In the midst of an election year, tied to a flurry of headlines about high prices for gasoline, there have been a lot of questions about the value and significance of government subsidies for the oil and gas industry. On a global scale, a closer look at the role of government subsidies reveals that they can be very much to blame for high oil prices.
Call them subsidies, or tax breaks, or credits, the question is not about semantics, it's about evaluating whether one of the most profitable industries in the world needs a government benefit. But if you look at this issue worldwide, the problem is much greater than the drama surrounding America's $4 billion payout.
The International Energy Agency estimates that in 2010, governments worldwide spent $409 billion subsidizing fossil fuels. In 2009, that number was $300 billion.
The IEA says that the single act of eliminating global fossil fuel subsidies would get the world nearly halfway to targeted greenhouse gas emissions reductions by 2020. In one fell swoop, the move would slash growing energy demand by 4.1%, and reduce oil demand by 3.7 million barrels per day. For comparison, that is equivalent to the rate of Canada's oil production in 2010.
Though I'd love to see a worldwide elimination of subsidies and the ensuing drop in greenhouse gas emissions, I'm not holding my breath. Foreign governments not only heavily subsidize the oil and gas industry, they frequently own the oil companies outright, and use the proceeds from operations to wield power and influence over their citizenry.
The large subsidies by these governments lead to massive overconsumption. Surging demand drives up the price of oil, and though we can debate the peak oil theory until the cows come home, the commodity is proving to be more elusive, and more expensive to produce, than ever before.
This same high price of oil, however, is also what ends up refilling government coffers, allowing administrations to maintain subsidies and fund other, non-energy-related activities. Saudi Arabia, for example, needs the price of oil to sit above $90 in order to fund increased social spending at a time when unrest has shaken much of the Arab world. It is in the Saudi government's best interest to keep oil prices up and uprisings down, and OPEC is sympathetic to this need.
That is the problem with state-owned companies. Unlike independent operators, they are a means to many different ends, which often exclude efficient exploration and production operations, and a commitment to reserve replacement, the very things that increase supply and drive down the price of oil.
State-owned entities dominate the energy industry, making up close to a 67% share in emerging markets as of June 2011. As of 2010, national oil companies, or NOCs, made up the top 10 of the largest oil and gas companies by reserves. ExxonMobil, in case you were wondering, is ranked 11th.
An incredible 73% of the world's reserves belong to state-owned entities such as the 10 listed here. And these 10 are just the beginning. Consider that many of the biggest "public companies" familiar to energy investors, PetroChina, Sinopec, Petrobras, and Statoil, are really NOCs.
By taking a closer look at a government and its NOC, we can develop a better understanding of the effect of the relationship on production and demand.
This small South American country is a perfect example of what can go right, and oh so very wrong, with a state-owned oil company. Venezuela went through the process of nationalizing its oil company in the 1970s. For a while, things were going well. With minimal government interference, Petroleos de Venezuela was one of the better operating NOCs, producing as much oil as Mexico's state-owned Pemex with one-third of the staff.
In the 1990s, the oil company decided to push the envelope and grow production to 6.5 million barrels per day. By increasing its own production, and encouraging foreign companies to develop its marginal fields (taking note of their superior technology), PDVSA set the stage for a successful future.
That all started to fall apart around 1998. PDVSA asked for government permission to increase investments in production. The government said no, and promptly slashed the oil company's budget to garner favor for an upcoming election. Production had reached 2.9 million barrels per day, but promptly fell and has not returned to that level since.
When Hugo Chavez came to power a year later, he cut the company's budget even further. Then he accused management of shady underhandedness, questioned its foreign expansion plans, and appointed top-level cronies to do his bidding. And then there was a strike!
Management and half of PDVSA's 40,000 employees stopped work for two months, devastating operations that required constant oversight. In a move that likely came up in a conversation that began "How can we make things worse?" Chavez fired all of the striking employees and filled the positions with a slew of incompetents.
Today, PDSVA is known not for efficiency, but for deteriorating operations, secrecy, an intimate relationship with the government, and a production level that is hundreds of thousands of barrels short of what it was 14 years ago.
Now that we have established the negative effect of an NOC on the supply side, let's flip our greasy coin and look at demand. Herein lies a catastrophe.
As is common in many countries with state-owned oil companies, the Venezuelan government subsidizes gasoline -- at a rate of 90%. Ninety percent! The immediate problem is that it drives consumption through the roof. Though Chavez has talked about reducing consumption, there is an election coming up in October. So Hummers and old American muscle cars from the 1970s roll through the streets mocking fuel efficiency, as drivers fill their tanks with gas that costs $0.07 to $0.18 cents a gallon.
Not all NOCs are as terribly operated as PDVSA. For example, Saudi Aramco is often touted as being incredibly efficient for an NOC. Regardless, Saudi Arabia also has a ridiculous gasoline subsidy policy. The country spends $13.3 billion a year on gas subsidies and $30 billion overall on fossil fuel subsidies. The price of gas in Saudi Arabia is about $0.48 per gallon. Mostly because of subsidies, oil producers in the Middle East will consume 1 million more barrels per day in 2015 than in 2010. If you think the oil market is tight now, just wait.
More importantly, as is clear in the case of Venezuela, well-run or not, today's efficiency is always subject to tomorrow's tyranny.
Government commitment to subsidy programs quite often keeps production down and drives up consumption. This two-headed beast wreaks havoc on worldwide energy prices, and while it is absolutely crucial to develop alternative energy sources, an equal emphasis should be placed on decreasing demand and increasing energy conservation. It is the best way to reduce American dependency on foreign oil and pry ourselves free from the tangled mess of the global energy picture.
This post originally appeared at The Motley Fool.


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redpajama_set_name: RedPajamaGithub


The more you excite your co-workers and keep them engaged, the more likely you are to win. And, the more fun you will have! Did we mention all the bragging rights you have throughout the year?! Here are some resources to help you!
Click here to download our weekly cheerleading guide complete with starter templates for eBlasts and posters for each phase of the event!
Want to maximize your team's level of participation and engagement? Consider these tips for winning!


text:
Aug 23, 2019 - Technology
Inside Honda's safety labs
Joann Muller, author of Axios What's Next
Honda engineer takes measurements ahead of a simulated crash test. Photo: Honda
In the ever-evolving world of car safety where billions are being spent on the most advanced lidar systems and deep learning algorithms, some companies are also focusing on how to improve occupant protection.
The big picture: Self-driving cars are a long way off, and newly available crash avoidance technologies like automatic emergency braking can't prevent all collisions.
Until cars are smart enough to avoid crashing entirely, automakers will keep working on ways to reduce the potential for injuries when they inevitably do.
There are many aspects of safety under the microscope, including mulling how to compensate for gender differences in crash tests, developing better seats for women and redesigning the air bag.
Driving the news: Honda is introducing a next-generation airbag designed like a giant catcher's mitt. It not only cushions the front passenger's head but also reduces twisting neck injuries that often lead to brain trauma.
This week, I got a rare peek behind the curtain at Honda's safety labs on the campus of the company's huge research and development facility in Raymond, Ohio.
Besides the new airbag, Honda showed journalists its work on improving pedestrian safety and its use of crumple zones and other structural engineering tricks to divert crash energy away from the occupant cabin.
From behind plexiglass we watched as technicians crashed a bright yellow 2019 Honda Civic coupe into a 90-metric-ton block of concrete.
Seeing a crash test in person is pretty horrifying. (Watch how high-speed cameras capture the impact from multiple angles. You can actually see the energy flow through the vehicle.)
The car was destroyed, but the crash test dummy inside fared pretty well.
"Any time something deforms, breaks or bends, it's a beautiful thing," because it means energy was directed away from the cabin, said Brian Bautsch, manager of crash safety at Honda R&D Americas.
How it works: Before Honda crashes its vehicles into a concrete barrier, it spends a lot of time in a simulation lab.
On a giant floor-to-ceiling screen, engineers simulate a crash again and again, examining every possible angle in slow motion, including the impact on components deep inside the vehicle.
Details: Honda's new passenger airbag is designed to reduce the potential for injuries that can occur in the real world, where crashes rarely happen directly head-on. Per Honda...
When a crash occurs at an angle, the lateral forces can cause an occupant's head to rotate severely or slide off the airbag, increasing the chance of serious brain injury.
Instead, its new airbag has multiple chambers that sandwich and cradle the passenger's head, mitigating the potential for injury. (Watch video here.)
The airbag, which will debut in 2020, was co-developed with Autoliv, Honda's new airbag supplier.
Flashback: Honda recalled 12.9 million Honda and Acura vehicles in the U.S. to replace faulty Takata airbag inflators in recent years.
The Takata airbag recall hit Honda the hardest, but has affected virtually all major automakers.
With 56 million defective air bags being recalled, it was the biggest auto safety recall in U.S. history, according to the NHTSA. (Check your car's status.)

redpajama_set_name: RedPajamaCommonCrawl


Scott Pruitt would like you to relax about the EPA cuts.
Pruitt said he opposes some of Trump's ideas for his agency—specifically Trump's idea to cut a number of popular grant programs for states. Pruitt said he'd like to preserve Brownfields grants, which help states clean up and reuse highly polluted industrial sites. He also cited federal grant programs that rebuild aging drinking water infrastructure, like the $100 million EPA grant that Flint, Michigan, received last year to improve its corroded pipes. "I want you to know that with the White House and also with Congress, I am communicating a message that the Brownfields Program, the Superfund program, water infrastructure … are essential to protect," Pruitt said.
Afterward, an EPA spokesperson declined to say if Pruitt opposed any other parts of Trump's reported slash-and-burn budget for the EPA, such as eliminating the Office of Environmental Justice, reducing the workforce from 15,000 to about 12,000, and zeroing out funds for climate change initiatives and native Alaskan villages.


