arXiv subjects

Irfan Alam

Publications and source records attributed to Irfan Alam.

7 recordsLinked to original sources

arXiv2025

Generalizing the de Finetti--Hewitt--Savage theorem

A sequence of random variables is called \textit{exchangeable} if its joint distribution is invariant under permutations of indices. The original formulation of de Finetti's theorem roughly says that any exchangeable sequence of $\{0,1\}$-valued random variables can be thought of as a mixture of independent and identically distributed sequences. Hewitt and Savage were able to obtain the same conclusion for exchangeable sequences of random variables taking values in more general state spaces under some topological conditions. Using tools from nonstandard analysis we prove that an exchangeable sequence of Radon-distributed random variables taking values in any Hausdorff state space must be representable as a mixture of sequences of independent and identically distributed random variables. Our presentation of this work follows the style of \textit{lecture notes} intended for broad graduate-level mathematical audiences -- the main body of the manuscript starts with a historically grounded introduction to the problem, foreshadowing our techniques that are developed via a series of appendices. These techniques are used to provide self-contained proofs of our main results in a short section following the introduction. We have provided a self-contained philosophically motivated introduction to nonstandard analysis in the first appendix, thus rendering first courses in measure theoretic probability and point-set topology as the only prerequisites for the work. This introduction aims to develop some new ideologies about the subject that might be of interest to mathematicians, philosophers, and mathematics educators alike. One highlight of the rest of the appendices is a new generalization of Prokhorov's theorem in the setting of the space of all probability measures on arbitrary Hausdorff spaces.

math.PR↗

arXiv2025

Matroid intersection and packing/covering conjectures are true in the class of finitary matroids

Given two finite matroids on the same ground set, a celebrated result of Edmonds says that the ground set can be partitioned into two disjoint subsets in a manner that there is a common independent set in both matroids whose intersection with the first subset spans that subset in the first matroid, and whose intersection with the second subset spans that subset in the second matroid. There is a longstanding conjecture regarding the situation of two matroids defined on the same infinite ground set. Infinite matroids were only recently axiomatized in the early 2010s in the work of Bruhn et al., while the conjecture had been proposed during the 1990s for the class of structures that are now called finitary matroids, which are matroids all of whose circuits are finite sets. The packing/covering conjecture, due to Bowler and Carmesin, is a related conjecture in the sense that it is true in the class of all matroids if and only if the matroid intersection conjecture is true in the class of all matroids. Given any family of matroids on the same ground set, the conjecture asks if it is possible to partition the ground set into two disjoint subsets in a way such that the corresponding family of matroids restricted to the first subset admits a packing while the family of matroids contracted to the second subset admits a covering. We prove both of these conjectures in the class of finitary matroids. Our main tool is nonstandard analysis, specifically the technique of iterated nonstandard extensions. Roughly, we first embed any infinite matroid inside a hyperfinite matroid defined on a subset of the nonstandard extension of the original ground set, and we iteratively nonstandardly extend the hyperfinite structure again in order to prove results in the internal universe that can be directly transferred to obtain results about the matroid(s) we started with.

math.CO↗

arXiv2023

Entropic exercises around the Kneser-Poulsen conjecture

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.

math.MG↗

arXiv2020

A nonstandard proof of de Finetti's theorem

We give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random variables. We use combinatorial arguments to show that this probability distribution is induced by a hyperfinite sample mean.

math.PR↗

arXiv2020

Limiting Probability Measures

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof. A review of the requisite nonstandard analysis is provided.

math.PR↗

arXiv2020

Limiting spherical integrals of bounded continuous functions

We use nonstandard analysis to study the problem of expressing a Gaussian integral in terms of the limiting behavior of a sequence of spherical integrals. Peterson and Sengupta proved that if a Gaussian measure $μ$ has full support on a finite-dimensional Euclidean space, then the expected value of a bounded measurable function on that domain can be expressed as a limit of integrals over spheres $S^{n-1}(\sqrt{n})$ intersected with certain affine subspaces of $\mathbb{R}^n$. This allows one to realize the Gaussian Radon transform of such functions as a limit of spherical integrals. Using nonstandard analysis, we study such limits in terms of Loeb integrals over a single hyperfinite dimensional sphere. This nonstandard geometric approach generalizes the known limiting result for bounded continuous functions to the case when the Gaussian measure is not necessarily fully supported. We also present an asymptotic linear algebra result needed in the above proof.

math.PR↗

arXiv2019

An introduction to the probabilistic method through the Lovász Local Lemma

We illustrate the use of probability theory in existential proofs, focusing on the Lovász Local Lemma. This result gives a lower bound for the probability of avoiding a suitable finite collection of events. We describe some applications of this result in hypergraph packing and Latin transversals.

math.CO↗