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Characteristics of the switch process and geometric divisibility

Part of: Stochastic processes Special processes Distribution theory - Probability

Published online by Cambridge University Press:  06 November 2023

Henrik Bengtsson Open the ORCID record for Henrik Bengtsson [Opens in a new window]
Henrik Bengtsson*
Affiliation:
Lund University
*
*Postal address: Department of Statistics, Box 743, 220 07 Lund. Email: Henrik.Bengtsson@stat.lu.se
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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 expected value of a switch process and the autocovariance function of the switch process stationary counterpart is obtained, leading to a new interpretation of the classical Pólya criterion for positive-definiteness.

Keywords

Renewal theorygeometric divisibilitybinary processes

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60G55: Point processes 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 8
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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