Mathematical Statistics And Stochastic Processes

Generally, books on mathematical statistics are restricted to the case of independent identically distributed random variables. In this book however, both this case AND the case of dependent variables, i.e. statistics for discrete and continuous time processes, are studied. This second case is very important for today’s practitioners. Mathematical Statistics and Stochastic Processes is based on decision theory and asymptotic statistics and contains up-to-date information on the relevant topics of theory of probability, estimation, confidence intervals, non-parametric statistics and robustness, second-order processes in discrete and continuous time and diffusion processes, statistics for discrete and continuous time processes, statistical prediction, and complements in probability. This book is aimed at students studying courses on probability with an emphasis on measure theory and for all practitioners who apply and use statistics and probability on a daily basis.

Topics in Stochastic Processes covers specific processes that have a definite physical interpretation and that explicit numerical results can be obtained. This book contains five chapters and begins with the L2 stochastic processes and the concept of prediction theory. The next chapter discusses the principles of ergodic theorem to real analysis, Markov chains, and information theory. Another chapter deals with the sample function behavior of continuous parameter processes. This chapter also explores the general properties of Martingales and Markov processes, as well as the one-dimensional Brownian motion. The aim of this chapter is to illustrate those concepts and constructions that are basic in any discussion of continuous parameter processes, and to provide insights to more advanced material on Markov processes and potential theory. The final chapter demonstrates the use of theory of continuous parameter processes to develop the Itô stochastic integral. This chapter also provides the solution of stochastic differential equations. This book will be of great value to mathematicians, engineers, and physicists.

Praise for the First Edition ". . . an excellent textbook . . . well organized and neatly written." —Mathematical Reviews ". . . amazingly interesting . . ." —Technometrics Thoroughly updated to showcase the interrelationships between probability, statistics, and stochastic processes, Probability, Statistics, and Stochastic Processes, Second Edition prepares readers to collect, analyze, and characterize data in their chosen fields. Beginning with three chapters that develop probability theory and introduce the axioms of probability, random variables, and joint distributions, the book goes on to present limit theorems and simulation. The authors combine a rigorous, calculus-based development of theory with an intuitive approach that appeals to readers' sense of reason and logic. Including more than 400 examples that help illustrate concepts and theory, the Second Edition features new material on statistical inference and a wealth of newly added topics, including: Consistency of point estimators Large sample theory Bootstrap simulation Multiple hypothesis testing Fisher's exact test and Kolmogorov-Smirnov test Martingales, renewal processes, and Brownian motion One-way analysis of variance and the general linear model Extensively class-tested to ensure an accessible presentation, Probability, Statistics, and Stochastic Processes, Second Edition is an excellent book for courses on probability and statistics at the upper-undergraduate level. The book is also an ideal resource for scientists and engineers in the fields of statistics, mathematics, industrial management, and engineering.

This book contains papers by participants in two seminars, one on martingales and statistics of stochastic processes, and one on sequential analysis, both of which were held at the Steklov Institute of the Russian Academy of Sciences. The papers develop the concepts of martingales and seminmartingales and stochastic calculus for them, as well as their applications in statistics and control of stochastic processes. The class of semimartingales - that is, the class of all processes which can be represented as a sum of a martingale and a process with bounded variation - is rather large. It contains such important processes as Brownian motion, Poisson processes, solutions of stochastic differential equations, and others. The papers treat theoretical aspects of statistics of stochastic processes as well as specific models of stochastic processes from the standpoint of their statistics and control. The collection is intended for undergraduate and graduate students and researchers in probability theory and mathematical statistics.

The book covers basic concepts such as random experiments, probability axioms, conditional probability, and counting methods, single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities; limit theorems and convergence; introduction to Bayesian and classical statistics; random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion; simulation using MATLAB and R.

Ideal for courses aiming to give examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models. It introduces the methods of probability model building and provides the reader with mathematically sound techniques as well as the ability to further study the theory of stochastic processes.

This volume contains papers presented at the Steklov Seminar on Statistics and Control of Stochastic Processes. For the past three decades, the seminar has determined the development, in a number of important directions, of the theory of random processes not only in the USSR (now Russia) but in the whole world. It was organised by A N Shiryaev in collaboration with N V Krylov and R Sh Liptser. It started off with optimal stopping and filtering with applications to engineering, and very soon extended its interests to more general problems of stochastic control, causal and anticipating stochastic calculus, limit theorems for semimartingales, martingale methods in queueing theory, foundations of statistics of random processes and, in recent years, mathematical finance. Many studies, for example of stochastic PDEs or extended stochastic integrals, anticipated largely Western works.The contributions in this book are devoted to the hottest topics and united by a martingale methodology which was the key idea of the seminar.

The second part (Chapters 4-6) provides a foundation of stochastic analysis, gives information on basic models of random processes and tools to study them. Here a certain familiarity with elements of functional analysis is necessary. Important material is presented in the form of examples to keep readers involved. Audience: This is a concise textbook for a graduate level course, with carefully selected topics representing the most important areas of modern probability, random processes and statistics.