text:
"One Son Resists" — fiction by Peter Anderson
Posted by Nick Sarno
Peter Anderson read the following story at The Parlor's 2nd Annual Emerging Writer's Festival. Look for the podcast soon at theparlorreads.com. In the meantime, check out Mr. Anderson's site for more of his work.
One Son Resists
Had Harold Lee Avery not died suddenly in April 1937, his weakened heart falling victim to too many years of rich meals, the knowledge of how his only son ended up after graduating from Elmhurst Prep would surely have killed him anyway. This would, one day, become the oft-repeated and bitterly heartfelt claim of Millicent Avery, wife of Harold Lee and mother of young Harold.
Such statements of disapproval and disappointment had become so common, so routine and familiar, that Harold barely heard them any longer. This afternoon she was voicing some similar lament, but Harold barely listened as the two sat far apart, across the broad distance of the Packard's rear seat, looking away from each other at differing street scenes outside. He didn't hear her specific words, but soberly grasped her meaning, which rarely varied.
"Well, never mind any of that now," Millicent concluded. "We have a fine evening ahead of us."
He merely grunted in reply while continuing to stare out the window, as the Packard made its smooth and leisurely pace along the tree-shaded and broad-lawned avenues of Magnolia Park, the finest neighborhood in Atlanta.
Millicent Avery always said she wanted only the best for her son, her only child, perhaps even more so than her husband had. After the latter's sudden death Millicent became the diligent caretaker of young Harold's future, intent on fulfilling all of the arrangements Harold Lee had made for his son. Four years at William & Mary and two years at Yale Law—the father's alma maters—and then an apprenticeship at Avery Fordham & Stearns, Attorneys at Law. A stellar early career as an attorney would be followed by a judgeship, both of which Harold Lee had himself attained, and then the United States Senate, a position to which Harold Lee had always aspired but ruefully never achieved.
Harold Lee—his middle name willfully given in tribute to the Confederate general, in proud defiance of Reconstruction—had been a quietly powerful and imperious man, scion of one of Atlanta's oldest families. For decades he methodically controlled the lives of scores of people around him, family and friends and colleagues alike, moving them around like pawns on a chess board, their every movement known to him in advance precisely as he had plotted out, each one serving his various needs and desires.
He had every reason to believe it would be no different with Harold, his own son. For Harold Lee saw in Harold the perpetuation of the Avery family destiny, one of greatness and of relentless ascension. As did Millicent.
The car rolled to a stop, driveway gravel crunching beneath the whitewalled tires. The right-side door promptly opened and Millicent briskly but elegantly stepped out. Harold heard a muffled but clearly subservient greeting, and saw a white-gloved hand resting politely atop the door frame, before he turned back to his own window.
"Harold?" Millicent said, bowing her head back inside the car. She spoke with a friendly air with an undercurrent which he immediately recognized as displeasure. "You are coming, aren't you, dear?"
Harold exhaled with irritation, then slid across the seat and climbed out. As Millicent gazed at her son he looked away, taking in the massive expanse of house before him. Two and a half stories with wings on each end and a broad veranda in front, with classic Atlanta red brick and freshly-painted white shutters, the surrounding grounds lushly and impeccably landscaped. Magnificent, even Harold had to admit to himself.
"Don't the Hutchinsons have a lovely home, dear?"
"Home," Harold replied. "Or house."
"Yes, a lovely home," Millicent repeated. She leaned in close, as if keeping a secret from the butler. "Still, it's not quite as nice as ours, now, is it?"
"Oh, it seems quite a bit like our house," he said, taking in the long line of large, spotless automobiles—Packards and Cadillacs and Lincolns—parked at the edge of the circular drive. "Quite a bit."
Despite Harold Lee Avery's diligent efforts, none of his beliefs or ambitions were ever successfully instilled in the young man. Harold was eased into elite private schools, where despite middling grades he was continually promoted upward as a favor to his father and his charitable largesse, and introduced into the loftiest and best social circles. But despite his father's well-intentioned prodding, the young man showed little interest in advancing academically or gaining the right friends.
Harold instead kept to himself and was rarely away from the family house, where he was usually found with his nose in a book—in his father's estimation, nothing but middling, populist tripe such as Dickens or Wilkie Collins—or otherwise idling with a menagerie of mongrel animals which he kept in an unused wing of the coach house.
It was just such simple and unambitious pursuits—reading popular literature, tending to cats and dogs and rabbits and a belligerent goose, or just being alone—which occupied Harold's thoughts as he hesitated before the Wendell Hutchinson house. Pursuits which he longed to returned to, and might have done so were it not for his mother's insistent pulling-forward.
Very well. He would do this for her, and for his father, and be done with it. Tonight, for a few more hours, he would observe and respect the formalities of refined society, the norms which his parents so revered. He would do so again, likely for the very last time. He gathered himself and followed his mother through the house's arched front doorway, their shoes tap-tapping across the marble floor, to the receiving line which awaited on the opposite side of the portrait-lined foyer.
"Harold, how good to see you," Mrs. Hutchinson said, with only the slightest suggestion of warmth. To Harold she seemed matronly, almost an old woman, even though she was the mother of his classmate, Emily, and couldn't have been much older than his own mother.
He nodded, purely out of polite habit, without a word, and shook her pallid hand.
"I'm so sorry to hear about your father," the woman continued. "That was such a shock, a terrible shock. How is your mother holding up, dear?"
It occurred to Harold that Mrs. Hutchinson could have asked his mother—then already past the receiving line and moving toward the ballroom—that very question herself, just seconds earlier. But he remembered that the woman was a notorious gossip, one presumably more interested in what other people thought of someone else rather than how that person actually was.
"She's holding up well, thank you," Harold replied, hurriedly, having quickly realized that his silent musing had caused a delayed response which might be construed as rudeness. Although being rude didn't particularly bother him, he wanted to be finished with this evening, and the entire weekend, with a minimum of repercussions from his mother. As tempting as it might be, he thought it best not to offend her society friends.
"That's good. It's wonderful that she has a fine son like you to look after her."
He nodded at her, gesturing his farewell in silence, and continued toward the ballroom, in his mother's blustering wake.
Oh, she's holding up just fine, he had wanted to reply. She's overcoming her grief by keeping Harold Lee alive through me, never mind what I might want for myself. But he refrained from saying so, as satisfying as it might have felt.
He moved with his mother through the milling throng, keeping just behind her, remaining on the distant fringes of her chattering conversations. There, but not entirely so. Everyone who struck up conversation with Harold began with some variant on the same words—"Sorry to hear about your father"—followed by subdued commiserations about Harold Lee's fine character, inquiries about his mother's well-being, and offers of whatever assistance might be needed, before Harold's subdued responses would send them along to another circle and a more inviting conversation.
After an hour had passed, Harold realized that no one—not one single person—had asked about him, how he was feeling, what he had been doing lately, what his plans were for after graduation. A few more well-wishers came and went as Harold went through the motions of polite discourse while darkly considering his newfound insight.
Do any of these people really know me? he thought, his frustration rising. Am I anything to them other than the son of Harold Lee Avery?
The answer, suddenly so obvious to him, came painfully, and with gritted teeth he readied himself to break free, to escape the crowd and its superficialities. He was about to set aside his empty punch cup and move toward the exit when he felt a finger gently tap his shoulder. He turned to see the smiling face of Emily Hutchinson and, just beyond at the edge of his vision, his mother looking on, beaming with approval.
"Ask a girl to dance?" Emily said, grinning.
"I don't think—"
"Of course he'll dance, Emily dear," his mother interjected, projecting her voice into the midst of their conversation.
"Come on, Harold, it's the waltz!" Emily gently prodded.
He sighed and nodded, allowing Emily to lead him by the hand to the dance floor. He would submit to this formality, not for Emily—who had always been kind to him—and certainly not for his mother, but to fulfill his obligation. To dance, and particularly to waltz, was such a deeply ingrained custom in such society that his refusal might bring about a minor scandal. Or perhaps he would do this partly for Emily, after all. Besides being kind to him, she was quite fetching—Easy on the eyes, his father might have said in one of his rare relaxed moments—and particularly so this evening. He might easily have been attracted to her, but had always sensed some emotional distance between them.
They began their waltz, stuttering to a start as Harold remembered only at the last moment to take the lead. He had never learn to dance properly. The activity seemed pointless to him—rigid couples flitting about in perfect but soulless circles, making only the lightest and emptiest of conversation—and his disinterest prevented him from mastering even the most basic steps. Which, when added to his lack of physical grace, always resulted in a generally sorry performance on the polished parquet floors of Atlanta's great houses, on those few occasions he had been enticed onto them. But he would give it one last chance. If nothing else, he would show once and for all that he didn't belong here.
As they moved about the floor—Emily lithe but Harold hopelessly stiff—she chatted, slowly drawing him out, showing interest in him which, while mild, was still greater than that shown by all of the other partygoers.
"We haven't decided yet where we're summering," she said, glancing up at Harold who stood a full foot taller than her. "Either Asheville or Ocean City. How about you?"
"Well, I don't know yet, but it surely won't be any place like that," he replied while further thinking No, not Asheville or Ocean City, and certainly not 'summering.'
"At any rate, at least you know where you'll be in the fall." Their feet shuffled in clumsy unison, one-two-three, one-two-three. "William & Mary. I'm so jealous of you. My father couldn't get me in there, no matter how hard he tried."
"Yes, my father certainly had influence," Harold grimly replied.
"Why, Harold Avery, you don't seem the least bit excited about any of this," she playfully admonished. "About graduation, the summer, college in the fall, and everything after that."
It was true. He wasn't excited about any of it, and he would no longer pretend otherwise. Within a few minutes he would leave the Hutchinsons' party, far earlier than his mother would have preferred, offering curt goodbyes to just a few people before moving swiftly through the arched doorway, down the spotless front steps and into the street for the long walk home, to his books and his animals and his solitude.
"No, I'm not really excited about any of that," Harold said to Emily as the waltz ended, just before his goodbye. "But I have something else in mind."
The graduation ceremony at Elmhurst Prep was every bit as grand as expected, the sons and daughters of the city's elite impeccably dressed and lavishly adorned, quietly glowing in anticipation of the glorious futures ahead of them. Harold, by contrast, sat stiffly throughout the ceremony, his face serious and his thoughts clearly elsewhere.
Millicent accompanied Harold home from the ceremony, the two of them stepping out of the Packard while the colored chauffeur held open the door. She had just opened her mouth to speak—brief, perfunctory congratulations prefacing another reaffirmation of the grand scheme for his future—when he interrupted her line of thought before it could once again be put into words.
"I'm not going to William & Mary, Mother," he said quietly before he had even removed the mortarboard from his head. He didn't look at her as he spoke, instead walking away from her, past the paneled oak front door and the stiffly silent butler.
She didn't follow him immediately, instead stopping in silent shock just outside the doorway. Her jaw dropped slightly before she composed herself and hurried after him.
"Not going? What do you mean, not going?"
"Just what I said," he replied, stopping and turning but barely returning her piercing gaze. "Not going."
"You most certainly are going, young man. It's all arranged."
"I don't care if it's arranged," he said, finally looking directly into her eyes. "I'm not going."
"And what, may I ask, are you planning to do?"
He hardened his stare, hoping it would strengthen his stance. "I'm joining the CCC."
"The CCC?" she responded, a sour look yellowing her face, her disgust readily apparent. "The Civilian…what's it called? Civilian Conservation Corps? That socialist outfit?"
"It's not socialist, Mother. It's a government program. The U.S. government, remember? The one in Washington, the one that Father wanted me to end up in?"
"Your father wanted you in Congress, in the Senate. Not digging ditches in Indiana or some such place."
"It's not just digging ditches, Mother," he said. His voice was quiet at first but steadily rose as he continued. "It's building roads, planting trees, stringing power lines. Most of all, the Corps is about giving of yourself, bettering mankind. It's President Roosevelt's project, and I admire him for it. He's a great man."
"Roosevelt, a great man," she spat, livid. "It's a good thing your dear father isn't alive to hear you talk like that. As it is, he's probably churning up the inside of his grave."
"Oh, Mother, you're so dramatic sometimes."
"Harold, son, listen to me," she said, calming herself, her tone softening. "I appreciate that you want to better mankind, as you put it. But the CCC isn't for people like us. It's for the poor, the unemployed, the veterans who have nowhere else to go. You're none of those things."
He said nothing, looking away.
"Your father, and your grandfather and great-grandfather before him, made sure of that," she continued. "They all saw to it that you would have every opportunity in life to succeed. And now the opportunity is right there, waiting for you—William & Mary, and Yale, and all the rest. Why in heavens would you give up all of that?"
"If you have to ask that question, Mother," Harold said bitterly as their eyes again met, "I guess you don't really know me."
She was silent as she considered his words, stunned, her quiet anger soon replaced by sadness. But he ignored her weakened state, her vulnerability, as he found himself unable to resist continuing.
"Father never did either. Neither of you ever understood me," he said in a low voice. "Never let me be myself. Never bothered getting to know me."
He walked away and ascended the staircase which curved upward toward the shadowed ceiling, never looking back at her, and moved toward the bedroom which would be his for only a short time longer.
"Hey, Hal, look at my new camera."
Despite the familiarity with which Goodman greeted him, Hal hadn't made any close new friends in the Corps, merely a few friendly acquaintances. Though he was on good terms with nearly everyone he met, he keep enough distance to reveal little about himself—about his upbringing, his legacy, and the opportunity he left behind to come to the dusty desolation of West Texas.
His colleagues liked him for what he was—a quiet, decent, hard-working young man—and not for what he could do for them. It was something his mother would never have understood.
"It's a Brownie. Come on, let me take your picture."
He knew a few things about Goodman, much more than he would ever allow to be known about himself. Goodman was from Brooklyn, working class, father deceased and mother still alive but endlessly worrying about him, Jewish but not practicing, garrulous and fearless, funny and casual and relaxed, poor but relentlessly happy. He admired Goodman, seeing in his friend many qualities which he himself longed to possess.
Hal's mother had been right about one thing: he was indeed digging ditches for the Corps. He was stationed at a work camp forty miles east of El Paso, helping to run irrigation pipe from the Rio Grande to a dry valley which, it was hoped, might someday take to cotton farming. A fanciful idea, perhaps, but what most appealed to him about the Corps was its willingness to try anything, to daringly risk failure, in the hope of bringing a better life to those in need.
"Sure, where?"
"Well, not in here. No flashbulb. Let's go outside."
He liked the idea of having his photograph taken, but didn't think it would look quite right to simply stand and pose. His Corps uniform looked good in person, neat and modest, but he knew enough about the art to realize the uniform might not photograph well, particularly under the harsh Texas sun.
He looked around for suitable props but soon realized that the Corps' humble accommodations might not provide much of visual interest. In harmony with its selfless mission, the Corps' provisions were sparse, giving him little to choose from. The barracks were austere, to the point of barrenness—a dozen plain cots in each room, each neatly made with a crisp sheet and a thin wool blanket, with a small wooden footlocker underneath in which each volunteer stowed his few belongings. The plywood walls were bare except for a few calendars showing the month of July and pastoral watercolor scenes of country lanes and suburban boulevards.
His eyes finally fell upon a frying pan. His mother thought cooking was beneath men of their class; that was the cook's work, and for the unfortunate lower classes that couldn't afford a cook it was the wife's. Yet during his short time in the Corps, he had already learned the simple joy of cooking, for himself and for the others. His mother would never have imagined such a thing.
Yes, he thought, the frying pan is perfect.
He looked around further, seeing little at first. But at last he spotted his drinking cup, into which he would regularly draw water from the common barrel. His mother wouldn't approve of sharing water with others, particularly the type of men who worked for the Corps. Nor would his father—a man who refused to share a bottle of bourbon with even a valued client—have approved. Not proper, they both would have said.
So the drinking cup was in, as well.
He stepped outside, looking for one more prop for the photograph, when across his path ambled an orange-striped tomcat which lived in the camp and reminded him of one he had kept back home. He casually reached down and scooped up the cat, which immediately began purring in the strength of his grasp.
He stood stiffly, facing away from the unpainted barracks wall which stood a dozen feet behind him, the tomcat held under one arm and the frying pan and drinking cup in the opposite hand. He didn't smile as Goodman released the camera's shutter, the tomcat looking more amiable than he did himself.
Weeks later, Goodman passed him the photograph, gesturing that it was his to keep. Hal looked at it, smiled, and reached down into his footlocker, searching for an envelope. He found one, wrote down the destination address and affixed a postage stamp to the upper right corner.
He stared at the photograph for another minute or two, trying to think of the right words, his lips moving almost imperceptibly. Finally he turned the photograph over and wrote out a simple message in elegant script on the reverse side.
Mother: Maybe someday you'll finally get to know me. –Hal
He read the message once more, just to be certain. Then he rose, sealed the envelope and walked toward the door. The outbound mail would be picked up in just a few minutes.
Filed in Fiction
Tagged: Pete Lit, Peter Anderson, The Parlor, The Parlor Emerging Writer's Festival


text:
\section{Introduction}
Consider random discrete distribution functions $F(y)=\sum_{k=1}^{\infty}P_{k}\mathbb{I}_{\{U_{k}\leq y\},}$ where we can assume the $(P_{\ell}):=(P_{\ell},\ell\ge 1)$  are placed in ranked order, and $(U_{\ell})$, independent of $(P_{\ell}),$ are a collection of iid $\mathrm{Uniform}[0,1]$ random variables. Formally we can say $(P_{\ell})\in \mathcal{P}_{\infty}=\{\mathbf{s}=(s_{1},s_{2},\ldots):s_{1}\ge s_{2}\ge\cdots\ge 0 {\mbox { and }} \sum_{i=1}^{\infty}s_{i}=1\},$ where $\mathcal{P}_{\infty}$ denotes the space of ranked mass partitions summing to $1.$ As discussed in~\cite{BerFrag,Kingman75,Pit06,PY97}, there has been considerable interest in the laws and interpretation of objects in
$\mathcal{P}_{\infty},$ within probability, statistics and related areas. The two most notable laws are the Poisson-Dirichlet law, say $\mathrm{PD}(\theta):=\mathrm{PD}(0,\theta)$ with parameter $\theta>0,$ see~\cite{Kingman75,Ewens, Pit96},and its two-parameter extension,$\mathrm{PD}(\alpha,\theta)$ with parameters $0\leq \alpha<1$ and $\theta>-\alpha.$
The latter, for $0<\alpha<1,$ was developed, see~\cite{BPY,PY92,PPY92,Pit96,PY97,Pit06}, where $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta)$ may be interpreted within the context of ranked lengths of excursions of Brownian motion and more general Bessel processes of dimension $2-2\alpha.$

