We give an elementary proof of the local central limit theorem for independent, non-identically distributed, integer valued and vector valued random variables.

An elementary proof of the local central limit theorem

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fermat and was originally formalized by Jakob Bernoulli, Abraham DeMoivre and Pierre-Simon Laplace. Here we describe the historical roots of the local limit theorem. We describe how it was supplanted by the central limit theorem in applications. Then we review the revival started by B. V. Gnedenko and we describe modern developments.

The Local Limit Theorem: A Historical Perspective

Equivalence of ensembles for two-species zero-range invariant measures

We consider the effect of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables from one lattice to another in the framework of the generalised allocation scheme.

On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme

Let G = (G, +) be a compact connected abelian group, and let μG denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound μG(A + B) ≥ min(μG(A) + μG(B), 1) whenever A and B are compact subsets of G, and A + B:= {a + b: a ∈ A, b ∈ B} denotes the sumset of A and B. Clearly one has equality when μG(A) + μG(B) ≥ 1. Another way in which equality can be obtained is when A = φ−1(I) and B = φ−1(J) for some continuous surjective homomorphism φ: G → ℝ/ℤ and compact arcs I, J ⊂ ℝ/ℤ. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then A and B are close to one of the above examples. We also give a more “robust” form of this theorem in which the sumset A + B is replaced by the partial sumset A +eB:= {1A * 1B ≥ e} for some small e > 0. In a subsequent paper with Joni Teravainen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.

A. B. Mukhin

Papers

Local Limit Theorems for Lattice Random Variables