Введение в стохастические процессы и их применение: теория, примеры и приложения

This book provides a self-contained introduction to stochastic processes, with emphasis on Markov chains (both on finite and countable state spaces), Markov chain Monte Carlo (MCMC) algorithms, the Poisson process, Markov jump processes, and queueing theory. The first chapter develops the theory of finite Markov chains, including absorbing chains, irreducibility, stationarity, and reversibility. The second chapter extends the theory to countable state spaces, studying recurrence, transience, periodicity, and stationary distributions. The third chapter applies these results to MCMC methods, in particular the Metropolis algorithm and simulated annealing. The fourth chapter reviews essential probability distributions (binomial, Poisson, normal, exponential). The fifth chapter constructs the Poisson point process via the counting function and waiting times, and discusses generalizations. The sixth chapter treats Markov jump processes, their generators, Kolmogorov equations, and stationary distributions. The final chapter is devoted to queueing theory, presenting Kendall notation, M/M/s queues, the PASTA property, and Little law. The text is intended for Master students and researchers who wish to master the fundamental tools of stochastic processes and their applications.