Perturbation approach to scaled type Markov renewal processes with infinite mean

TL;DR

This paper investigates the long-term behavior of scaled type Markov renewal processes, especially focusing on cases with infinite mean renewal times, revealing how the asymptotic distribution depends on the finiteness of expectations and the tail behavior of renewal laws.

Contribution

It introduces a perturbation approach to analyze the asymptotic distribution of Markovian parameters in scaled renewal processes with infinite mean, extending classical results to regularly varying laws.

Findings

Finite mean case: explicit limit distribution derived.

Infinite mean case: limits depend on regular variation exponent.

Results highlight the role of tail behavior in asymptotics.

Abstract

Scaled type Markov renewal processes generalize classical renewal processes: renewal times come from a one parameter family of probability laws and the sequence of the parameters is the trajectory of an ergodic Markov chain. Our primary interest here is the asymptotic distribution of the Markovian parameter at time t \to \infty. The limit, of course, depends on the stationary distribution of the Markov chain. The results, however, are essentially different depending on whether the expectations of the renewals are finite or infinite. If the expectations are uniformly bounded, then we can provide the limit in general (beyond the class of scaled type processes), where the expectations of the probability laws in question appear, too. If the means are infinite, then - by assuming that the renewal times are rescaled versions of a regularly varying probability law with exponent 0 \leq alpha \leq 1 - it is the exponent a which emerges in the limits.