Since the early work of Ferguson~\cite{Ferg73} on the Dirichlet process,$F(y),$ and by-products arising from sampling from such distributions,  have served as major components in Bayesian nonparametric statisics and machine learning applications. When $(P_{\ell})\sim \mathrm{PD}(0,\theta)$ $F(y):=F_{0,\theta}(y)$ is a Dirichlet process.  $F(y):=F_{\alpha,\theta}(y),$ with $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta),$ is often referred to as a Pitman-Yor process. However, due to the generality intractability of $(P_{\ell})\sim\mathrm{PD}(\alpha,\theta),$ the $(P_{\ell})$ are rarely used directly in statistical applications. Instead, other remarkable properties of $F_{0,\theta}$ and $F_{\alpha,\theta},$ including generalized Chinese restaurant processes and tractable stick-breaking representations~(arising from the size biased sampling re-arrangement of $(P_{\ell})$), are employed,(see~\cite{Broderick2,IJ2001,IJ2003,Pit96,Sethuraman}). These can be described within the context of species sampling (sequential capturing and tagging animals appearing in an eco-system) or in effect sampling $(Y_{1},\ldots,Y_{n})$ conditionally iid from $F_{\alpha,\theta}.$ Our goal in this paper is to provide interpretation and derive explicit stick-breaking representations for  $F_{\alpha,\theta}$ conditioned on the number of animals that have appeared up to a time $\lambda,$ where this number is modeled by a mixed Poisson process.
These interpretations are obtained from an unpublished mansucript of Pitman~\cite{PitmanPoissonMix}.
Details will be explained below. As a by-product we obtain an explicit stick-breaking representation, in terms of tractable random variables,  for the popular $F(y)$ following a \textit{normalized generalized gamma process,} see~\cite{LPD}, which equates to the case of $F_{\alpha,0}$ given that $m=0$ animals have been trapped up till some time $\lambda.$ This representaton in the normalized generalized gamma case, without the conditional  interpretation with respect to $F_{\alpha,0}$  was first given in the unpublished work of~\cite{JamesPGarxiv}. Subsequent stick-breaking representations for the normalized generalized gamma process, that is $m=0,$ for general $\alpha,$ appear in~\cite{FavStick1,FavStick2,LauStick}. See also \cite{Fav1} for a description of $\alpha=1/2,m=0$ corresponding to the Normalized Inverse Gaussian process. In Section~\ref{m5} of this paper we shall present results for $\alpha=1/2$ and general $m=0,1,2,\ldots,$ based on the results of \cite{AldousPit,PPY92} as discussed in \cite{Pit02,Pit06}.

\begin{rem}See \cite[Section 5,2]{Pit02} for the work of  McCloskey~\cite{McCloskey} on the (normalized) generalized gamma process within a species sampling context.
\end{rem}

We first recount some aspects of species sampling models, and random distribution functions based on~\cite{Pit96}, (see also \cite{BerFrag,IJ2001,IJ2003,PPY92,Pit02,Pit06,PY92,PY97}). Let $F_{0,\theta},$ for $\theta>0,$ denote a random discrete distribution following a Dirichlet process law as in Ferguson~\cite{Ferg73}.
Pitman~\cite{Pit96}, starting from a Bayesian viewpoint of the Blackwell-MacQueen Polya urn and its relation to  conditionally iid sampling $(Y_{1},\ldots,Y_{n})$ from $F_{0,\theta}$, presents a broad view of Fisher's model for species sampling, whereby within an eco-system animals/individuals are trapped and tagged/classified, one by one in a sequential fashion over time. The first $n$ individuals are classified into $K_{n}\leq n$ species with iid unique tags $(Y^{*}_{1},\ldots,Y^{*}_{K_{n}})$ and membership, in order of appearance, denoted by the set ${\{C_{1},\ldots,C_{K_{n}}\}},$ with $C_{j}:={\{i:Y_{i}=Y^{*}_{j}\}}$ having random multiplicities $|C_{j}|:=N_{j,n},$ constituting a random partition of the integers $[n]:=\{1,\ldots,n\}.$  In the case of $F_{0,\theta},$ the law of ${\{C_{1},\ldots,C_{K_{n}}\}},$ corresponds to the one parameter Chinese restaurant process with law denoted as $\mathrm{CRP}(\theta):=\mathrm{CRP}(0,\theta).$ Furthermore the relative frequencies $(N_{j,n}/n)$ placed in ranked order converge to  $(P_{\ell})\in \mathcal{P}_{\infty}=\{\mathbf{s}=(s_{1},s_{2},\ldots):s_{1}\ge s_{2}\ge\cdots\ge 0 {\mbox { and }} \sum_{i=1}^{\infty}s_{i}=1\}$ having a Poisson-Dirichlet law, denoted as $(P_{\ell})\sim \mathrm{PD}(\theta).$ While $(P_{\ell})$ has a fairly intractable distributional form, the limit of the relative frequencies $(N_{j,n}/n)$ converge
to the size biased re-arrangement of $(P_{\ell}),$ described as $(\tilde{P}_{\ell}=(1-W_{\ell})\prod_{j=1}^{\ell-1}W_{i},\ell\ge 1)$ where $(W_{i})$ are iid $\mathrm{Beta}(\theta,1),$ and $\tilde{P}_{1}=1-W_{1}$ is the first size biased pick from $(P_{\ell}).$ $(\tilde{P}_{\ell})$ is said to have a $\mathrm{GEM}(\theta):=\mathrm{GEM}(0,\theta)$ distribution~\cite{Ewens}, and leads to the tractable stick-breaking representation of $F_{0,\theta},$ in particular, for $(U_{\ell}), (\tilde{U}_{\ell})$ iid $\mathrm{Uniform}[0,1]$ independent of $(P_{\ell}),(\tilde{P}_{\ell})$,
\begin{equation}
F_{0,\theta}(y)=\sum_{k=1}^{\infty}P_{k}\mathbb{I}_{\{U_{k}\leq y\}}= \sum_{k=1}^{\infty}\tilde{P}_{k}\mathbb{I}_{\{\tilde{U}_{k}\leq y\}}
\label{stickDP}
\end{equation}
\cite{Pit96} shows that one may extend the framework above by replacing $F_{0,\theta}$ with general $F$ defined as in~(\ref{stickDP}) by placing more general distributions on $(P_{\ell})\in \mathcal{P}_{\infty} $ and hence $(\tilde{P}_{\ell}).$ Of particular interest to us, as highlighted in~\cite{Pit96}, is the case where $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta),$  where $\mathrm{PD}(\alpha,\theta)$ denotes a two parameter  Poisson Dirichlet distribution, for
$0\leq \alpha<1, \theta>-\alpha,$ with $\mathrm{PD}(0,\theta)=\mathrm{PD}(\theta),$ with origins and applications to for instance excursion thoeory, random tree and graph models as described in \cite{BerFrag,PPY92,PY97,Pit06}. Its size-biased permutation $(\tilde{P}_{\ell}):=(\tilde{P}_{\ell}=(1-W_{\ell})\prod_{j=1}^{\ell-1}W_{i},\ell\ge 1)$ is such that the $W_{i}$ are now independent $\mathrm{Beta}(\theta+i\alpha,1-\alpha),$ for $i=1,2,\ldots.$ One may express this distribution as
$(\tilde{P}_{\ell})\sim \mathrm{GEM}(\alpha,\theta).$ Conversely $\mathrm{Rank}((\tilde{P}_{\ell}))\sim \mathrm{PD}(\alpha,\theta),$ where $\mathrm{Rank}(\cdot)$ denotes the ranked rearrangement of the sequence into $\mathcal{P}_{\infty}.$

Remarkably, the two-parameter family $\mathrm{PD}(\alpha,\theta)$ is the only case where its size-biased permutation consists of independent $(W_{i})$,~\cite{Pit95,Pit96}.  The corresponding $F:=F_{\alpha,\theta},$ is now commonly referred to as the  Pitman-Yor process, as named in \cite{IJ2001}, and is regularly employed in complex applications in Bayesian statistics and machine learning with particular utility in cases involving power law behavior.
\begin{rem}In relation to the above statements, one can construct many $F,$ such that its mass probabilities are of the form $(1-W_{k})\prod_{i=1}^{k-1}W_{i},$ for $(W_{i})$ independent or iid random variables. However, they do not have the interpretation as the size biased permutation of their ranked counterpart. See \cite{Doksum,GnedinPitman2,JamesNTR} for the case of $F(y)=\int_{0}^{\infty}S(t-)
\Lambda(dt,y)$ derived from the masses of a Neutral to the Right $(S(t):t>0)$  process under a homogeneous subordinator.
\end{rem}
\subsection{Species sampling waiting times and counts}
Hidden from the species sampling schemes above, are random mechanisms describing the time of appearance of individuals, or the sample size up till a given time. Pitman\cite{PitmanPoissonMix} points out that the early work of Fisher\cite{Fisher}, McCloskey~\cite{McCloskey} and other authors, already provide a rich mixed Poisson process framework to describe such mechanisms in a species sampling context.
\cite{PitmanPoissonMix}, offers a fresh exposition on mixed Poisson models within this context which points out links to this older work and more recent works in the literature, in particular the recent appearance of mixed Poisson models in Bayesian non-parametric statistics and machine learning.  This, for instance connects to recent work on frequency of frequency (FoF) distributions~\cite{ZhouFoF}, where the population size is modelled by mixed Poisson random variables, as well as gives an interpretation of conditioning on latent mixed gamma variables, $(T_{r})$ described below, as employed in~\cite{JLP2} and elsewhere, in terms of waiting/arrival times.

The following descriptions, including section~\ref{PK}, can be read from~\cite{PitmanPoissonMix}. The distributional expressions below are based on elementary conditioning arguments.
Let $A:=\sum_{k=1}^{\infty}A_{k},$ denote a non-negative almost surely finite (random)sum, such that one may set $(P_{k}:=A_{k}/A)\in\mathcal{P}_{\infty}.$ For $r=1,2\ldots,$ let $G_{r}:=\sum_{j=1}^{r}\mathbf{e}_{j}:=G_{r-1}+\mathbf{e}_{r}$ denote increasing sums of independent standard exponential variables. For $A$ independent of the sequence $(G_{r}),$ define, for each $r,$ $T_{r}=G_{r}/A,$ whence $(T_{r})$ may be interpreted as the sequence of waiting times of a mixed Poisson process $(N_{A}(t) ;t\ge 0)$ defined as
$$
N_{A}(t)=\sum_{r=1}^{\infty}\mathbb{I}_{\{T_{r}\leq t\}}.
$$
That is, $T_{r}=\inf{\{t:N_{A}(t)=r\}}$, for $r=1,2,\ldots$. Within the species sampling context, $A$ has the interpretation as the \textit{total species abundance}, with $(A_{k})$ corresponding to abundance of type. $N_{A}(t)$ is number of animals/individuals appearing up till time $t,$ and for each $n=1,2,3,\ldots$ $T_{n}$ represents the time of appearance/trapping of the $n$-th individual.
There is the following description of the conditional distribution  of $A$ given $N_{A}(\lambda)=0,$ that is to say given that no individuals have been observed up till time $\lambda\ge 0,$
\begin{equation}
\mathbb{P}(A\in da|N_{A}(\lambda)=0)=\mathbb{P}(A\in da|T_{1}>\lambda)=
\frac{{\mbox e}^{-\lambda a}\mathbb{P}(A\in da)}{\mathbb{E}[{\mbox e}^{-\lambda A}]}.
\label{AconPoisson}
\end{equation}
where (\ref{AconPoisson}) corresponds to that of a random variable $A_{0}(\lambda)$ formed by exponential tilting the distribution of $A$
and for $m=1,2,\ldots,$ and $\lambda>0$
\begin{equation}
\mathbb{P}(A\in da|N_{A}(\lambda)=m)=\mathbb{P}(A\in da|T_{m}=\lambda)=
\frac{a^{m}{\mbox e}^{-\lambda a}\mathbb{P}(A\in da)}{\mathbb{E}[A^{m}{\mbox e}^{-\lambda A}]}.
\label{AconPoisson2}
\end{equation}
Denote  a random variable specified by~(\ref{AconPoisson}) and (\ref{AconPoisson2}) as $A_{m}(\lambda).$ Furthermore, for brevity set $\gamma^{[m]}_{\lambda}(da):=\mathbb{P}(A\in da|N_{A}(\lambda)=m),$ for $m=0,1,2,\ldots$

