An Introduction to Probability and Stochastic Processes / by Marc A. Berger.

Format:

Book

Author/Creator:

Berger, Marc A., Author.

Series:

Springer Texts in Statistics, 2197-4136

Language:

English

Subjects (All):

Probabilities.

Probability Theory.

Local Subjects:

Probability Theory.

Physical Description:

1 online resource (XII, 205 p.)

Edition:

1st ed. 1993.

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1993.

Language Note:

English

Summary:

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.

Contents:

I. Univariate Random Variables

Discrete Random Variables

Properties of Expectation

Properties of Characteristic Functions

Basic Distributions

Absolutely Continuous Random Variables

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

III. Limit Laws

Law of Large Numbers

Weak Convergence

Bochner’s Theorem

Extremes

Extremal Distributions

Large Deviations

IV. Markov Chains—Passage Phenomena

First Notions and Results

Limiting Diffusions

Branching Chains

Queueing Chains

V. Markov Chains—Stationary Distributions and Steady State

Stationary Distributions

Geometric Ergodicity

Examples

VI. Markov Jump Processes

Pure Jump Processes

Poisson Process

Birth and Death Process

VII. Ergodic Theory with an Application to Fractals

Ergodic Theorems

Subadditive Ergodic Theorem

Products of Random Matrices

Oseledec’s Theorem

Fractals

Bibliographical Comments

References

Solutions (Sections I–V).

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

ISBN:

1-4612-2726-7