Non-Conventional Limits of Random Sequences Related to Partitions of Integers

TL;DR

This paper investigates the limit distribution of normalized sums related to restricted integer partitions, revealing that, unlike typical cases, the limit can be a bounded non-normal random variable with explicitly characterized properties.

Contribution

The paper demonstrates that the normalized sum of certain partition-related random variables converges to a bounded distribution, expanding understanding beyond the classical normal limit.

Findings

Limit distribution is bounded, not necessarily normal.

Explicit range and properties of the limit distribution are derived.

Tools include moment generating functions and cumulants.

Abstract

We deal with a sequence of integer-valued random variables $\{Z_N\}_{N=1}^{\infty}$ which is related to restricted partitions of positive integers. We observe that $Z_N=X_1+ \ldots + X_N$ for independent and bounded random variables $X_j$'s, so $Z_N$ has finite mean ${\bf E}Z_N$ and variance ${\bf Var}Z_N$. We want to find the limit distribution of ${\hat Z}_N=\left(Z_N-{\bf E}Z_N\right)/{\sqrt{{\bf Var}Z_N}}$ as $N \to \infty.$ While in many cases the limit distribution is normal, the main results established in this paper are that ${\hat Z}_N \overset{d}{\to} Z_{*},$ where $Z_{*}$ is a bounded random variable. We find explicitly the range of values of $Z_*$ and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments and cumulants of the random variables involved. Useful related topics are also discussed.