Properties and conjectures regarding discrete renewal sequences

TL;DR

This paper explores properties of discrete renewal sequences derived from positive integer-valued random variables, focusing on their extrema and proposing a conjecture about polynomial bounds.

Contribution

It introduces a conjecture about universal polynomial envelopes for the minima of renewal sequences and proves it in a special case.

Findings

Minima and maxima occur within the support of the underlying distribution.

A conjecture about polynomial envelopes bounding the minima.

Proof of the conjecture in a specific case.

Abstract

In this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of the underlying random variable. In particular we study the minima and the maxima of these sequences and prove that they are attained for indices of the sequences smaller or equal than the support of the underlying random variable. Noting that the minimum itself is a minimum of multi-variate polynomials we conjecture that one universal polynomial envelopes the minimum from below and that it is maximal in some sense and largest in another. We prove this conjecture in a special case.