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This book explores advanced topics in probability theory and stochastic processes. It begins with large deviations in Poisson approximation and examines weak interaction limits for stochastic particle systems. The text delves into convergence rates in maximum and minimum limit transfer theorems and the geometry of state spaces in quantum probability. It discusses the transition from diffusions to jump processes and presents the central limit theorem in Skorohod spaces, along with the asymptotic strength distribution of fiber bundles. Further, it covers ergodicity in weak convergence, equilibrium-type identities, and large deviations for Markov chains, as well as second-order properties of mixing random sequences and fields. Optimal couplings are applied to Riemannian geometry, and empirical processes are analyzed in Banach function spaces. The book also addresses multi-time parameter processes in change-point analysis and Bergström-type asymptotic expansions related to Kolmogorov's theorem. Additional topics include differentiation of heat semigroups, classical Markov processes as quantum flows, unimodular eigenvalues, and function-indexed partial-sum processes. It discusses spectral equations for empirical covariance matrices, conditionally exponential families, and the nonuniform approximation of sums of independent variables. Other highlights include Kolmogorov's strong law, cycle processes, density-dependent branching
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Probability Theory and Mathematical Statistics, B. Grigelionis
- Langue
- Année de publication
- 1999
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