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An Introduction to Probability Theory and its applications- Volume 2
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CHAPTER
I-THE EXPONENTION AND THE UNIFROM DENSITIES
1-Introduction
2-Densities.Convolutions
3-The Exponential Density
4-Waiting Times Paradoxes.The Poisson Process
5-The Persistence of Bad Luck
6-Waiting Time and Order Statistics.
7-The Unifrom Disribution
8-Random Splittings
9-Convolution and Covering Theorens
10- Random Directions
11-The Ues of Lebesgue Measure
12-Empirical Distributions
13-Problems for Solution
CHAPTER
II-SOECIAL DENSITIES .RANDOMIZATION
1-Noteations and Conventions
2-Gamma Distribution
*3-Related Distributions of Statistics
4-Some Common Densities
5-Randomization and Mixitures
6-Discrete Distributions
7-Bessel Functions and Randim Walks
8-Distributions on a Circle
9-Problems for Solution
CHAPTER
III-DENSITIES IN HIGHER DIMENSIONS. NORMAL DENSITIES AND PROCESSES
1-Densities
2-Conditionnal Distributions
3-Return to the Exponential and the Unifrom Distributions
4-A Caracterization of the Normal Distributions
5-Matrix Nitation.The Covariance Matrix
6-Normal Densities and Distribution
*7-Stationary Normal Processes
8-Markovian Normal Densition
9-Problems for Solution.
CHAPTER
IV-PROBABILITY MEASURES AND SPACS
1-Baire Funcition
2-Interval Functions and Integrals in R'
3-Q-Algebras .Measurability
4-Probability Spaces .Random Variables
5-The Extension Theorem
6-Product Spaces. Sequences of Independent Variables
7-Null Sets.Completion
CHAPTER
V-PROBABILITY DISTRIBUTION IN R
1-Distribution and Expectations
2-Preliminaries
3-Densities
4-Convolutions
5-Symmetrization
6-Integration by Parts Existence of Moments
7-Chebyshev's Inequality
8-Further Inequality
9-Simple Conditional Distrional Distribution. Mixtures
*10-Conditional Distribution
*11-Conditional Expectations
12-Problems for Solution
CHAPTER
VI-A SURVEY OF SOME IMPORTANT DISTRIBUTIONS AND PROCESSES..169
1-Stable Distributions in R1
2-Examples
3-Infinitely Divisible Distribution in R1
4-Processes with Independent Increments
*5-Ruin Problems in Compound Poisson Processes
6-Renewal Processes
7-Examples and Problems
8-Random Walks
9-The Queuing Process
10-Persistent and Transient Random Walks
11-General Markov Chains
12-Martingales
13-Problems for Solution
CHAPTER
VII-LAWS OF LARGE NUMBERS.APPLICATION IN ANALYSIS
1-Main Lemma and Notations
2-Bernstein Pplynomials,Absolutely Monotone Functuions
3-Moment Problems
*4-Application to Exchangeable Variables
*5-Generalized Tarlor Formula and Semi-Groups
6.Inversion Formulas for Laplace Tarnsforms
*7-Laws of Large Number for Identically Distributeb Variables
*8-Strong Laws
9-Generalization to Martingales.
10-Problems for Solution
CHAPTER
VIII-THE BASIC LIMIT THEOREMS
1-Convergence of Measures
2-Special Properation
3-Distribution as Operators
4-The Central Limit Theorem
*5-Infinite Convolutions
6-Selection Theorems
*7-Ergodic Theorems for Markov Chains
8-Regular Variation
*9-Asymptiotic Propreties of Regularly Varying Functions
10-Problems for Solution
CHAPTER
IX-INFINITELY DIVISIBLE DISTRIBUTIONS AND SAMI-GROUPS
1-Orienation.
