A nonmeasure theoretic introduction to stochastic processes. Considers its diverse range of applications and provides readers with probabilistic intuition and insight in thinking about problems. This revised edition contains additional material on compound Poisson random variables including an identity which can be used to efficiently compute moments; a new chapter on Poisson approximations; and coverage of the mean time spent in transient states as well as examples relating to the Gibb's sampler, the Metropolis algorithm and mean cover time in star graphs. Numerous exercises and problems have been added throughout the text.
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assume Azuma's inequality birth and death Brownian Bridge compute conditional distribution Consider continuous-time Markov chain converge convex convex function Corollary counting process cycle death process define denote the number density distributed with mean distribution F equivalently Example expected number exponential with rate exponentially distributed finite given hazard rate function Hence identically distributed increasing functions independent and identically independent random variables integrable interarrival distribution large numbers law of large Lemma limiting probabilities Markov process martingale number of customers number of events number of transitions obtain occurs parameters player Poisson approximation Poisson distributed Poisson process Poisson random variable population Problem process with rate process X(t Proof Let Proposition prove queue random walk recurrent reversed chain Section semi-Markov process sequence server stationary probabilities stationary process Statistical stochastic process suppose Taking expectations transition probabilities urns variable with mean variance verify visited Wald's equation yields