Characteristics of asymmetric switch processes with independent switching times

Abstract

The asymmetric switch process is a binary stochastic process that alternates between the values one and minus one, where the distribution of the time in these states may differ. In this sense, the process is asymmetric, and this paper extends previous work on symmetric switch processes. Two versions of the process are considered: a non-stationary one that starts with either the one or minus one at time zero and a stationary version 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.