\subsection{Poisson Kingman distributions given $N_{A}(\lambda)=m$}\label{PK}
If we assume that $(A_{k})$ are the ranked jumps of a subordinator say $(A(y):y\ge 0),$ then one may set $A:=A(1),$ satisfying $\mathbb{E}[{\mbox e}^{-\lambda A}]={\mbox e}^{-\psi(\lambda)},$ where $\psi(\lambda)=\int_{0}^{\infty}
(1-{\mbox e}^{-\lambda s})\rho(s)ds ,$ and $\rho(s)$ is a L\'evy density which we further assume satisfies $\int_{0}^{\infty}\rho(s)ds=\infty.$ Then, as defined in~\cite[Definition 3]{Pit02}(see also \cite{Kingman75}), $(P_{\ell})$ has the law of a \textit{Poisson-Kingman distribution with Levy density} $\rho,$ denoted $\mathrm{PK}(\rho).$ Furthermore, the law of $(P_{\ell})|A=t,$ is denoted a $\mathrm{PK}(\rho|t),$ and for a probability distribution $\gamma(dt)$ on $(0,\infty),$
$$
(P_{\ell})\sim \mathrm{PK}(\rho,\gamma)=\int_{0}^{\infty}\mathrm{PK}(\rho|t)\gamma(dt).
$$
where $\mathrm{PK}(\rho,\gamma)$ is a said to be a \textit{Poisson-Kingman distribution with L\'evy density $\rho$ and mixing distribution $\gamma$.} Although one may create an infinite number of distributions over $\mathcal{P}_{\infty},$
it is not obvious how to interpret the meaning of $(P_{\ell})$ with respect to $\gamma.$
However, in the present setting, since the law of $N_{A}$ depends on $(P_{\ell})$ only through $A,$  the law of $(P_{\ell})|A=t,N_{A}=m$ is equivalent to that of $(P_{\ell})|A=t\sim \mathrm{PK}(\rho|t).$  It is evident that $(P_{\ell})|N_{A}(\lambda)=m,$ for each $m=0,1,2,\ldots$ have Poisson-Kingman distributions with mixing distributions $\gamma^{[m]}_{\lambda}.$ That is the distributions $\mathrm{PK}(\rho,\gamma^{[m]}_{\lambda}).$
The above species sampling framework then allows one to interpret these laws.

\begin{rem} (\ref{AconPoisson2}) indicates that $(P_{\ell})|T_{m}=\lambda,$ has the same law as $(P_{\ell})|N_{A}(\lambda)=m,$  for $m=1,2,\ldots.$ Furthermore $\mathrm{PK}(\rho,\gamma^{[0]}_{\lambda})=\mathrm{PK}(\rho_{\lambda}),$
for $\rho_{\lambda}(s)={\mbox e}^{-\lambda s}\rho(s).$
\end{rem}
\begin{rem}General descriptions of the sized biased permutations of $(P_{\ell})\sim \mathrm{PK}(\rho,\gamma^{[m]}_{\lambda}),$ can be obtained by employing the results of~\cite{PPY92}.
\end{rem}

\subsection{Distributional results for $-\infty<\varrho_{1}<\varrho_{2}<\infty$}
We now describe a randomization relationship between general $\varrho$ biased random variables, and hence corresponding $(P_{\ell}).$ The results shows that one needs to only consider the cases $m=0,1,2,\ldots,$ and a suitable randomization to recover all possible laws.
\begin{prop}\label{transfer}
Suppose that for a real number $\varrho,$ $A_{\varrho}(\lambda)$ is a positive random variable with density
$$
\mathbb{P}(A_{\varrho}(\lambda)\in da)=
\frac{a^{\varrho}{\mbox e}^{-\lambda a}\mathbb{P}(A\in da)}{\mathbb{E}[A^{\varrho}{\mbox e}^{-\lambda A}]}.
$$
and consider for $-\infty<\varrho_{1}<\varrho_{2}<\infty,$ the random variable $Y_{\varrho_{2},\varrho_{1}}(\lambda)\overset{d}=G_{\varrho_{2}-\varrho_{1}}/{A_{\varrho_{1}}(\lambda)}.$ Then for $-\infty<\varrho_{1}<\varrho_{2}<\infty,$
$$
A_{\varrho_{2}}(\lambda+Y_{\varrho_{2},\varrho_{1}}(\lambda))\overset{d}=A_{\varrho_{1}}(\lambda)
$$
\end{prop}
\begin{proof}The density of $Y_{\varrho_{2},\varrho_{1}}(\lambda)$ can be expressed as
$$
\mathbb{P}(Y_{\varrho_{2},\varrho_{1}}(\lambda)\in dy)/dy=\frac{y^{\varrho_{2}-\varrho_{1}-1}\mathbb{E}[A^{\varrho_{2}}{\mbox e}^{-(\lambda+y)A}]}{\Gamma(\varrho_{2}-\varrho_{1})\mathbb{E}[A^{\varrho_{1}}{\mbox e}^{-\lambda A}]}
$$
It follows that $\mathbb{P}(A_{\varrho_{1}}(\lambda)\in da)=\int_{0}^{\infty}\mathbb{P}(A_{\varrho_{2}}(\lambda+y)\in da)\mathbb{P}(Y_{\varrho_{2},\varrho_{1}}(\lambda)\in dy)$
\end{proof}
\section{The $\mathrm{PD}(\alpha,\theta)$ case given $N_{S_{\alpha,\theta}}(\lambda)$}
If $(A(y):y\ge 0)$ is a stable subordinator, such that $S_{\alpha}:=A(1):=A$ is a stable random variable with Laplace transform $\mathbb{E}[{\mbox e}^{-\lambda S_{\alpha}}]={\mbox e^{-\lambda^{\alpha}}},$ with density denoted as $f_{\alpha}(t),$ then $(P_{\ell}:=A_{\ell}/S_{\alpha},\ell\ge 1)\sim \mathrm{PD}(\alpha,0).$ The corresponding  L\'evy density is given by $\rho_{\alpha}(s)=\alpha s^{-\alpha-1}/\Gamma(1-\alpha),$ see~\cite[eq. (54)]{Pit02}.
Hence $\mathrm{PD}(\alpha,0):=\mathrm{PK}(\rho_{\alpha}).$ The law of $(P_{\ell})|S_{\alpha}=t$ is denoted as $\mathrm{PD}(\alpha|t):=\mathrm{PK}(\rho_{\alpha}|t),$ and $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta)=\int_{0}^{\infty}\mathrm{PD}(\alpha|t)f_{\alpha,\theta}(t)dt,$ where $f_{\alpha,\theta}(t)=t^{-\theta}f_{\alpha}(t)/\mathbb{E}[S^{-\theta}_{\alpha}]$ corresponds to the density of the random variable $S_{\alpha,\theta}.$ It follows that for $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta),$
the law of $(P_{\ell})|N_{S_{\alpha,\theta}}(\lambda)=m,$ can be expressed as
\begin{equation}
\mathbb{P}^{[m-\theta]}_{\alpha}(\lambda)=\int_{0}^{\infty}\mathrm{PD}(\alpha|t)\mathbb{P}(S_{\alpha,\theta}\in dt|N_{S_{\alpha,\theta}}(\lambda)=m),
\end{equation}
where $\mathbb{P}(S_{\alpha,\theta}\in dt|N_{S_{\alpha,\theta}}(\lambda)=0)=\mathbb{P}(S_{\alpha,\theta}\in dt|\frac{G_{1}}{S_{\alpha,\theta}}>\lambda),$ and ,for $m=1,2,\ldots$ there is the equivalence,
$$\mathbb{P}(S_{\alpha,\theta}\in dt|N_{S_{\alpha,\theta}}(\lambda)=m)=\mathbb{P}(S_{\alpha,\theta}\in dt|\frac{G_{m}}{S_{\alpha,\theta}}=\lambda),$$
and otherwise for each $m=0,1,2,\ldots,$ has the density
\begin{equation}
f^{[m-\theta]}_{\alpha}(t|\lambda)=\frac{{\mbox e}^{-\lambda t}t^{m-\theta}f_{\alpha}(t)}{\mathbb{E}[S^{m-\theta}_{\alpha}{\mbox e}^{-\lambda S_
{\alpha}}]},
\label{condden}
\end{equation}
for $\theta>-\alpha.$ Set, $\varrho:=m-\theta$ which can be any real number, and let
$S_{\alpha,\varrho}(\lambda)$ denote a random variable with density~(\ref{condden}). $S_{\alpha,\varrho}(\lambda)$ is well defined for all $\lambda\ge 0$ and $\varrho=-\theta<\alpha,$ with $S_{\alpha,-\theta}(0)=S_{\alpha,\theta},$ and otherwise defined for any $\varrho$ and $\lambda>0.$ A special case of Proposition~\ref{transfer} leads to the next result.
\begin{cor}\label{PDrecovery}Let for $\varrho_{2}>\varrho_{1}$, $Y_{\varrho_{2},\varrho_{1}}(\lambda)\overset{d}=G_{\varrho_{2}-\varrho_{1}}/S_{\alpha,\varrho_{1}}(\lambda)$ then
$$\mathbb{E}[\mathbb{P}^{[\varrho_{2}]}_{\alpha}(\lambda+Y_{\varrho_{2},\varrho_{1}}(\lambda))]=
\mathbb{P}^{[\varrho_{1}]}_{\alpha}(\lambda).
$$
As special cases, set $\varrho_{1}=-\theta,$ for $\theta>-\alpha,$ then $Y_{\varrho_{2},-\theta}(0)\overset{d}=G_{\varrho_{2}+\theta}/S_{\alpha,\theta}$ and
$$\mathbb{E}[\mathbb{P}^{[\varrho_{2}]}_{\alpha}(Y_{\varrho_{2},-\theta}(0))]=
\mathrm{PD}(\alpha,\theta).
$$
Conversely if $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta)$ then $(P_{\ell})|G_{\varrho_{2}+\theta}/S_{\alpha,\theta}=\lambda\sim \mathbb{P}^{[\varrho_{2}]}_{\alpha}(\lambda).$
\end{cor}
\subsection{The normalized generalized gamma case, and size biased mixing distributions}
Throughout, let $(\tau_{\alpha}(y):y\ge 0)$ denote a generalized gamma subordinator, such that the density of
$S_{\alpha,0}(\lambda)\overset{d}=\tau_{\alpha}(\lambda^{\alpha})/\lambda$ is equivalent to $f^{[0]}_{\alpha}(t|\lambda)=
{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda t}f_{\alpha}(t),$ and a normalized generalized gamma process can be expressed as
\begin{equation}
\frac{\tau_{\alpha}(y\lambda^{\alpha})}{\tau_{\alpha}(\lambda^{\alpha})}\overset{d}=\sum_{k=1}^{\infty}P^{[0]}_{\ell}(\lambda)\mathbb{I}_{\{U_{k}\leq y\}}
\end{equation}
where $(P^{[0]}_{\ell}(\lambda))\sim\mathbb{P}^{[0]}_{\alpha}(\lambda).$ Furthermore setting $\rho_{\alpha,\lambda}(s)={\mbox e}^{-\lambda s}\rho_{\alpha}(s),$ there is the dual representation
$$
\mathbb{P}^{[0]}_{\alpha}(\lambda)=\int_{0}^{\infty}\mathrm{PD}(\alpha|t)f^{[0]}_{\alpha}(t|\lambda)dt:=\mathrm{PK}(\rho_{\alpha,\lambda}).
$$
The generalized gamma process and its normalized version appear in many applications. While $(\tau_{\alpha}(y):y\ge 0)$ is known to arise from the operation of exponentially tilting a stable subordinator, we see that within the species sampling context it arises from conditioning a $(P_{\ell})\sim\mathrm{PD}(\alpha,n)$ on the event $N_{S_{\alpha,n}}(\lambda)=n,$ for each $n=0,1,2,\ldots$
\begin{rem}Random variables having the distribution of $S_{\alpha,0}(\lambda)\overset{d}=\tau_{\alpha}(\lambda^{\alpha})/\lambda$ can be generated by the methods in \cite{DevroyeGen}.
\end{rem}
\subsection{Special properties of the $\mathbb{P}^{[m]}_{\alpha}(\lambda)$ distribution}
As in, \cite[pages 64-66]{Pit06}, see also\cite{Pit99}, let $K_{m}$ for $m=1,2,\ldots,$  denote the number of blocks in a random partition of $[m]=\{1,2,\ldots,m\}$ under a $\mathrm{PD}(\alpha,0)$ Chinese restaurant process scheme. Its probability mass function can be expressed as,
$$
\mathrm{P}_{\alpha}^{(m)}(k):=\mathbb{P}_{\alpha,0}(K_m = k)=\frac{\alpha^{k-1}\Gamma(k)}{\Gamma(m)} S_\alpha(m,k),
$$
with $S_\alpha(m,k) =  \frac{1}{\alpha^k k!} \sum_{j=1}^k (-1)^j \binom{k}{j} (-j\alpha)_m$ denoting the generalized Stirling number of the second kind. Hereafter set $\Omega_{0}(\lambda^{\alpha})=\lambda^{-\alpha}/\alpha,$ $\Omega_{1}(\lambda^{\alpha})=1$ and generally for $m=1,2,\ldots$
$$
\Omega_{m}(\lambda^{\alpha})=\Gamma(m)\times \sum_{\ell=1}^{m}\mathrm{P}^{(m)}_{\alpha,0}(\ell)
\frac{{(\lambda^{\alpha})}^{\ell-1}}{\Gamma(\ell)}.
$$

