Renewal and Regenerative Processes: short review

TL;DR

This paper reviews key results in the theory of stochastic processes, focusing on Markov, renewal, regenerative, and stationary processes, highlighting their properties, conditions for stationarity, and applications across various fields.

Contribution

It provides a concise overview of fundamental results in regenerative and renewal processes, emphasizing their importance and applications, serving as a foundation for further research.

Findings

Identification of regeneration points and stationarity conditions

Analysis of Markov and regenerative process properties

Highlighting applications in cryptography, queueing, and Monte Carlo methods

Abstract

This document presents a compilation of results related to the theory of stochastic processes, with a specific focus on Markov processes, regenerative processes, renewal processes, and stationary processes. The relevance of these topics lies in the ability to identify regeneration points and the necessary conditions to ensure the stationarity of the process. The study begins with a review of Markov chains and continues with the analysis of processes that satisfy the strong Markov property. Subsequently, it delves into renewal processes, regenerative processes, and finally, stationary regenerative processes, highlighting the results presented by Thorisson [22]. This work is not intended to be exhaustive but aims to provide a solid foundation for further deepening the knowledge of these processes, given their broad range of applications in cryptography [16], queueing theory [15], and Monte Carlo methods [23]. Additionally, the importance of Poisson-type processes is emphasized due to their numerous applications (see [8]).