Probability Theory, Random Processes and Mathematical Statistics

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1 Probability Theory, Random Processes and Mathematical Statistics

2 Mathematics and Its Applications Managing Editor: M.HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 344

3 Probability Theory, Random Processes and Mathematical Statistics by Yu. A. Rozanov Steklov Mathematicallnstitute, Moscow, Russia 1iII... " SPRINGER SCIENCE+BUSINESS, MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / This is a completely revised and updated translation of the original Russian work of the same title, Moscow, Nauka, 1980 (second edition). Translated by the author. Printed on acid-free paper Ali Rights Reserved 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint of the hardcover 2nd edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner.

5 CONTENTS Preface Annotation IX XI Chapter 1. Introductory Probability Theory 1 1. The Notion of Probability Equiprobable outcomes Examples Conditional probability Independent events Probability and frequency 9 2. Some Probability Models Trials with countable outcomes Bernoulli trials Limit Poisson distribution Finite number of events The general model of probability theory Some examples Random Variables Probability distributions Joint probability distribution Independent random variables Conditional distributions Functions of random variables Random variables in the general model of probability theory Mathematical Expectation Mean value of discrete variable Limit mean values Some limit properties Conditional expectation Correlation Variance and correlation Normal correlations Properties of the variance and the law of large numbers Characteristic Functions Some examples Elementary analysis of characteristic functions 78

6 vi 6.3. The inverse formula of probability distributions Weak convergence of distributions The Central Limit Theorem Some limit properties of probabilities The central limit theorem 87 Chapter 2. Random Processes Random Processes with Discrete State Space The Poisson process and related processes The Kolmogorov equations Example (Branching processes) The (limit) stationary probability distribution Random Processes with Continuous States The Brownian motion Trajectories of the Brownian motion Maxima and hitting times Diffusion processes 126 Chapter 3. An Introduction to Mathematical Statistics Some Examples of Statistical Problems and Methods Estimation of the success probability in Bernoulli trials Estimation of parameters in a normal sample Chi-square criterion for probability testing Sequential analysis of alternative hypotheses Bayesian approach to hypotheses testing and parameters estimation Maximum likelihood method Sample distribution function and the method of moments The method of least squares Optimality of Statistical Decisions The most powerful criterion Sufficient statistics Lower bound for the mean square error Asymptotic normality and efficiency of the maximum likelihood estimate 166 Chapter 4. Basic Elements of Probability Theory General Probability Distributions Mappings and a-algebras Approximation of events law Mathematical expectation as the Lebesgue integral cp-spaces Conditional Probabilities and Expectations 187

7 CONTENTS vii 2.1. Preliminary remarks Conditional expectation and its properties Conditional probability Conditional Expectations and Martingales General properties 194 Chapter 5. Elements of Stochastic Analysis and Stochastic Differential Equations Stochastic Series Series of independent random variables Three series' criterion Stochastic Integrals Random functions (Preliminary remarks) Integration in l-space Stochastic integrals in 2-space Stochastic Ito integral in 2-space Stochastic Integral Representations Canonical representations Spectral representation of a stationary process and its applications Stochastic integral representation of a process with independent increments Stochastic Differential Equations Stochastic differentials Linear stochastic differential equations Linear differential equations with constant coefficients The Kalman-Bucy filter 246 Subject Index 253

8 Preface Probability Theory, Theory of Random Processes and Mathematical Statistics are important areas of modern mathematics and its applications. They develop rigorous models for a proper treatment for various 'random' phenomena which we encounter in the real world. They provide us with numerous tools for an analysis, prediction and, ultimately, control of random phenomena. Statistics itself helps with choice of a proper mathematical model (e.g., by estimation of unknown parameters) on the basis of statistical data collected by observations. This volume is intended to be a concise textbook for a graduate level course, with carefully selected topics representing the most important areas of modern Probability, Random Processes and Statistics. The first part (Ch. 1-3) can serve as a self-contained, elementary introduction to Probability, Random Processes and Statistics. It contains a number of relatively simple and typical examples of random phenomena which allow a natural introduction of general structures and methods. Only knowledge of elements of real/complex analysis, linear algebra and ordinary differential equations is required here. The second part (Ch. 4-6) provides a foundation of Stochastic Analysis, gives information on basic models of random processes and tools to study them. Here a familiarity with elements of functional analysis is necessary. Our intention to make this course fast-moving made it necessary to present important material in a form of examples. Yu. Rozanov

9 Annotation The book consists of two parts which differ one from another in their contents and the style of exposition. The first one discusses many relatively simple problems which lead to different models of probability and random processes, as well as basic methods of mathematical statistics, including typical applications. The second part presents elements of general analysis of random processes.

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