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.

Inhalt

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).