It is known through the relation between cumulants and moments of a generalized gamma variable that
\begin{equation}
\mathbb{E}[S^{m}_{\alpha}{\mbox e}^{-\lambda S_
{\alpha}}]=\alpha{\mbox e}^{-\lambda^{\alpha}}\lambda^{\alpha-m}\times {\Omega}_{m}(\lambda^{\alpha})
\label{momentrep}.
\end{equation}
with $\mathbb{E}[S_{\alpha}{\mbox e}^{-\lambda S_
{\alpha}}]=\alpha\lambda^{\alpha-1}{\mbox e}^{-\lambda^{\alpha}}.$ Then for $m=1,2,\ldots,$ there are the representations for the density of $S_{\alpha}|N_{S_{\alpha}}(\lambda)=m,$
\begin{equation}
{f}^{[m]}_{\alpha}(t|\lambda)=\frac{\lambda^{m-\alpha}{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda t}t^{m}f_{\alpha}(t)}{\alpha\Omega_{m}(\lambda^{\alpha})}=\frac{t^{m} \times (
{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda t}f_{\alpha}(t))}{\mathbb{E}\left[{\left(\frac{\tau_{\alpha}(\lambda^{\alpha})}{\lambda}\right)}^{m}\right]},
\label{condSU}
\end{equation}
which is the $m$-th size biased density of $\tau_{\alpha}(\lambda^{\alpha})/\lambda$. Furthermore there is the representation of the corresponding random variable
\begin{equation}
S_{\alpha,m}(\lambda)\overset{d}=\frac{\tau_{\alpha}\left(\lambda^{\alpha}+G_{\frac{m}{\alpha}-K_{m}(\lambda)}\right)}{\lambda}\overset{d}=\frac{\tau_{\alpha}(\lambda^{\alpha})+G_{{m}-K_{m}(\lambda)\alpha}}{\lambda}.
\label{Sm}
\end{equation}
where $K_{m}(\lambda)$ is the discrete random variable corresponding to the conditional distribution of $K_{m}|N_{S_{\alpha}}(\lambda)=m,$ which has probability mass function
\begin{equation}
\mathbb{P}_{\alpha,0}(K_{m}(\lambda)=k)=\frac{\mathrm{P}^{(m)}_{\alpha,0}(k)\frac{\lambda^{k\alpha}}{\Gamma(k)}}{\sum_{\ell=1}^{m}\mathrm{P}^{(m)}_{\alpha,0}(\ell)\frac{\lambda^{\ell\alpha}}{\Gamma(\ell)}}
\label{condKu}
\end{equation}
We close with an expression for the density of $Y_{m,m-k}(\lambda)/\lambda$ which will play a role in the description of the size biased ordering of $(P_{\ell})\sim\mathbb{P}^{[m]}_{\alpha}(\lambda).$
\begin{lem}\label{Ymmk}
Consider $Y_{m,m-k}(\lambda)\overset{d}=G_{k}/S_{\alpha,m-k}(\lambda),$ satisfying $\mathbb{E}[\mathbb{P}^{[m]}_{\alpha}(\lambda+Y_{m,m-k}(\lambda))]=\mathbb{P}^{[m-k]}_{\alpha}(\lambda)$ for each $k=1,\ldots,m.$ Then $Y_{m,m-k}(\lambda)/\lambda$ has density
\begin{equation}
\frac{\Omega_{m}(\lambda^{\alpha}{(1+y)}^{\alpha})}{\Gamma(k)\Omega_{m-k}(\lambda^{\alpha})}y^{k-1}{(1+y)}^{\alpha-m}{\mbox e}^{-[\lambda^{\alpha}{(1+y)}^{\alpha}-\lambda^{\alpha}]}.
\label{Ydensity}
\end{equation}
\end{lem}
\begin{rem}Except for parametrization, the law of $K_{m}(\lambda),$~(\ref{condKu}), appears in~\cite[eq. (15), p. 1629]{ZhouFoF}, although $\lambda$'s relation to time is not considered in that context. This is also a special case of the distribution $\mathbb{P}(n(\pi)=k|U_{m}=\lambda)$ as described in~\cite[p. 85]{JLP2}, where $U_{m}\overset{d}=T_{m}$ in the general $\mathrm{PK}(\rho)$ setting.
\end{rem}

In the next sections we will describe results for the size biased re-arrangement of $(P_{\ell})|N_{S_{\alpha}}=m,$
where $(P_{\ell})\sim \mathrm{PD}(\alpha,0).$ Applications of Corollary~\ref{PDrecovery} will allow for representations in the general cases. We focus on the case of $m=0,$ corresponding to a normalized generalized gamma process, in Section~\ref{gg}. The case of $m=1$ in Section~\ref{sizegg} and general $m$ in Section~\ref{genm}.

\section{Stick-breaking representations for $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta)$ given $N_{S_{\alpha,\theta}}(\lambda)=m$}

Suppose that $(\tilde{P}_{\ell})\sim\mathrm{GEM}(\alpha,\theta),$ then $F_{\alpha,\theta}(y):=\sum_{k=1}^{\infty}\tilde{P}_{\ell}\mathbb{I}_{\{\tilde{U}_{\ell}\leq y\}}$ is a Pitman-Yor process with parameters $(\alpha,\theta).$ The result in \cite{PPY92,Pit96,PY97} lead to the following exact decomposition
\begin{equation}
F_{\alpha,\theta}(y)=W_{1}F_{\alpha,\theta+\alpha}(y)+(1-W_{1})\mathbb{I}_{\{\tilde{U}_{1}\leq y\}}
\label{gemdecomp}
\end{equation}
where $F_{\alpha,\theta+\alpha}(y)=\sum_{k=2}^{\infty}(\tilde{P}_{\ell}/W_{1})\mathbb{I}_{\{\tilde{U}_{\ell}\leq y\}},$
and $(\tilde{P}_{\ell}/W_{1},\ell\ge 2)\sim\mathrm{GEM}(\alpha,\theta+\alpha),$ independent of the first size biased pick $(1-W_{1})\sim \mathrm{Beta}(1-\alpha,\theta+\alpha).$ Furthermore $\mathrm{Rank}((\tilde{P}_{\ell}/W_{1},\ell\ge 2))\sim\mathrm{PD}(\alpha,\theta+\alpha).$ This operation of \textit{size biased deletion} and ranking the normalized components is described in \cite[Chap 6]{PY97}. Repeated application of this procedure leads to a collection of mass partitions with laws
$(\mathrm{PD}(\alpha,\theta+(k-1)\alpha)),k\ge 1).$ It suffices to  focus on the $\mathrm{PD}(\alpha,0)$ case where $\theta=0.$ These facts, as established in~\cite{PPY92}, may be derived from the corresponding  sequence of random variables $(S_{\alpha,(k-1)\alpha},k\ge 1),$ forming a Markov Chain satisfying $W_{k}= S_{\alpha,k\alpha}/S_{\alpha,(k-1)\alpha}$ and where for any $n,$
$(W_{1},\ldots,W_{n})$ is independent of $S_{\alpha,n\alpha},$ with $S_{\alpha}:=S_{\alpha,0}=S_{\alpha,n\alpha}/\prod_{j=1}^{n}W_{i}.$ In other words the joint law of $(W_{1},\ldots,W_{n},S_{\alpha,n\alpha})$ is given by,
\begin{equation}
\left[\prod_{k=1}^{n}f_{B_{k}}(w_{k})\right]f_{\alpha,n\alpha}(s)ds
\label{JointWn}
\end{equation}
where $f_{B_{k}}$ denotes the density of a $\mathrm{Beta}(k\alpha,1-\alpha)$ variable, and
$f_{\alpha,n\alpha}(s)=s^{-n\alpha}f_{\alpha}(s)/\mathbb{E}[S^{-n\alpha}_{\alpha}]$ is the density of $S_{\alpha,n\alpha}.$
Precisely (\ref{JointWn}) can be written as
$$
\frac{\alpha^{n-1}\Gamma(n)}{[\Gamma(1-\alpha)]^{n}\Gamma(n\alpha)}\prod_{k=1}^{n}w^{k\alpha-1}_{k}{(1-w_{k})}^{-\alpha}\times \frac{\alpha\Gamma(n\alpha)}{\Gamma(n)}s^{-n\alpha}f_{\alpha}(s).
$$
\begin{rem}See \cite[Section 5]{JamesLamp} for interpretations of~(\ref{gemdecomp}) in connection with the results in
\cite[Theorem 1.3.1]{PY92} and \cite[Theorem 3.8, Lemma 3.11]{PPY92}.
\end{rem}
\subsection{An expression for the density of $(W_{1},\ldots,W_{n}),S_{\alpha,n\alpha}|N_{S_{\alpha}}(\lambda)=m$}
\begin{lem}\label{Wdensity}
Suppose that $(P_{\ell})\sim \mathrm{PD}(\alpha,0)$ with corresponding $(W_{1},\ldots,W_{n},S_{\alpha,n\alpha})$ having joint density in~(\ref{JointWn}). Then for each $m=0,1,2,\ldots,$
\begin{enumerate}
\item[(i)] $(P_{\ell})|N_{S_{\alpha}}(\lambda)=m$ has law $\mathbb{P}^{[m]}_{\alpha}(\lambda)=\int_{0}^{\infty}\mathrm{PD}(\alpha|t)f^{[m]}_{\alpha}(t|\lambda)dt.$
\item[(ii)]The (conditional) joint density of $(W_{1},\ldots,W_{n},S_{\alpha,n\alpha})|N_{S_{\alpha}}(\lambda)=m,$ can be expressed as, for each $n\ge 1,$
\begin{equation}
q_{m}(s/\prod_{i=1}^{n}w_{i}|\lambda)\left[\prod_{k=1}^{n}f_{B_{k}}(w_{k})\right]f_{\alpha,n\alpha}(s)
\label{jointVGm}
\end{equation}
where $q_{m}(t|\lambda)f_{\alpha}(t)=f^{[m]}_{\alpha}(t|\lambda).$ That is,
\begin{equation}
q_{m}(t|\lambda)=\frac{\lambda^{m-\alpha}t^{m}{\mbox e}^{-[\lambda t-\lambda^{\alpha}]}}
{\alpha\Omega_{m}(\lambda^{\alpha})}.
\label{mweight}
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof}This is a consequence of the results in \cite{PPY92}, and the extension of those results to the Poisson Kingman distributions with mixing, in the $\alpha$-stable case, as described in \cite{Pit02}. Where the mixing distribution is
$q_{m}(t|\lambda)f_{\alpha}(t)=f^{[m]}_{\alpha}(t|\lambda).$ Otherwise, the inhomogeneous Markovian structure of
$S_{\alpha,k\alpha}|S_{\alpha,(k-1)\alpha}=s$ for $k=1,2,\ldots,$ as established in~\cite{PPY92}, leads to a description of $(W_{1},\ldots,W_{n},S_{\alpha,n\alpha})|S_{\alpha}=s,N_{S_{\alpha}}(\lambda)=m,$ which just depends on $S_{\alpha}=s.$ The result is realized by then mixing over  $f^{[m]}_{\alpha}(s|\lambda),$ which is the density of $S_{\alpha}|N_{S_{\alpha}}(\lambda)=m.$
\end{proof}
\subsection{The variables $(R_{1}(\lambda),\dots,R_{n}(\lambda),n \ge 2)$}
Throughout, let $(\mathbf{e}_{i})$ denote a sequence of independent $\mathrm{exponential}~(1)$ variables.
Define partial sums $\tilde{G}_{k}=\sum_{i=1}^{k}\mathbf{e}_{i},$ with $\tilde{G}_{0}=0.$
Define for $k=1,2,\dots,$
\begin{equation}
R_{k}(\lambda):={\left(\frac{\tilde{G}_{k-1}+\lambda^{\alpha}}{\tilde{G}_{k}+\lambda^{\alpha}}\right)}^{\frac{1}{\alpha}}
\label{Rrep}
\end{equation}
The next result provides an important component to the stick-breaking representation and analysis. The proof follows  from elementary conditioning arguments and is omitted.
\begin{lem}\label{Rdist}Let for $k\ge 1,$ $R_{k}:=R_{k}(\lambda)$ be random variables described in~(\ref{Rrep}), then
given any $\lambda>0$
\begin{enumerate}
\item[(i)]$R_{1}$ has density
$$
f_{R_{1}}(r_{1}|\lambda)=\alpha\lambda^{\alpha}{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda^{\alpha}/r^{\alpha}_{1}}r^{-\alpha-1}_{1}.
$$
\item[(ii)]$R_{j}|R_{1}=r_{1},\ldots,R_{j-1}=r_{j-1}$ has density $f_{R_{1}}(r_{j}|\lambda/\prod_{i=1}^{j-1}r_{i})$
\item[(iii)]Hence $(R_{1},\dots,R_{n})$ has joint density,
$$
f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})=\alpha^{n}\lambda^{n\alpha}{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda^{\alpha}/(\prod_{l=1}^{n}r_{l})^{\alpha}}\prod_{l=1}^{n}r^{-(n-l+1)\alpha-1}_{l}
$$
\end{enumerate}
Furthermore,
$
\lambda^{\alpha}/{\prod_{l=1}^{k}R^{\alpha}_{l}}=\tilde{G}_{k}+\lambda^{\alpha}.
$
\end{lem}

\subsection{Case $m=0,$ $\mathbb{P}^{[0]}_{\alpha}(\lambda),$the normalized generalized gamma case}\label{gg}
In the case where $(P_{\ell})|N_{S_{\alpha}}(\lambda)=0\sim \mathbb{P}^{[0]}_{\alpha}(\lambda),$ having the law of the normalized jumps of a generalized gamma process, the joint density of
$(W_{1},\ldots,W_{n},S_{\alpha,n\alpha})|N_{S_{\alpha}}(\lambda)=0,$ can be expressed as,
\begin{equation}
\frac{{\mbox e}^{-\frac{\lambda s}{\prod_{i=1}^{n}w_{i}}}{\mbox e}^{\lambda^{\alpha}}\alpha^{n}}{[\Gamma(1-\alpha)]^{n}}\prod_{k=1}^{n}w^{k\alpha-1}_{k}{(1-w_{k})}^{-\alpha}\times s^{-n\alpha}f_{\alpha}(s).
\label{joint1}
\end{equation}
The key to obtaining a simple description of the $(W_{i})$ in terms of random variables is facilitated by the following identity.

