Occupation time distributions for the telegraph process

TL;DR

This paper derives explicit occupation time distributions for the one-dimensional telegraph process, including long-term limits that follow the arcsine law, and extends these results to more general occupation functionals.

Contribution

It provides explicit formulas for occupation time distributions of the telegraph process and extends limit theorems to broader occupation-type functionals.

Findings

Explicit distribution of occupation time derived

Long-term limits follow the arcsine law

Results extended to general occupation functionals

Abstract

For the one-dimensional telegraph process, we obtain explicit distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of sub-normal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.