The associated random walk and martingales in random walks with stationary increments

TL;DR

This paper generalizes the concepts of associated random walk and Wald martingale from i.i.d. increments to stationary ergodic increments, including Markovian and Gaussian cases, with applications in queueing theory.

Contribution

It introduces a broader framework for associated random walks and Wald martingales in stationary ergodic settings, extending previous i.i.d. assumptions.

Findings

Extended the notion of associated random walk to stationary ergodic increments.

Provided examples with Markovian and Gaussian increments.

Applied the theory to queueing systems.

Abstract

We extend the notion of the associated random walk and the Wald martingale in random walks where the increments are independent and identically distributed to the more general case of stationary ergodic increments. Examples are given where the increments are Markovian or Gaussian, and an application in queueing is considered.