Ebook: An Introduction to Probability and Stochastic Processes

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.

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This is a textbook which will provide students with a straightforward introduction to the mathematical theory of probability. It is written with the aim of presenting the central results and techniques of the subject in a complete and self-contained account. The emphasis is on giving results in simple forms with clear proofs and to eschew more powerful forms of theorems which require technically involved proofs. Any student who has a familiarity with calculus and basic algebra will be able to use this text and throughout there are a wide variety of exercises to illustrate and to develop ideas. A highlight of the book is an account of random iterated function systems which is a fascinating area of current research.

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This is a textbook which will provide students with a straightforward introduction to the mathematical theory of probability. It is written with the aim of presenting the central results and techniques of the subject in a complete and self-contained account. The emphasis is on giving results in simple forms with clear proofs and to eschew more powerful forms of theorems which require technically involved proofs. Any student who has a familiarity with calculus and basic algebra will be able to use this text and throughout there are a wide variety of exercises to illustrate and to develop ideas. A highlight of the book is an account of random iterated function systems which is a fascinating area of current research.
Content:
Front Matter....Pages i-xii
Univariate Random Variables....Pages 1-26
Multivariate Random Variables....Pages 27-44
Limit Laws....Pages 45-77
Markov Chains—Passage Phenomena....Pages 78-100
Markov Chains—Stationary Distributions and Steady State....Pages 101-120
Markov Jump Processes....Pages 121-138
Ergodic Theory with an Application to Fractals....Pages 139-172
Back Matter....Pages 173-206

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This is a textbook which will provide students with a straightforward introduction to the mathematical theory of probability. It is written with the aim of presenting the central results and techniques of the subject in a complete and self-contained account. The emphasis is on giving results in simple forms with clear proofs and to eschew more powerful forms of theorems which require technically involved proofs. Any student who has a familiarity with calculus and basic algebra will be able to use this text and throughout there are a wide variety of exercises to illustrate and to develop ideas. A highlight of the book is an account of random iterated function systems which is a fascinating area of current research.
Content:
Front Matter....Pages i-xii
Univariate Random Variables....Pages 1-26
Multivariate Random Variables....Pages 27-44
Limit Laws....Pages 45-77
Markov Chains—Passage Phenomena....Pages 78-100
Markov Chains—Stationary Distributions and Steady State....Pages 101-120
Markov Jump Processes....Pages 121-138
Ergodic Theory with an Application to Fractals....Pages 139-172
Back Matter....Pages 173-206
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