On Sums of Monotone Random Integer Variables

TL;DR

This paper investigates the probability that sums of independent monotone integer variables equal a specific value, providing sharp estimates near the mean and extending results to tail probabilities under stronger conditions.

Contribution

It introduces estimates for point probabilities of sums of monotone variables without requiring identical distribution, and extends to tail estimates using exponential tilting under strong monotonicity.

Findings

Sharp estimates near the mean for sum probabilities

Extension to tail probabilities with strong monotonicity

Applicable to various common discrete distributions

Abstract

We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,\pi]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the trick of \emph{exponential tilting}, we obtain sharp estimates for the point probabilities in the tail under a slightly stronger assumption on the random integer variables which we call strong monotonicity.