Characteristics of the switch process and geometric divisibility

TL;DR

This paper explores the relationship between switching time distributions and the expected value function in a switch process, providing explicit relations and applications to statistical physics.

Contribution

It derives explicit relations between switching time distributions and expected value functions under monotonicity assumptions, linking geometric divisibility to non-negative decreasing functions.

Findings

Geometric divisible switching times correspond to non-negative decreasing expected value functions.

Explicit relation established between switching time distribution and autocovariance function.

Results applicable to approximation methods in statistical physics.

Abstract

The switch process alternates independently between 1 and -1, with the first switch to 1 occurring at the origin. The expected value function of this process is defined uniquely by the distribution of switching times. The relation between the two is implicitly described through the Laplace transform, which is difficult to use for determining if a given function is the expected value function of some switch process. We derive an explicit relation under the assumption of monotonicity of the expected value function. It is shown that geometric divisible switching time distributions correspond to a non-negative decreasing expected value function. Moreover, an explicit relation between the switching time distribution and the autocovariance function of the switch process stationary counterpart is obtained, which allows parallel results for the stationary switch process and its covariance function. These results are applicable to approximation methods in statistical physics, and an example is presented.