\begin{prop}\label{integral1}
The integral
$$
\int_{0}^{w_{1}}\cdots\int_{0}^{w_{n}}
\lambda^{n\alpha}
{\mbox e}^{-\frac{\lambda s}{(\prod_{l=1}^{n}r_{l})}}\prod_{l=1}^{n}(w_{l}-r_{l})^{\alpha-1}r^{-(n-l+1)\alpha-1}_{l}dr_{n}\ldots dr_{1}
$$
is equal to
$$
{[\Gamma(\alpha)]}^{n}s^{-n\alpha}{\mbox e}^{-\frac{\lambda s}{(\prod_{l=1}^{n}w_{l})}}\prod_{l=1}^{n}w^{l\alpha-1}_{l}.
$$

\end{prop}
\begin{proof}
For positive quantities $(v,t,\lambda,w)$ there is the integral identity,
$$
\Gamma(\alpha)v^{\alpha}t^{-\alpha}w^{\alpha-1}{\mbox e}^{-\frac{\lambda t}{vw}}=
\lambda^{\alpha}\int_{0}^{w}{(w-r)}^{\alpha-1}{\mbox e}^{-\frac{\lambda t}{vr}}r^{-\alpha-1}dr,
$$
which can be obtained by the change of variables $r=1/y$ and then $s=rx-1$ leading to a gamma integral.
The result follows by repeated usage of this identity starting with $r_{n}$ and initially setting $v=\prod_{i=1}^{n-1}r_{i}.$ Then integrate with respect to $r_{n-1}$, setting $v=w_{n}\prod_{i=1}^{n-2}r_{i}$ and so on.
\end{proof}
\subsubsection{Descriptions of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=0$}
\begin{lem}\label{jointWzero}The joint density of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=0,$ can be expressed as
$$
\left[\prod_{k=1}^{n}\frac{{(1-w_{k})}^{\alpha-1}}{\Gamma(1-\alpha)\Gamma(\alpha)}\right]\int_{0}^{w_{1}}\cdots\int_{0}^{w_{n}}
\prod_{l=1}^{n}(w_{l}-r_{l})^{\alpha-1}f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})dr_{n}\ldots dr_{1}
$$
\end{lem}
\begin{proof} Replace $s^{-n\alpha}{\mbox e}^{-\frac{\lambda s}{(\prod_{l=1}^{n}w_{l})}}\prod_{l=1}^{n}w^{l\alpha-1}_{l}$ in (\ref{joint1}) with the integral in Proposition~\ref{integral1}. Integrate with respect to $f_{\alpha}(s)$ and rearrange terms.

\end{proof}
\begin{thm}\label{Wcase00}Let $(P^{[0]}_{\ell}(\lambda))\sim\mathbb{P}^{[0]}_{\alpha}(\lambda),$ corresponding in distribution to the ranked masses in $\mathcal{P}_{\infty}$ of the normalized jumps of a generalized gamma subordinator . Then the joint distribution of $(\tilde{P}_{\ell}(\lambda)),$ the size biased rearrangement of $(P^{[0]}_{\ell}(\lambda)),$ can be expressed as
$$
(\tilde{P}_{1}(\lambda),\tilde{P}_{2}(\lambda),\ldots)=(1-W_{1},(1-W_{2})W_{1},\ldots)
$$
that is $\tilde{P}_{\ell}(\lambda)=(1-W_{\ell})\prod_{i=1}^{\ell-1}W_{i},$ where the  $(W_{k})$ are dependent random variables represented as
\begin{equation}
W_{k}=1-\beta^{(k)}_{({1-\alpha},\alpha)}[1-R_{k}(\lambda)]
\label{weightszero}
\end{equation}
for $(\beta^{(k)}_{(1-\alpha,\alpha)})$ iid $\mathrm{Beta}(1-\alpha,\alpha)$ variables independent of the $R_{k}(\lambda)$ defined  in (\ref{Rrep}) and Lemma~\ref{Rdist}.
\end{thm}
\begin{proof}Augmenting the integral expression in Lemma~\ref{jointWzero} shows that the $W_{1},\ldots,W_{n}|R_{1}(\lambda)=r_{1},\ldots, R_{n}(\lambda)=r_{n},N_{S_{\alpha}}(\lambda)=0$ has joint density
$$
\left[\prod_{k=1}^{n}\frac{{(1-w_{k})}^{\alpha-1}}{\Gamma(1-\alpha)\Gamma(\alpha)}\right]
\prod_{l=1}^{n}(w_{l}-r_{l})^{\alpha-1}$$
for $r_{i}<w_{i}<1,$ $i=1,\ldots,n.$ Which leads to the description in~(\ref{weightszero}).
\end{proof}
\begin{cor}
For each fixed $k,$ $1-W_{k}$ has the distribution of the first size-biased pick from a
$\mathbb{E}[\mathbb{P}^{[0]}_{\alpha}({(\tilde{G}_{k-1}+\lambda^{\alpha})}^{1/\alpha})]$ mass partition
$$(P_{l,k-1}(\lambda),\ell\ge 1):=\mathrm{Rank}((\tilde{P}_{\ell+k-1}(\lambda)/\prod_{i=1}^{k-1}W_{i},\ell\ge 1)),$$ with $W_{0}:=1,$ and $(P_{\ell,0}(\lambda),\ell\ge 1)\overset{d}=(P^{[0]}_{\ell}(\lambda))\sim\mathbb{P}^{[0]}_{\alpha}(\lambda).$
\end{cor}
\begin{proof}This follows from the description of the $(W_{k},R_{k}(\lambda))$ in Theorem \ref{Wcase00}, which leads to
$(P_{\ell,1}(\lambda),\ell\ge 1)\sim \mathbb{E}[\mathbb{P}^{[0]}_{\alpha}(\lambda/R_{1}(\lambda))]$ and more generally
$(P_{\ell,k-1}(\lambda),\ell\ge 1)\sim \mathbb{E}[\mathbb{P}^{[0]}_{\alpha}(\lambda/\prod_{i=1}^{k-1}R_{i}(\lambda))].$
\end{proof}

\subsubsection{Recovering the Pitman-Yor- $\mathrm{PD}(\alpha,\theta)$ case for $\theta
\ge 0$}
We now show how to recover the stick-breaking result in the (Pitman-Yor) $\mathrm{PD}(\alpha,\theta)$ case for $\theta\ge 0,$ in terms of $(R_{k}).$
\begin{cor}When $\lambda^{\alpha}=G_{\theta/\alpha}$ for $\theta\ge0,$
$$
\mathbb{E}[\mathbb{P}^{[0]}_{\alpha}(G^{1/\alpha}_{\theta/\alpha})]=\mathrm{PD}(\alpha,\theta)
$$
The corresponding,
$$
R_{k}(G^{1/\alpha}_{\theta/\alpha}):={\left(\frac{\tilde{G}_{k-1}+G_{\theta/\alpha}}{\tilde{G}_{k}+G_{\theta/\alpha}}\right)}^{\frac{1}{\alpha}}.
$$
are now collections of independent beta distributed variables where one can set,
$R_{k}(G^{1/\alpha}_{\theta/\alpha}):=\beta_{\theta+(k-1)\alpha,1}\sim \mathrm{Beta}(\theta+(k-1)\alpha,1).$
Hence
$$
1-W_{k}:=\beta^{(k)}_{1-\alpha,\alpha}[1-\beta_{\theta+(k-1)\alpha,1}]=\beta_{1-\alpha,\theta+k\alpha}
$$
and are independent. Recall that $\mathrm{PD}(\alpha,0)=\mathbb{P}^{[0]}_{\alpha}(0)$, in this case $R_{1}(0)=0,$ and $R_{k}(0)=\beta_{(k-1)\alpha,1},$ for $k\ge 2.$
\end{cor}
\begin{proof} Since $\mathbb{P}^{[0]}_{\alpha}(\lambda)$ corresponds to the generalized gamma case this can be seen as an immediate consequence of \cite[Proposition 21]{PY97}. However from Corollary~\ref{PDrecovery}, we can also use the fact that
$Y_{0,-\theta}(0):=G_{\theta}/S_{\alpha,\theta}\overset{d}=G^{1/\alpha}_{\theta/\alpha}.$
\end{proof}

\subsection{Case $m=1,$  $(P^{[1]}_{\ell}(\lambda))\sim\mathbb{P}^{[1]}_{\alpha}(\lambda)$}\label{sizegg}
For $(P_{\ell})\sim \mathrm{PD}(\alpha,0)$ recall that $(P_{\ell})|N_{S_{\alpha}}(\lambda)=1$ or equivalently $(P_{\ell})|G_{1}/S_{\alpha}=\lambda,$ has the law of a mass partition $(P^{[1]}_{\ell}(\lambda))\sim\mathbb{P}^{[1]}_{\alpha}(\lambda),$ where
$$
\mathbb{P}^{[1]}_{\alpha}(\lambda)=\frac{\lambda^{1-\alpha}}{\alpha}{\mbox e}^{\lambda^{\alpha}}\int_{0}^{\infty}\mathrm{PD}(\alpha|t)t{\mbox e}^{-\lambda t}f_{\alpha}(t) dt
$$

Since it is known that $Y_{1,0}(0):=G_{1}/S_{\alpha}\overset{d}=\mathbf{e}^{1/\alpha}_{1},$ it follows that
$(P^{[1]}_{\ell}(\mathbf{e}^{1/\alpha}_{1}))\sim \mathrm{PD}(\alpha,0).$ The next result shows how $\mathbb{P}^{[0]}_{\alpha}(\lambda)$ can be expressed in terms of $\mathbb{P}^{[1]}_{\alpha},$ in general

\begin{prop}\label{onetozero}Let $Y_{1,0}(\lambda)=\lambda G_{1}/\tau_{\alpha}(\lambda^{\alpha}),$ then
\begin{enumerate}
\item[(i)]$Y_{1,0}(\lambda)\overset{d}={(\lambda^{\alpha}+\mathbf{e}_{1})}^{1/\alpha}-\lambda.$
\item[(ii)]
$
\mathbb{E}[\mathbb{P}^{[1]}_{\alpha}({(\lambda^{\alpha}+\mathbf{e}_{1})}^{1/\alpha})]=\mathbb{P}^{[0]}_{\alpha}(\lambda).
$
\item[(iii)]$(P^{[1]}_{\ell}({(\lambda^{\alpha}+\mathbf{e}_{1})}^{1/\alpha}))\sim \mathbb{P}^{[0]}_{\alpha}(\lambda).$
\item[(iv)]Furthermore $\lambda/R_{1}(\lambda)\overset{d}={(\lambda^{\alpha}+\mathbf{e}_{1})}^{1/\alpha}$
\end{enumerate}
\end{prop}
\begin{proof}Using Laplace transforms, $\mathbb{P}(Y_{1,0}(\lambda)>y)={\mbox e}^{-[{(\lambda+y)}^{\alpha}-\lambda^{\alpha}]},$ yielding [(i)]. Statements [(ii)] and [(iii)] are equivalent and follow from Corollary~\ref{PDrecovery} since $\lambda+ Y_{1,0}(\lambda)\overset{d}= {(\lambda^{\alpha}+\mathbf{e}_{1})}^{1/\alpha}.$
\end{proof}

\begin{thm}\label{TheoremP1} If $(P^{[1]}_{\ell}(\lambda))\sim\mathbb{P}^{[1]}_{\alpha}(\lambda),$ then the sequence $(\tilde{P}_{\ell}(\lambda)),$ obtained by size-biased sampling from $(P^{[1]}_{\ell}(\lambda)),$
can be represented as $\tilde{P}_{k}(\lambda)=(1-{W}_{k})\prod_{l=1}^{k-1}W_{l},$
with the following properties;
\begin{enumerate}
\item[(i)]The  $(W_{k})$ are generally dependent random variables represented as
\begin{equation}
W_{k}:={[\beta^{(k)}_{1-\alpha,\alpha}[(1-R_{k}(\lambda))/R_{k}(\lambda)]+1]}^{-1}
 \label{weightsone}
\end{equation}
for $(\beta^{(k)}_{(1-\alpha,\alpha)})$ iid $\mathrm{Beta}(1-\alpha,\alpha)$ variables independent of the $R_{k}(\lambda)$ defined  in (\ref{Rrep}).
\item[(ii)] Hence given $(R_{1}=r_{1},R_{2}=r_{2},\ldots),$ the $(W_{k})$ are conditionally independent. In particular $W_{k}|R_{1}=r_{1},\ldots,R_{k}=r$ equates in distribution to $1/(\beta^{(k)}_{1-\alpha,\alpha}[(1-r)/r]+1)$ with density,
\begin{equation}
\frac{\Gamma(1)}{\Gamma(1-\alpha)\Gamma(\alpha)}{w^{-1}{(1-w)}^{-\alpha}{(w-r)}^{\alpha-1}r^{1-\alpha}},
\label{1overb}
\end{equation}
 $0<r<w<1.$
\end{enumerate}
\end{thm}
\begin{proof}The result follows as an immediate consequence of the forthcoming Lemma~\ref{jointWm} in the case of $m=1.$
\end{proof}

\begin{cor}
For each fixed $k,$ $1-W_{k}$ has the distribution of the first size-biased pick from a
$\mathbb{E}[\mathbb{P}^{[1]}_{\alpha}({(\tilde{G}_{k-1}+\lambda^{\alpha})}^{1/\alpha})]$ mass partition

$$(P_{\ell,k-1}(
\lambda),\ell\ge 1):=\mathrm{Rank}(\tilde{P}_{\ell+k-1}(\lambda)/\prod_{i=1}^{k-1}W_{i}),$$

with $W_{0}:=1.$ Furthermore,
$$
\mathbb{E}[\mathbb{P}^{[1]}_{\alpha}({(\tilde{G}_{k-1}+\lambda^{\alpha})}^{1/\alpha})]:=\mathbb{E}[\mathbb{P}^{[0]}_{\alpha}({(\tilde{G}_{k-2}+\lambda^{\alpha})}^{1/\alpha})],
$$
for $k=2,3,\ldots.$ In particular $(P_{\ell,0},\ell\ge 1)\sim\mathbb{P}^{[1]}_{\alpha}(\lambda)$ and $(P_{\ell,1},\ell\ge 1)\sim\mathbb{P}^{[0]}_{\alpha}(\lambda).$
\end{cor}
\begin{proof} Proof follows from the structure of $(W_{k},R_{k}(\lambda))$ in Theorem~\ref{TheoremP1}. The relation to $\mathbb{P}^{[0]}(\lambda)$ follows from
Porposition~\ref{onetozero}.
\end{proof}

