On the variance to mean ratio for random variables from Markov chains and point processes

Timothy C. Brown, Kais Hamza, Aihua Xia

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3 Citations (Scopus)


Criteria are determined for the variance to mean ratio to be greater than one (overdispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

Original languageEnglish
Pages (from-to)303-312
Number of pages10
JournalJournal of Applied Probability
Issue number2
Publication statusPublished - 1 Jan 1998
Externally publishedYes


  • Conditional intensity
  • Infinitesimal generator
  • Markov chain
  • Positive and negative correlations
  • Transition rates

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