#probability-theory — Public Fediverse posts

#paperOfTheDay is "Correlation functions and zeros of a Gaussian power series and Pfaffians" from 2013.
This paper is a generalisation of the study of random polynomials: They consider random power series, i.e. polynomials with infinitely many terms. These have (almost always) a radius of convergence of unity, so that it only makes sense to study them in the domain (-1,1). There is an accumulation of zeros (=roots) close to the boundaries of this interval.
Given that the coefficients of the power series are random, so are the locations of zeros. The positive locations form an infinite sequence of random numbers, a point process. As such, one can ask about the mean, variance, and all other correlation functions. The main result of the article is that these quantities are given by a Pfaffian (which is an algebraic object similar to a determinant) of some explicitly known matrices.
I got interested in this observation because Pfaffians also show up in #quantumFieldTheory . For example, Isserlis theorem (sometimes called Wicks theorem by physicists) says that the expectation of a product of Gaussian variables is the Pfaffian of their covariances. Or, Pfaffians show up as the integrands in #FeynmanIntegral s in topological field theories.
#mathematics #probabilityTheory
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-18/issue-none/Correlation-functions-for-zeros-of-a-Gaussian-power-series-and/10.1214/EJP.v18-2545.full

Lehrer and Lobachevsky

I couldn’t resist adding a little anecdote by way of a postscript to yesterday’s item about the the late Tom Lehrer. I didn’t know anything about this story until yesterday when I saw it as a thread on Bluesky (credit to @opalescentopal). The whole thread can be read here, so I’ll just give you a short summary and add a bit of context.

Tom Lehrer’s debut album, Songs by Tom Lehrer, released in 1953, contained a number called Lobachevsky. If you don’t know the song then you can listen to it, for example, here. This song contains this verse:

I am never forget the day
I am given first original paper to write
It was on "Analytic and Algebraic Topology
Of Locally Euclidean Metrizations
Of Infinitely Differentiable Riemannian Manifolds"
Bozhe moi!

That’s relevant to what follows.

In 1957, while he was still working as a mathematician, Lehrer co-wrote a paper for the U.S. National Security Agency, with R.E. Fagan, with the title Gambler’s Ruin With Soft-Hearted Adversary, the full text of which can be found here. For those of you unaware, the Gambler’s Ruin problem is an important problem in the theory of probability. The paper was an internal document but was unclassified. It was later published, in 1958, with some modifications under the title Random Walks with Restraining Barrier as Applied to the Biased Binary Counter.

The 1957 paper was filed away, attracting little attention until 2016 when the person who wrote the Bluesky thread looked at it and noticed something strange. The reference list contains six papers, indexed numerically. References [1], [2] and [4] are cited early on in the paper, and references [5] and [6] somewhat later. But nowhere in the text is there any mention of reference [3]. So what is reference [3]? Here it is:

(It’s a pity about the spelling mistake, but there you go.) Although the song Lobachevsky had been written a few years before the Gambler’s Ruin paper, and had proved very popular, nobody had spotted the prank until 2016. This is episode is testament to Lehrer’s mischievous sense of humour, and to his patience. He made a joke and then kept quiet about it for almost 60 years, waiting for the payoff!

P.S. The Lobachevsky reference was omitted from the modified paper published in 1958.

#AnalyticAndAlgebraicTopologyOfLocallyEuclideanMetrizationsOfInfinitelyDifferentiableRiemannianManifolds_ #GamblerSRuin #NicolaiLobachevsky #ProbabilityTheory #SongsByTomLehrer #TomLehrer

An Introduction to Stochastic Calculus

https://bjlkeng.io/posts/an-introduction-to-stochastic-calculus/

#HackerNews #StochasticCalculus #Introduction #MathFinance #ProbabilityTheory #Education

2025 is looking like a great year for work at the intersection of #categorytheory, #systemstheory, #controltheory, #machinelearning and #probabilitytheory, this thread will be a very biased collection of works (in no specific order) I'm hoping to read as soon as possible!

Starting with:
"Logical Aspects of Virtual Double Categories"
https://mastoxiv.page/@arXiv_mathCT_bot/113921693949589956

I don't always share #videos of a #dragqueen giving a #tierlist regarding the #films and #games found on #OsamaBinLaden laptop, but when I do, it's Sunday December 29th 2024... I'm not expecting this to happen again. The #ProbabilityTheory be damned.

This #toot brought to you by the #WokeDetector. Say "up yours, you #woke #moralists" to games that try to groom your kids with the trans liberal agenda.

I Ranked Every Movie AND Show On Osama Bin Laden's Hard Drive
https://youtube.com/watch?v=FIM_aG8ZsVA

Given that I am surrounded by Mathematicians here, let me ask for help for what should be a simple problem I can't seem to be able to solve:
Assume you have n fair dice with m faces (i.e. each can roll an integer from 1 to m with a uniform probability). You roll all n, and keep the k (with 0<k<=n) highest results. What is the probability that the sum of the k dice you kept is X?
(If one keeps all the dice, probability-generating functions give the answer straightforwardly. If I roll 2 dice and keep 1 I can easily enumerate the outcomes and calculate the probabilities, but I am stumped by the general case).

#ProbabilityTheory #IShouldKnowHowToSolveThisButIDoNot 😞

#FinishedReading #Boole 's Laws of Thought, whose final chapters partly shift attention from #logic to #probabilityTheory and #philosophyOfScience . I don't have a strong sense of his historical standing in either discipline, although for probability there is https://en.m.wikipedia.org/wiki/Boole%27s_inequality . The attraction of probability is clear, with its range of values from 0 to 1, use of '1 -' for negation, multiplication for conjunction (of independent events) etc. There are rhymes here with his logic at least

There are many situations in the real world where small initial differences can easily grow into very large differences just out of pure chance.
Since we are on a social network, let's create a toy model* where a number of posts all have the same probability to be reposted/shared/boosted by any person seeing them. Since the more people see a post, the more people have a chance of boosting it, the posts with more visibility are also the ones that are likely to gain more visibility. So small initial fluctuations (just one or two extra boosts at the beginning) can lead a post to skyrocket in popularity, even though it is not intrinsically "better" than any of the other.
If we simulate this process numerically and make a histogram of the result, we see that the distribution of how many boosts a post had rapidly grows a tail, with most posts having no visibility whatsoever, and a few having a LOT more than the average.
#ITeachPhysics #ProbabilityTheory #ToyModel

* In the #Physics jargon, a "toy model" is a very simple (often unrealistic) model, which nevertheless capture the essence of the problem, without being burdened by all the real world complications. If you ever heard about spherical cows in vacuum, that is a toy model!

James Bernoulli on #ProbabilityTheory and the application of #statistics

"I cannot conceal the fact here that in the specific application of these rules, I foresee many things happening which can cause one to be badly mistaken if he does not proceed cautiously." -1713, pt 4, cpt III

1In 1654, letters between Blaise Pascal and Pierre de Fermat established the basis of probability theory. #Science #History #Poetry #Mathematics #ProbabilityTheory (https://sharpgiving.com/thebookofscience/items/p1654.html)