Asymptotic Stochastic Comparison of Random Processes

TL;DR

This paper introduces the concept of asymptotic stochastic order for random processes, defining four types and exploring their properties, relations, and conditions for specific stochastic processes.

Contribution

It extends stochastic comparison methods to random processes by defining asymptotic orders and analyzing their properties and interrelations.

Findings

Four asymptotic stochastic orders are defined.

Properties and interrelations of these orders are discussed.

Sufficient conditions for the orders to hold are derived.

Abstract

Several methods are available in the literature to stochastically compare random variables and random vectors. We introduce the notion of asymptotic stochastic order for random processes and define four such orders. Various properties and interrelations of the orders are discussed. Sufficient conditions for these orders to hold for certain stochastic processes, evolving from some statistical entities of interest, are derived.