An Introduction to Probability and Stochastic Processes

ISBN-13: 9781461276432 / Angielski / Miękka / 2011 / 205 str.

Marc A. Berger

Opis

These notes were written as a result of my having taught a "nonmeasure theoretic" course in probability and stochastic processes a few times at the Weizmann Institute in Israel. I have tried to follow two principles. The first is to prove things "probabilistically" whenever possible without recourse to other branches of mathematics and in a notation that is as "probabilistic" as possible. Thus, for example, the asymptotics of pn for large n, where P is a stochastic matrix, is developed in Section V by using passage probabilities and hitting times rather than, say, pulling in Perron Frobenius theory or spectral analysis. Similarly in Section II the joint normal distribution is studied through conditional expectation rather than quadratic forms. The second principle I have tried to follow is to only prove results in their simple forms and to try to eliminate any minor technical com putations from proofs, so as to expose the most important steps. Steps in proofs or derivations that involve algebra or basic calculus are not shown; only steps involving, say, the use of independence or a dominated convergence argument or an assumptjon in a theorem are displayed. For example, in proving inversion formulas for characteristic functions I omit steps involving evaluation of basic trigonometric integrals and display details only where use is made of Fubini's Theorem or the Dominated Convergence Theorem."

Spis treści

I. Univariate Random Variables.- Discrete Random Variables.- Properties of Expectation.- Properties of Characteristic Functions.- Basic Distributions.- Absolutely Continuous Random Variables.- Basic Distributions.- Distribution Functions.- Computer Generation of Random Variables.- Exercises.- II. Multivariate Random Variables.- Joint Random Variables.- Conditional Expectation.- Orthogonal Projections.- Joint Normal Distribution.- Multi-Dimensional Distribution Functions.- Exercises.- III. Limit Laws.- Law of Large Numbers.- Weak Convergence.- Bochner’s Theorem.- Extremes.- Extremal Distributions.- Large Deviations.- Exercises.- IV. Markov Chains—Passage Phenomena.- First Notions and Results.- Limiting Diffusions.- Branching Chains.- Queueing Chains.- Exercises.- V. Markov Chains—Stationary Distributions and Steady State.- Stationary Distributions.- Geometric Ergodicity.- Examples.- Exercises.- VI. Markov Jump Processes.- Pure Jump Processes.- Poisson Process.- Birth and Death Process.- Exercises.- VII. Ergodic Theory with an Application to Fractals.- Ergodic Theorems.- Subadditive Ergodic Theorem.- Products of Random Matrices.- Oseledec’s Theorem.- Fractals.- Bibliographical Comments.- Exercises.- References.- Solutions (Sections I–V).