a separate post on Tuesday, Love also revealed that she is "touched" by Cyrus' take on the Hole classic. The Instagram post includes a throwback video of Kurt Cobain telling an audience that Love was "the best fuck in the world." Then, Nirvana head straight into their performance of "Smells Like Teen Spirit."
Read more: Trapt only have one place left to post after Twitter finally suspends them
Love also takes the time to explain just how "Doll Parts" was created. She wrote the song in 20 minutes while in a bathroom in Massachusetts. After she ran out of paper, Love wrote most of the song's lyrics on her arm in a Sharpie. As it turns out, the classic song is about Kurt Cobain.
"The song 'Doll Parts' is a homage I wrote in 20 minutes in a girl named Joyce's bathroom in Cambridge, Massachusetts," Love says. "I had to write most of the lyrics on my arm in sharpie as I ran out of paper. People were pounding on the door as I wrote it. It was played for the first time about an hour later, at the Virgin megastore in Boston. It was about a boy, whose band had just left town, who I'd been sleeping with, who I heard was sleeping with 2 other girls, it was my way of saying 'You're a fucking idiot if you don't choose ME, and here is all the desire and fury and love that I feel for you.' Good songs don't always come in 20 minutes but the force was strong and that one did. Anyway I married that guy."
Read more: Hear Every Time I Die return with their first new songs in four years
During a recent interview with Apple Music's Zane Lowe, Cyrus revealed that she is already working on her next record. As well, she previously confirmed that she has a Metallica covers album in the works. Cyrus credits the metal band as one of the artists who influenced the sound on Plastic Hearts.
As for Hole, Love recently teased that a Hole reunion could happen in the future. Last month, she told NME that she would love to reunite the band again. In fact, she's even taking guitar lessons in preparation for the event.
"Before I came back I actually had Melissa [Schemel] and Patty [Auf der Maur] come with our tech to this old-world rehearsal studio," she said. "We had a good session, but it takes a bit of time to get back into the rhythm of it all. It's something I'd love to do and been taking guitar lessons over Zoom during lockdown and I'm writing again so we'll see!"
Read more: Here's how BMTH inspired YUNGBLUD long before their collab on "Obey"
Along with keeping her fingers crossed for a Hole reunion, Love is also working on new solo music.
"It's early days and I've recorded several new songs whilst I've been in London," Love said. "I've also taken this time to enjoy rediscovering really good songwriters like Aimee Mann who's from LA but went to Juilliard and is a genius and overlooked. Frances sends me playlists, which are fantastic and full of artists that I hadn't come across before too."
Hole's "Doll Parts" is available to stream below.
What are your reactions to Miley Cyrus' Hole cover? Do you like it more than the original? Let us know in the comments below.
Filed Under: courtney love, courtney love hole, doll parts, hole, hole cover, howard stern, live through this, miley cyrus, miley cyrus courtney love, miley cyrus cover, miley cyrus hole, miley cyrus hole cover, miley cyrus hole performance, plastic hearts, the howard stern show


Robert is signed up for all of his Iditarod qualifiers this year. It is an exciting time at Team Ineka. We have a core team that is roaring and ready to go and several up-and-coming pups that will add depth to the team.


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When your group faces a pivotal moment, a future search is a big, bold way to chart a path. Consensus has extensive experience with future search conferences, as well as with more traditional strategic planning processes.
A future search gets people reflecting the whole system in the room for about two days. There are no speeches or experts other than the participants. The facilitators guide the process, but participants create the content and decide what it means. The result of a future search is an action plan built upon common ground.
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text:
Progress slow on virus relief bill as negotiations continue
Progress is coming slowly as lawmakers craft another major response to the coronavirus pandemic
By ANDREW TAYLOR Associated Press
White House Chief of Staff Mark Meadows speaks during a press briefing in the James Brady Press Briefing Room at the White House, Friday, July 31, 2020, in Washington. (AP Photo/Alex Brandon)
WASHINGTON -- Negotiators on a huge coronavirus relief bill reported slight progress after talks resumed Monday afternoon in the Capitol, with issues like food for the poor and aid to schools struggling to reopen safely assuming a higher profile in the talks.
