On the moments of a self-correcting process

The existence of ordinary and exponential moments of a point process with conditional intensity of the form is deduced from a Markov chain representation for t – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.

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

Journal of Applied Probability ◽

10.1239/jap/1032192849 ◽

1998 ◽

Vol 35 (2) ◽

pp. 303-312 ◽

Cited By ~ 4

Author(s):

Timothy C. Brown ◽

Kais Hamza ◽

Aihua Xia

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Point Process ◽

Continuous Time ◽

Infinitesimal Generator ◽

Point Processes ◽

Random Variables ◽

Simple Point ◽

Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) 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.

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

Journal of Applied Probability ◽

10.1017/s0021900200014960 ◽

1998 ◽

Vol 35 (02) ◽

pp. 303-312 ◽

Cited By ~ 1

Author(s):

Timothy C. Brown ◽

Kais Hamza ◽

Aihua Xia

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Point Process ◽

Continuous Time ◽

Infinitesimal Generator ◽

Point Processes ◽

Random Variables ◽

Simple Point ◽

Conditional Intensity

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) 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.

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Corrigendum to: Exponential Moments and Piecewise Thinning for the Bessel Point Process

International Mathematics Research Notices ◽

10.1093/imrn/rnab037 ◽

2021 ◽

Author(s):

Christophe Charlier

Keyword(s):

Point Process ◽

Exponential Moments

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Laws of large numbers for the annealing algorithm

Stochastic Processes and their Applications ◽

10.1016/0304-4149(90)90009-h ◽

1990 ◽

Vol 35 (2) ◽

pp. 309-313 ◽

Cited By ~ 3

Author(s):

Nina Gantert

Keyword(s):

Laws Of Large Numbers ◽

Large Numbers ◽

Annealing Algorithm

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On the volume of parallel bodies: a probabilistic derivation of the Steiner formula

Advances in Applied Probability ◽

10.2307/1428098 ◽

1995 ◽

Vol 27 (1) ◽

pp. 97-101 ◽

Cited By ~ 4

Author(s):

Richard A. Vitale

Keyword(s):

Convex Bodies ◽

Laws Of Large Numbers ◽

Large Numbers ◽

Steiner Formula ◽

Random Convex

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

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