A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments

TLDR

The paper studies a counting process whose increments are non‑negative random variables whose distribution depends only on the cumulative sum of previous increments. The authors aim to extend ordinary renewal theory by analyzing the expected counting function \(H(t)=\mathbb{E}[N(t)]\). They show that the derivative \(h(t)=dH/dt\) satisfies an extended renewal equation, examine its existence and properties, and illustrate the theory with several examples. They prove that, under suitable conditions, \(h(t)\) exists uniquely, may be constant, monotone decreasing or increasing, and describe its asymptotic behavior as \(t\to\infty\).

Abstract

Let N ( t ) be a counting process associated with a sequence of non-negative random variables ( X j ) 1 ∞ where the distribution of X n +1 depends only on the value of the partial sum S n = Σ j=1 n X j . In this paper, we study the structure of the function H ( t ) = E [ N ( t )], extending the ordinary renewal theory. It is shown under certain conditions that h ( t ) = ( d/dt ) H ( t ) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h ( t ) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h ( t ) and H ( t ) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.