Multiple obstacles remain, including an impasse so far on extending a $600-per-week pandemic jobless benefit, funding for the Postal Service, and aid to renters facing eviction. Democratic negotiators spoke of progress at almost the very moment that top Senate Republican Mitch McConnell was slamming them for taking a hard line in the talks.
All sides predict a long slog ahead. Several more days of talks are expected, if not more, as lawmakers seek to deliver what will likely be the final legislative response to the pandemic before the November election.
"We are really getting an understanding of each side's position. And we're making some progress on certain issues moving closer together," said Senate Minority Leader Chuck Schumer, D-N.Y. "There are a lot of issues that are still outstanding. But I think there is a desire to get something done as soon as we can."
After the meeting, Pelosi told her Democratic colleagues on a call that she's hopeful a deal could be reached this week, but doesn't know if it's possible, according to a Democratic aide who was granted anonymity to describe the private discussion.
Treasury Secretary Steven Mnuchin, a lead negotiator for President Donald Trump, said afterward that "we continue to make a little bit of progress" and that the administration is not insistent on a small-bore approach centered on extending the supplemental unemployment benefit and leaving other items for later. A GOP move to advance a slimmed-down relief package has been a recent point of conflict, with Democrats insisting there must be a comprehensive deal.
"We're open to a bigger package if we can reach an agreement," Mnuchin said.
On the Senate floor, McConnell, R-Ky., re-upped his complaint that Democrats are taking too tough a line. McConnell is not a direct participant in the talks but is likely to be an important force in closing out any potential agreement.
"The Speaker of the House and the Democratic Leader are continuing to say 'our way or the highway' with a massive wish list for left-wing lobbyists that was slapped together a few weeks ago called a coronavirus bill," McConnell said.
Speaking to reporters after the two-hour session, Democratic negotiators pressed the case for additional food aid, funding for the Postal Service, and the $600-per-week jobless benefit that lapsed last week. The benefit has helped prop up the economy and family budgets as the coronavirus has wrought havoc.
The White House is seeking opportunities to boost Trump, like providing another round of $1,200 stimulus payments and extending the supplemental jobless benefit and partial eviction ban. Pelosi appears intent on an agreement as well, but she's made it clear she needs big money for state and local governments, unemployment benefits and food aid.
"It was productive, we're moving down the track. We still have our differences, we are trying to have a clearer understanding of what the needs are, and the needs are that millions of children in our country are food insecure," Pelosi said. "Millions of people in our country are concerned about being evicted. Tens of millions of people are on unemployment insurance."
Most members of the Democratic-controlled House have left Washington and won't return until there is an agreement to vote on, but the GOP-held Senate is trapped in the capital. Several senators said Monday evening that they expect the process to drag into next week, when they are scheduled to leave for the August recess.
"I can't see how we can go home and tell people we've failed, so I think that's going to be a lot of pressure on everybody to come up with something," said Sen. John Cornyn, R-Texas. "It really is a matter of will. It's not a matter of substance at this point. This is just a painful period between people finally deciding OK, we want a deal, and then what that deal will ultimately look like."
Areas of agreement already include the $1,200 direct payment and changes to the Paycheck Protection Program to permit especially hard-hit businesses to obtain another loan under generous forgiveness terms.
But the terms and structure of the unemployment benefit have remained a sticking point, along with the money for state and local governments. Democrats want almost $1 trillion for localities grappling with pandemic-related revenue losses.
Speaking to reporters Monday evening, Trump made clear the two sides were still far apart on that issue.
"They want to bail out cities and states that have been in trouble for years of bad management," Trump said. "In all cases Democrat-run cities, and we don't think that's fair."
On unemployment, Pelosi said she'd consider reducing the $600 benefit for people in states with lower unemployment rates. Republicans want to cut the benefit to encourage beneficiaries to return to work and say it is bad policy since it pays many jobless people more money than they made at their previous jobs.