We now show how to recover the stick-breaking result in the $\mathrm{PD}(\alpha,\theta)$ case for $\theta>-\alpha,$ in terms of $(R_{k}).$ The next result follows from Corollary~\ref{PDrecovery} using $Y_{1,-\theta}(0)=G_{1+\theta}/S_{\alpha,\theta}\overset{d}=G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}$
\begin{cor}There are the following results for $\theta>-\alpha.$
\begin{enumerate}
\item[(i)]$(P^{[1]}_{\ell}(G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}))\sim \mathrm{PD}(\alpha,\theta)$
\item[(ii)] The $(R_{k}(G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}))$ are independent $\mathrm{Beta}(\theta+k\alpha,1)$ variables.
\item[(iii)]Hence,
$$
\beta^{(k-1)}_{1-\alpha,\alpha}\frac{1-R_{k}(G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}))}{R_{k}(G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}))}\overset{d}=\frac{\gamma_{1-\alpha}}{\gamma_{\theta+k\alpha}},
$$
a ratio of independent $\mathrm{Gamma}(1-\alpha,1)$ and $\mathrm{Gamma}(\theta+k\alpha,1)$ variables. Hence the variables $W_{k}$ in (\ref{weightsone}) are independent with distribution,
$$
1-W_{k}=\beta_{1-\alpha,\theta+k\alpha}.
$$
\end{enumerate}
\end{cor}
\begin{rem}One may compare the $R_{k}(G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha})),$ with variables denoted $R_{1},R_{2},\ldots$ in \cite{PY97}. See in particular \cite[Theorem 15]{PY97}.
\end{rem}
\begin{rem} $(P_{\ell,k-1}(
G_{{\frac{\theta+\alpha}{\alpha}}}^{1/\alpha}),\ell\ge 1)\sim\mathrm{PD}(\alpha,\theta+(k-1)\alpha)$ for $k=1,2,\ldots.$
This equates to the $\mathrm{PD}(\alpha,\theta)$ Markov Chain on the space of mass partitions by the action of\textit{ size biased deletion} as described in \cite[Chap 6]{PY97}.
\end{rem}

\section{Stick-breaking for $(P_{\ell}^{[m]}(\lambda))\sim \mathbb{P}^{[m]}_{\alpha}(\lambda)$}\label{genm}
We now discuss the case where $m=1,2,3,\ldots$ is a general positive integer, corresponding to $(P_{\ell})|N_{S_{\alpha}}(\lambda)=m,$ when $(P_{\ell})\sim \mathrm{PD}(\alpha,0).$

The joint density of
$(W_{1},\ldots, W_{n},S_{\alpha,n\alpha})|N_{S_{\alpha}}(\lambda)=m,$ as described in Lemma~\ref{Wdensity}, can be expressed as
\begin{equation}
\frac{\lambda^{m-\alpha}{\mbox e}^{-\frac{\lambda s}{\prod_{i=1}^{n}w_{i}}}{\mbox e}^{\lambda^{\alpha}}\alpha^{n}}{[\Gamma(1-\alpha)]^{n}\alpha \Omega_{m}(\lambda^{\alpha})}\prod_{k=1}^{n}w^{-m}_{k}w^{k\alpha-1}_{k}{(1-w_{k})}^{-\alpha}\times s^{m}s^{-n\alpha}f_{\alpha}(s).
\label{jointm}
\end{equation}
\subsection{Description of joint density of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=m$}
\begin{lem}\label{jointWm}The joint density of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=m$ can be expressed
in terms of the following conditional joint density of variables $(W_{1},\ldots,W_{n},R_{1,m},\ldots,R_{n,m})$
$$
\frac{\Omega_{m}(\lambda^{\alpha}/\prod_{l=1}^{n}r^{\alpha}_{l})}{\Omega_{m}(\lambda^{\alpha})}\left[\prod_{i=1}^{n}\frac{w^{-m}_{i}{(1-w_{i})}^{-\alpha}{(w_{i}-r_{i})}^{\alpha-1}r^{m-\alpha}_{i}}{\Gamma(1-\alpha)\Gamma(\alpha)}\right]
f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})
$$
for $0<r_{i}<w_{i}<1, i=1,\ldots,n.$ Specifically a description of the density of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=m$
is obtained by integrating over $(r_{1},\ldots,r_{n}).$
\end{lem}
\begin{proof} Replace $s^{-n\alpha}{\mbox e}^{-\frac{\lambda s}{(\prod_{l=1}^{n}w_{l})}}\prod_{l=1}^{n}w^{l\alpha-1}_{l}$ in (\ref{jointm}) with the integral in Proposition~\ref{integral1}. Integrate with respect to $s^{m}f_{\alpha}(s)$
leading to the evaluation of
$$
\mathbb{E}[S^{m}_{\alpha}{\mbox e}^{-\frac{\lambda}{\prod_{i=1}^{n}r_{i}}S_{\alpha}}]
=\alpha{\mbox e}^{-\frac{\lambda^{\alpha}}{(\prod_{l=1}^{n}r^{\alpha}_{l})}}\lambda^{\alpha-m}\prod_{i=1}^{n}r^{m-\alpha}_{i}\times \Omega_{m}(\lambda^{\alpha}/\prod_{l=1}^{n}r^{\alpha}_{l}).
$$
An augmentation and arrangement of terms leads to the indicated joint density.
\end{proof}
Setting $m=1,$ in Lemma~\ref{jointWm} verifies the results in Theorem~\ref{TheoremP1}.
Furthermore if  $X\overset{d}=1/(\beta^{(k)}_{1-\alpha,\alpha}[(1-r)/r]+1)$
\begin{equation}
\mathbb{E}[X^{-(m-1)}]=r^{-(m-1)}\mathrm{E}[{(\beta_{1-\alpha,\alpha}(1-r)+r)}^{m-1}]
\label{normalising}
\end{equation}
which is the same as
$$
\int_{r}^{1}\frac{w^{-m}{(1-w)}^{-\alpha}{(w-r)}^{\alpha-1}r^{1-\alpha}}{\Gamma(1-\alpha)\Gamma(\alpha)}dw.
$$
This identifies the normalizing constants in Lemma~\ref{jointWm} leading to further descriptions of the
$(W_{1},\ldots,W_{n})$  and $(R_{1,m},\ldots,R_{n,m}).$
\subsection{Description of the random vector $(R_{1,m}(\lambda),\ldots,R_{n,m}(\lambda))$}

Let $(R_{1,m}(\lambda),\ldots,R_{n,m}(\lambda)):=(R_{1,m},\ldots,R_{n,m})$ denote the joint vector of random variables appearing in Lemma~\ref{jointWm}. It then follows from~(\ref{normalising}), that the joint density can be expressed as
\begin{equation}
f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})\frac{\Omega_{m}(\lambda^{\alpha}/\prod_{l=1}^{n}r^{\alpha}_{l})}{\Omega_{m}(\lambda^{\alpha})}
\prod_{i=1}^{n}\mathrm{E}[{(\beta_{1-\alpha,\alpha}(1-r_{i})+r_{i})}^{m-1}]
\label{jointr}
\end{equation}
where, $f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})$ is as in Lemma~\ref{Rdist}
$$
f_{R_{1},\ldots,R_{n}}(\mathbf{r}_{n})=\alpha^{n}\lambda^{n\alpha}{\mbox e}^{\lambda^{\alpha}}{\mbox e}^{-\lambda^{\alpha}/(\prod_{l=1}^{n}r_{l})^{\alpha}}\prod_{l=1}^{n}r^{-(n-l+1)\alpha-1}_{l}.
$$
Furthermore, from~(\ref{jointr}), it follows that $R_{1,m}(\lambda)$ has marginal density
  $$
f_{m}(r_{1}|\lambda)=\frac{\Omega_{m}(\lambda^{\alpha}/r^{\alpha}_{1})}{\Omega_{m}(\lambda^{\alpha})}
\mathrm{E}[{(\beta_{1-\alpha,\alpha}(1-r_{1})+r_{1})}^{m-1}]
f_{R_{1}}(r_{1}|\lambda)
$$
and for each $\ell \ge 1,$ $R_{\ell,m}|R_{1,m}=r_{1},\ldots,R_{\ell-1,m}=r_{\ell-1},\lambda$ has density
$f_{m}(r_{\ell}|\lambda/\prod_{i=1}^{\ell-1}r_{i}).$

\begin{prop}\label{PropPGWm1}
Let $R_{1,m}(\lambda),$ denote a random variable taking values in $[0,1],$ with density,
$$
f_{m}(r|\lambda)=\frac{\Omega_{m}(\lambda^{\alpha}/r^{\alpha})}{\Omega_{m}(\lambda^{\alpha})}
\mathrm{E}[{(\beta_{1-\alpha,\alpha}(1-r)+r)}^{m-1}]
f_{R_{1}}(r|\lambda).
$$
\begin{enumerate}
\item[(i)]Then by augmentation there exists a discrete random variable $N^{(1)}_{m}(\lambda),$ such that given $R_{1,m}(\lambda)=r$
its conditional probability mass function $\mathbb{P}(N^{(1)}_{m}(\lambda)=k|R_{1,m}(\lambda)=r):=q_{m}(k|r),$ can be expressed as
\begin{equation}
q_{m}(k|r)=\frac{\frac{\Gamma(k-\alpha)}{\Gamma(k)\Gamma(1-\alpha)}
{m-1\choose k-1}{(1-r)}^{k-1}r^{m-k}}{\mathrm{E}[{(\beta_{1-\alpha,\alpha}(1-r)+r)}^{m-1}]}
\label{mpmf}
\end{equation}
for $k=1,\ldots,m,$ not depending on $\lambda.$
\item[(ii)]For each $\lambda>0,$ the marginal distribution of $N^{(1)}_{m}(\lambda)$ can be expressed as
$$
q_{k,m}(\lambda):=\mathbb{P}(N^{(1)}_{m}(\lambda)=k)
=\frac{\frac{\Gamma(k-\alpha)}{\Gamma(1-\alpha)}
{m-1\choose k-1}\Omega_{m-k}(\lambda^{\alpha})\alpha\lambda^{\alpha}}{\Omega_{m}(\lambda^{\alpha})}
$$
for $k=1,\ldots,m$
\item[(iii)] $R_{1,m}(\lambda)|N^{(1)}_{m}(\lambda)=k$ has the density of $\lambda/(\lambda+Y_{m,m-k}(\lambda)),$ given by
\begin{equation}
f_{m,k}(r|\lambda):=\frac{\Omega_{m}(\lambda^{\alpha}/r^{\alpha})}{\Gamma(k)\Omega_{m-k}(\lambda^{\alpha})\alpha\lambda^{\alpha}}{(1-r)}^{k-1}{r}^{m-k}f_{R_{1}}(r|\lambda).
\label{rkmdensity}
\end{equation}
\item[(iv)]That is,
$
\lambda/R_{1,m}(\lambda)\overset{d}=\lambda+Y_{m,m-N^{(1)}_{m}(\lambda)}(\lambda).
$
\end{enumerate}
\end{prop}
\begin{proof} The description is a consequence of Lemma~\ref{jointWm},Lemma~\ref{Ymmk} and the density of $Y_{m,m-k}(\lambda)/\lambda$ in (\ref{Ydensity}). These can be used to verify calculations of integrals and normalizing constants above. The results then follow from standard augmentation arguments.
\end{proof}
Let $(N^{(j)}_{m}(x_{j}),x_{j}>0, j=1,2,\ldots),$ denote for fixed $(x_{j})$ a collection of independent random variables having the same marginal distribution as $N^{(j)}_{m}(x_{j})\overset{d}=N^{(1)}_{m}(x_{j})$ as described in Proposition~\ref{PropPGWm1}. In addition let $(Y^{(j)}_{m,\ell}(x_{j}),\ell=0,\ldots,m-1,j=1,2,\ldots)$ denote a collection of independent random variables such that

$$
Y^{(j)}_{m,\ell}(x_{j})\overset{d}=Y_{m,\ell}(x_{j})\overset{d}=\frac{x_{j} G_{m-\ell}}{\tau_{\alpha}\left(x^{\alpha}_{j}+G_{\frac{\ell}{\alpha}-K_{\ell}(x_{j})}\right)}
$$

\begin{cor}Let $(R_{1,m}(\lambda),\ldots,R_{n,m}(\lambda))$ denote the joint vector with distribution described by (\ref{jointr}) or equivalently by $\prod_{\ell=1}^{n}f_{m}(r_{\ell}|\lambda/\prod_{i=1}^{\ell-1}r_{i}).$ Set $\lambda_{0}=\lambda,$ and $\lambda_{j-1}=\lambda/\prod_{i=1}^{j-1}R_{i,m}(\lambda)$ for $j=2,3,\ldots,$ then one may set
$$
\lambda_{j-1}/R_{j}(\lambda)=\lambda_{j-1}+Y^{(j)}_{m,m-N^{(j)}_{m}(\lambda_{j-1})}(\lambda_{j-1})
$$
for $j=1,\ldots,n.$
\end{cor}

