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This SpringerBriefs employs a novel approach to obtain the precise asymptotic behavior at infinity of a large class of permanental sequences related to birth and death processes and autoregressive Gaussian sequences using techniques from the theory of Gaussian processes and Markov chains.
The authors study alpha-permanental processes that are positive infinitely divisible processes determined by the potential density of a transient Markov process. When the Markov process is symmetric, a 1/2-permanental process is the square of a Gaussian process. Permanental processes are related by the Dynkin isomorphism theorem to the total accumulated local time of the Markov process when the potential density is symmetric, and by a generalization of the Dynkin theorem by Eisenbaum and Kaspi without requiring symmetry. Permanental processes are also related to chi square processes and loop soups.
The book appeals to researchers and advanced graduate students interested in stochastic processes, infinitely divisible processes and Markov chains.
Preface
Introduction
General Results
Applications
Birth and Death Processes
Birth and Death Processes with Emigration
Birth and Death Processes with Emigration Related to First Order Gaussian Autoregressive Sequences
Markov Chains with Potentials That Are the Covariances of Higher Order Gaussian Autoregressive Sequences
Relating Permanental Sequences to Gaussian Sequences
Permanental Sequences with Kernels That Have Uniformly Bounded Row Sums
Uniform Markov Chains
Appendix Bibliography
Index