Another sticking point is that Republicans want to give more school aid to systems that are restarting with in-school learning, even as Dr. Deborah Birx, Trump's top coronavirus adviser, cautioned that schools in areas with spikes in cases should delay reopening.
Pelosi said Monday's talks featured extensive discussion on reopening schools. Schumer said the discussion got into the details on the dollar figures involving school aid and food aid.
The House passed a $3.5 trillion measure in May, but Republicans controlling the Senate have demanded a slower approach, saying it was necessary to take a "pause" before passing additional legislation. Since they announced that strategy, however, coronavirus caseloads have spiked and the economy has absorbed an enormous blow.
Associated Press writer Mary Clare Jalonick contributed to this report.
NOTIFIED: Jan. 18, 2021
Newborn giraffe dies when mother accidentally steps on its neck after giving birth


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Wolchik, S. A., West, S. G., Sandler, I. N., Tein, J.-Y., Coatsworth, D., Lengua, L.,...Griffin, W. A. (2000). An experimental evaluation of theory-based mother and mother-child programs for children of divorce. Journal of Consulting and Clinical Psychology, 68, 843856.
Kidpsych is an excellent website for young children (https://www.kidpsych.org).
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How do you stop a murder that's already happened? The Hardcastle family is hosting a masquerade at their home, and their daughter Evelyn Hardcastle will die. She will die everyday until Aiden Bishop is able identify her killer and break the cycle. But every time the day begins again, Aiden wakes up each day in a different body as one of the guests. Aiden's only escape is to solve Evelyn Hardcastle's murder and conquer the shadows of an enemy he struggles to even comprehend. But nothing and no one are quite what they seem. Deeply atmospheric and ingeniously plotted, The 7½ Deaths of Evelyn Hardcastle is a highly original debut that's an Agatha Christie mystery in a Groundhog Day Loop, with a bit of Quantum Leap to it. Perfect for fans of Kate Atkinson and Claire North.
Great review for a great book. I read it earlier in the year. I do wonder why it is called 7 deaths in UK and 7 1/2 in other countries.
I wondered that too. The covers are also exactly alike too. I'm still thinking about the 1/2 and what it could stand for.
This book sounds fabulous!!! It sounds like one of those murder mystery party games in book form!
It is in a way except the same person gets to play different characters. It's so good!
This sounds tasty - I wants it!
Such a good book. You should totally read it. Highly recommend it.
Awesome review! This sounds like a lovely book.
It was awesome. Best read of the year so far.


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Rock's Most Hated Records
Michael Gallucci
First off, let's point out the difference between a list of Rock's Worst Records and Rock's Most Hated Records. There's crossover between them, for sure, as you'll see in our above list of Rock's Most Hated Records, but there's a special place for records that aren't just bad but totally despised too.
Terrible records can be ignored. They often disappear after their short shelf lives have expired. But hated records? They can live on forever. Younger music fans may not know a single thing about Paul Anka's gloopy '70s ballad "(You're) Having My Baby," but they can probably tell you that Starship's "We Built This City" from the following decade is all kinds of awful. And that they really, really hate it.
Same goes for Neil Young's Trans album. The singer-songwriter has shifted gears dozens of times over his five-decade career, recording all-in, speaker-shredding rock 'n' roll songs and spare, acoustic country ballads -- often on the same album. He's made LPs dedicated to rockabilly, big-band, R&B and grunge music. Many of his albums are adored; several are ignored. Trans, an exploration into synthesizer-enhanced sounds, is downright hated because of its whiplash-inducing shift.
That's the sort of thing you'll find on our list of Rock's Most Hated Records. And we made it easy for you by combining albums and singles in one place, so you don't have to look too far for full-length dives into ego-inflating excess (remember that album Gregg Allman and Cher made together?) and four-minute slices of head-shaking confusion (hello, "Mr. Roboto"!). You many not break into Hulk-sized fits of anger while revisiting these records, but we advise a calming drink beforehand all the same.
Next: More Than 100 Times Rock's Biggest Bands Let Us Down
Source: Rock's Most Hated Records
Categories: Lists, Original Features