\subsection{Description of $(W_{1},\ldots,W_{n})|N_{S_{\alpha}}(\lambda)=m$}

Hereafter, for $j=1,2,\ldots,$ set
$$
N^{(j)}_{m}(\lambda_{j-1})=N_{j,m}(\lambda).
$$

\begin{thm}\label{TheoremPGWm} If $(P^{[m]}_{\ell}(\lambda))\sim\mathbb{P}^{[m]}_{\alpha}(\lambda),$ then the sequence $(\tilde{P}_{k}(\lambda)),$ obtained by size-biased sampling from $(P^{[m]}_{\ell}(\lambda)),$
can be represented as $\tilde{P}_{k}(\lambda)=(1-{W}_{k})\prod_{l=1}^{k-1}W_{l},$
with the following properties;
\begin{enumerate}
\item[(i)] Given $(R_{1,m}(\lambda)=r_{1},R_{2,m}(\lambda)=r_{2},\ldots),$ the $(W_{\ell})$ are conditionally independent,
with respective densities $B_{m,\alpha}(w_{\ell}|r_{\ell}),$ for $\ell=1,\ldots$ expressible as
$$
B_{m,\alpha}(w_{\ell}|r_{\ell})=\frac{1}{\Gamma(1-\alpha)\Gamma(\alpha)}\frac{w^{-m}_{\ell}{(1-w_{\ell})}^{-\alpha}{(w_{\ell}-r_{\ell})}^{\alpha-1}r^{m-\alpha}_{\ell}}{\mathbb{E}[{(\beta_{1-\alpha,\alpha}(1-r_{\ell})+r_{\ell})}^{m-1}]}
$$
for $r_{\ell}<w_{\ell}<1,$ not depending on $\lambda.$
\item[(ii)] The density $B_{m,\alpha}(w|r)$ can be expressed in terms of an m-component mixture and it follows that, given $(R_{1,m}(\lambda)=r,N_{1,m}=k)$ for $k=1,\ldots,m.$
\begin{equation}
W_{1}\overset{d}={[\beta^{(1)}_{({k-\alpha},\alpha)}{(1-r)}/{r}+1]}^{-1}
  \label{weights3}
\end{equation}
with density, for $r<w<1,$
$$
\frac{\Gamma(k)}{\Gamma(k-\alpha)\Gamma(\alpha)}{w^{-k}{(1-w)}^{k-\alpha-1}{(w-r)}^{\alpha-1}r^{k-\alpha}{(1-r)}^{1-k}}.
$$
\item[(iii)] Hence, the $(W_{\ell})$ are generally dependent random variables represented as
\begin{equation}
W_{\ell}:={[\beta^{(\ell)}_{N_{\ell,m}(\lambda)-\alpha,\alpha}[(1-R_{\ell,m}(\lambda))/R_{\ell,m}(\lambda)]+1]}^{-1}
 \label{weights2}
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}These results follow from Lemma~\ref{jointWm} and the descriptions in Proposition~\ref{PropPGWm1}.
\end{proof}
\begin{cor}\label{number}Consider $(N_{\ell,m}(\lambda), \ell\ge 1),$ as described in Theorem~\ref{TheoremPGWm} and (\ref{weights2}). Then  $(N_{\ell,m}(\lambda), \ell\ge 1)|((W_{\ell},R_{\ell,m}(\lambda)),\ell\ge 1)$ are conditionally independent such that each $N_{\ell,m}(\lambda)-1$ has a $\mathrm{Binomial}(m-1, (1-W_{\ell}))$ distribution.
\end{cor}

\subsection{Recovering $\mathrm{GEM}(\alpha,\theta)$}
The $\mathrm{GEM}(\alpha,\theta)$ distribution is recovered by mixing with respect to the distribution of
$Y_{m,-\theta}(0)\overset{d}=G_{m+\theta}/S_{\alpha,\theta}\overset{d}=G^{1/\alpha}_{\theta/\alpha+K_{m}},$ which has density, for $y>0,$
\begin{equation}
\frac{\alpha\Omega_{m}(y^{\alpha})}{\Gamma(m+\theta)\mathbb{E}[S^{-\theta}_{\alpha}]}y^{\alpha+\theta-1}{\mbox e}^{-y^{\alpha}}.
\label{GEMm}
\end{equation}
It follows from Corollary~\ref{number} that $(N_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}), \ell\ge 1),$  are independent with respective probability mass functions for each $\ell\ge 1,$
$$
{m-1\choose k-1}\mathbb{E}[{(1-W_{\ell})}^{k-1}W_{\ell}^{m-k}]
$$
where $W_{\ell}~\sim \mathrm{Beta}(\theta+\ell\alpha,1-\alpha),$ see also~\cite{Pit95}. $(W_{\ell},\ell\ge 1)|(N_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}), \ell\ge 1),$ are independent such that $W_{\ell}|N_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta})=k,$ has a $\mathrm{Beta}(\theta+\ell\alpha+m-k,k-\alpha)$ distribution. $(R_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}), \ell\ge 1)|(N_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}), \ell\ge 1)$ are independent such that  $R_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta})|N_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta})=k,$ has a $\mathrm{Beta}(\theta+\ell\alpha+m-k,k)$ distribution.
\subsubsection{Distribution of $(R_{\ell,m},\ell\ge 1)$ in the $\mathrm{GEM}(\alpha,\theta)$ case}
It is interesting to highlight the distribution of
$(R_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}),\ell \ge 1),$ which we don't believe is well known even in this ideal setting.
\begin{cor}Let $(R_{1,m}(\lambda),\ldots,R_{n,m}(\lambda)),$ for each $n,$ be the variables with joint density in~
(\ref{jointr}). Then in the $\mathrm{GEM}(\alpha,\theta)$ where the $(W_{\ell})$ are independent $\mathrm{Beta}(\theta+\ell\alpha,1-\alpha),$ One has that the collection of variables $(R_{\ell,m}(G_{m+\theta}/S_{\alpha,\theta}),\ell \ge 1),$
are independent with respective distributions on $[0,1],$
$$
\frac{\alpha\Gamma(m+\theta+\ell \alpha)\mathbb{E}[S^{-(\theta+\ell\alpha)}_{\alpha}]}{\Gamma(m+\theta+(\ell-1)\alpha)\mathbb{E}[S^{-(\theta+(\ell-1)\alpha)}_{\alpha}]}r^{\theta+\ell\alpha-1}_{\ell}\mathbb{E}[{(\beta_{1-\alpha,\alpha}(1-r_{\ell})+r_{\ell})}^{m-1}]
$$
where $\mathbb{E}[S^{-\theta}_{\alpha}]=\Gamma(\theta/\alpha+1)/\Gamma(\theta+1).$
\end{cor}
\begin{proof}The is obtained by integrating~(\ref{jointr}) with respect to (\ref{GEMm}) and deducing the right normalization which is obscured by telescoping.
\end{proof}
\subsection{Results for $\alpha=1/2, \mathbb{P}^{[m]}_{1/2}(\lambda)$}\label{m5}
When $\alpha=1/2,$ $(P_{\ell})\sim \mathrm{PD}(1/2,0),$  corresponds to the laws of the ranked lengths of Brownian motion. $S_{1/2}(1)=S_{1/2}\overset{d}=1/(4G_{1/2})$ and hence
$S_{1/2,0}(\lambda)\overset{d}=\tau_{1/2}(\sqrt{\lambda})/\lambda$ has an Inverse Gaussian distribution, denoted $\mathrm{IG}(\frac{1}{2\lambda},\frac{1}{2})$ with explicit density ${\mbox e}^{\sqrt{\lambda}}{\mbox e}^{-\lambda t-1/(4t)}t^{-3/2}/(2\sqrt{\pi}).$ Furthermore $S_{1/2,m}(\lambda),$ in addition to its representation in~(\ref{Sm}), which allows it to be easily sampled, has, as can be read from \cite[Section 4]{Pit99}, a Generalized Inverse Gaussian distribution with density, satisfying for all $\theta>-1/2,$ where $S_{1/2,\theta}\overset{d}=1/(4G_{\theta+1/2}),$
\begin{equation}
f^{[m]}_{1/2}(t|\lambda)=\mathbb{P}(\frac{1}{4G_{\theta+1/2}}\in dt|4G_{\theta+1/2}G_{m+\theta}=\lambda)/dt.
\label{halfdensity}
\end{equation}
where
\begin{equation}
Y_{m,-\theta}(0)\overset{d}=4G_{\theta+1/2}G_{m+\theta}\overset{d}=G^{2}_{2\theta+K_{m}},
\label{halfmap}
\end{equation}
and $K_{m}$ is the number of blocks in a partition of $[m]$ under a $\mathrm{PD}(1/2,\theta)$ distribution.

Following \cite[Section 8]{Pit02} and \cite[Section 4.5]{Pit06}, let $\sqrt{2}L_{1}\overset{d}=S^{-1/2}_{1/2},$ where $L_{1}$
denotes the local time of Brownian motion starting at $0$ up till time $1$, and let $B_{1}$ denote  Brownian motion at time 1, which has a standard Normal distribution. Then $(P_{\ell})|L_{1}=s$ has the same distribution as $(P_{\ell})|S_{1/2}=s^{-2}/2$ which is $\mathrm{PD}(1/2|s^{-2}/2).$ Although perhaps not well known in the statistical literature, the result of Aldous and Pitman~\cite[Corollary 5]{AldousPit} leads to an explicit description of the sequence which we denote as $(\hat{P}_{\ell}(s),\ell\ge 1),$ which is the size biased ordering of a  mass partition with distribution
$\mathrm{PD}(1/2|s^{-2}/2).$
In particular the first size biased pick $\hat{P}_{1}(s),$ satisfies
$$
\hat{P}_{1}(s)\overset{d}=\frac{B^{2}_{1}}{B^{2}_{1}+s^{2}}
$$
and, for each $\ell\ge 1$,
\begin{equation}
\hat{P}_{\ell}(s)=\frac{s^{2}}{Q_{\ell-1}+s^{2}}-\frac{s^{2}}{Q_{\ell}+s^{2}},
\label{condstick}
\end{equation}
where $Q_{0}=0,$ and $Q_{\ell}=\sum_{i=1}^{\ell}X_{i}$ for $X_{i}$ independent with common distribution equivalent to $B^{2}_{1}.$ The representations in~(\ref{condstick}) lead easily to representations of the size biased ordering of any Poisson Kingman distribution with mixing derived from the case of a $1/2$-stable subordinator, see~\cite{Pit02}.  Our relevant result takes the form described below.

\begin{cor}\label{cor5}Let $(P_{\ell}(\lambda))\sim \mathbb{P}^{[m]}_{1/2}(\lambda)$, with corresponding $1/2$-diversity $\sqrt{2}L_{1,m}(\lambda)\overset{d}=S^{-1/2}_{1/2,m}(\lambda).$ Then for $m=0,1,2,\ldots$ its size-biased re-arrangement
$(\tilde{P}_{\ell}(\lambda))$ is equivalent in (joint) distribution to
$$
(\hat{P}_{\ell}(L_{1,m}(\lambda)); \ell\ge 1),
$$
which is specified by~(\ref{condstick}). Furthermore, as indicated by~(\ref{halfdensity}) and (\ref{halfmap}),
 $(\hat{P}_{\ell}(L_{1,m}(G^{2}_{2\theta+K_{m}})))\sim \mathrm{GEM}(1/2,\theta)$.
\end{cor}

\begin{rem}The case of $m=0,$ in Corollary~\ref{cor5}, as well as our results in Theorem~\ref{Wcase00} with $\alpha=1/2$ and $m=0,$ corresponds to the Normalized Inverse Gaussian(NIG) case.  One may compare these with the representations first obtained in~\cite{Fav1}.
\end{rem}

redpajama_set_name: RedPajamaArXiv


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PPL, Belgium's Elia form offshore wind transmission team in New England
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Pennsylvania-based energy provider PPL Corporation (NYSE:PPL) and Belgium's Elia Group SA (EBR:ELI) intend to collaborate on the development, construction and operation of onshore transmission assets for offshore wind farms in the New England region of the US.
The two companies' respective subsidiaries, PPL TransLink Inc and WindGrid, have entered into a memorandum of understanding that outlines their plan to combine their expertise in order to offer innovative transmission solutions, according to an announcement on Monday.
Moreover, PPL and WindGrid are preparing to jointly respond later this month to a request for information (RFI) issued by the states of Massachusetts, Connecticut, Rhode Island, Maine and New Hampshire. These states seek feedback from interested stakeholders in relation to transmission system changes and upgrades required to integrate renewables into their grids in the future. Recent studies cited in the RFI suggest that up to 30 GW of offshore wind capacity will be deployed off the coast of New England by 2050.
While PPL is experienced in building and operating large-scale onshore transmission in the US, having spent more than USD 9 billion (EUR 9.25bn) on such projects over the past decade, its Belgian partner has been involved in the connection to its onshore grid of 14 offshore wind farms in Europe to date. At present, Elia operates three high-voltage direct current (HVDC) subsea cable interconnectors and is developing the world's first energy islands.
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