Characteristics of asymmetric switch processes with independent switching times

TL;DR

This paper investigates the properties of asymmetric switch processes with independent switching times, focusing on their expected values, covariance, and conditions for valid interval distributions, with implications for modeling random switching attempts.

Contribution

It introduces a stochastic representation linking expected value monotonicity to interval distributions, providing new insights into asymmetric switch processes with independent switching times.

Findings

Expected value functions' monotonicity relates to interval distribution representations.

A stochastic model with geometric number of summands explains switching attempts.

Conditions for valid interval distributions are derived for applications.

Abstract

The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distributions of the time in these states may differ. Two versions of the process are considered: a non-stationary version that starts with a switch at time zero and a stationary one constructed from the non-stationary one. Characteristics of these two processes, such as the expected values and covariance, are investigated. The main results show an equivalence between the monotonicity of the expected value functions and the distribution of the intervals having a stochastic representation in the form of a sum of random variables, where the number of terms follows a geometric distribution. This representation has a natural interpretation as a model in which switching attempts may fail at random. From these results, conditions are derived when these characteristics lead to valid interval distributions, which is vital in applications.