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Full Description
Compound renewal processes (CRPs) are among the most ubiquitous models arising in applications of probability. At the same time, they are a natural generalization of random walks, the most well-studied classical objects in probability theory. This monograph, written for researchers and graduate students, presents the general asymptotic theory and generalizes many well-known results concerning random walks. The book contains the key limit theorems for CRPs, functional limit theorems, integro-local limit theorems, large and moderately large deviation principles for CRPs in the state space and in the space of trajectories, including large deviation principles in boundary crossing problems for CRPs, with an explicit form of the rate functionals, and an extension of the invariance principle for CRPs to the domain of moderately large and small deviations. Applications establish the key limit laws for Markov additive processes, including limit theorems in the domains of normal and large deviations.
Contents
Introduction; 1. Main limit laws in the normal deviation zone; 2. Integro-local limit theorems in the normal deviation zone; 3. Large deviation principles for compound renewal processes; 4. Large deviation principles for trajectories of compound renewal processes; 5. Integro-local limit theorems under the Cramér moment condition; 6. Exact asymptotics in boundary crossing problems for compound renewal processes; 7. Extension of the invariance principle to the zones of moderately large and small deviations; A. On boundary crossing problems for compound renewal processes when the Cramér condition is not fulfilled; Basic notation; References; Index.
