# An Introduction to Probability and Stochastic Processes

• Marc A. Berger
Textbook

Part of the Springer Texts in Statistics book series (STS)

1. Front Matter
Pages i-xii
2. Marc A. Berger
Pages 1-26
3. Marc A. Berger
Pages 27-44
4. Marc A. Berger
Pages 45-77
5. Marc A. Berger
Pages 78-100
6. Marc A. Berger
Pages 101-120
7. Marc A. Berger
Pages 121-138
8. Marc A. Berger
Pages 139-172
9. Back Matter
Pages 173-206

### Introduction

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.

### Keywords

Ergodic theory Law of large numbers Markov chain Normal distribution Poisson process Random variable Stochastic processes ergodicity jump process stochastic process

#### Authors and affiliations

• Marc A. Berger
• 1
1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-2726-7
• Copyright Information Springer-Verlag New York 1993
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-7643-2
• Online ISBN 978-1-4612-2726-7
• Series Print ISSN 1431-875X
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