Convexity and monotonicity of the probability mass function of the Poisson distribution of order $k$

TL;DR

This paper investigates the convexity and monotonicity properties of the probability mass function of the Poisson distribution of order k, providing bounds and conjectures for the parameter lambda that ensure decreasing pmf behavior.

Contribution

It establishes the absolute monotonicity of the pmf in the first block and analyzes the decreasing and concave nature in the second block, proposing bounds for lambda.

Findings

Elements in the first block form an absolutely monotonic sequence.

For small lambda, the second block sequence is decreasing and concave.

Numerical methods suggest an optimal bound for lambda ensuring decreasing pmf.

Abstract

This note focuses on the properties of two blocks of elements of the probability mass function (pmf) of the Poisson distribution of order $k\ge2$. The first block is the elements for $n\in[1,k]$ and the second block is the elements for $n\in[k+1,2k]$. It is proved that elements in the first block form an ``absolutely monotonic sequence'' by which is meant that all the finite differences of the sequence are positive. Next, the properties of the elements in the second block are analyzed. It is shown that for sufficiently small $\lambda>0$, the sequence of elements for $n\in[k+1,2k]$ is strictly decreasing and also concave. The purpose of the analysis is to help determine a supremum value for $\lambda$, such that the pmf of the Poisson distribution of order $k\ge2$ decreases strictly for all $n \ge k$. A conjectured criterion for the supremum is given. Numerical calculations indicate it is the optimal bound, i.e.~the supremum. In addition, a simple expression is proposed, based on numerical calculation, which is sufficient (but not necessary) and is a good approximation for the supremum.