2-Convolution Semi-Groups
3-Preparatory Lemmas
4-Finite Variances
5-The Main Theorems
6-Example :Stable Semi-Groups
7-Triangular Arrays with Identical Distribution
8-Dommains of Attraction
9-Variable Distribution .The Three-Series Theorem
10-Problems for Solution
CHAPTER
X-MARKOV PROCESSES AND SEMI -GROUPS
1-The Pseudo- Poisson Type
2-A Variant:Linear Increments
3-Jump Processes
4-Diffusion Processes in R1
5-The Forward Equation. Boundary Conditions
6-Diffusion in Higher Dimensions
7-Subordinated Processes
8-Markov Processes and Semi-Groups
9-The "Exponential Formula "of Semi-Groups
10-Generators .The Backward Equation
CHAPTER
XI-RENEWAL THEORY
1-The Renewal Theorem
2-Proof of the Renewal Theorem
3-Refinements
4-Persistent Renewal Processes
5-The Number N of Renewal Epochs
6-Terminating (Teansient)Processes
7-Diverse Application
8-Existence of Limit in Stochastic Orocesses
*9-Renewal Theory on the Whole Line
10-Problems for Solution
CHAPTER
XII-RANDOM WALKS IN R1
1-Basic Concepts and Notations
2-Duality .Types of Random Walk
3-Distribution of Ladder Heights. Weiner-Hopf Factorization
3a-The Wiener -Hopf Integral Equation
4-Examples
5-Applications
6-A Combinatorial Lemma
7-Distribution of Ladder Epochs
8-The Are Sine Laws
9-Miscellaneous Complements
10-Problems for Solution
CHAPTER
XIII-LAPLACE TRANSFROMS.TAUBERIAN THEOREMS.RESOLVENTS
1-Definition.The Continuity Theorem
2-Elementary Properties
3-Examples
4-Completely Monotone Functions .Inversion Formulas
5-Tauberian Theorems
*6-Stable Dsitributions
*7-Infinitely Divisible Distributions
*8-Higher Dimensions
9-Laplace Transfoms for Semi-Group
10-The Hille-Yosida Theorem
11-Problems for Solution
CHAPTER
XIV-APPLICATION OF LA0LACE TRANSFOME
1-The Renewel Equation : Theory
2-Renewal -Type Equation:Examples
3-Limit Theorems Invoving Arc Sine Distributions
4-Busy Periods and Related Branching Processes
5-Diffusion Processes
6-Birth-and-Death Processes and Random Walks
7-The Kolmogorow Differential Equations
8- Example : The Pure Brith Proces
9-Calulation of Ergodic Limit and of First-Passage Time
10-Problems for Solution
CHAPTER
XV-CHARACTERSTIC FUNCTIONS
1-Definition.Basic Properties
2-Special Distributions. Mixtures
2a-Some Unexpected Phenomena
3-Uniqueness.Inversion Formulas
4-Regularity Properties
5-The Central Limit Theorem For Equal Componets
6-The Lindederg Conditions
7-Characteristic Functions in Higher Dimension
*8-Two Characterizations of The Normal Distribution
9-Problems for Solution
CHAPTER
XVI*-EXPANSIONS RELATED TO THE CENTERAL LIMIT THEOREM
1-Notations
2-Expansions for Disnsities
3-Smoothing
4-Expansions for Theorems
5-The Berry -Esseen Theorems
6-Expansions in the Case of Varying Components
7-Large Deviations
CHAPTER
XVII-INFINITELY DIVISBLE DISTRIBUTION
1-Infinitely Divisible Distributions
2-Canonical Forms. The Main Limit Theorem
2a-Derivatives of Characteristic Funnctions
3-Examples and Special Properties
4-Special Properties
5-Stable Distribution and Their Domains of Attraction
*6-Stable Densities
7-Triangular Arrays
*8- The Class L
*9-Partial Attraction. ''Universal Laws''
10-Infinite Convolutions
11-Higher Dimensions
12-Problems for Solution
CHAPTER
XVIII-APPLICTIONS OF FOURIER METHHODS TO RANDOM WOLKS
1-The Basic Identity
*2-Finite Interval .Wald's Approximation
3-The Winer-Hopf Factorization
4-Implications and Application
5-Two Deeper Theorems
6-Criteria for Persistency
7-Problems for Solution
CHAPTER
XIX-HARMONIC ANALYSIS
1-The Parseval Relation
2-Positive Definite Functions
3-Stationary Processes
4-Fourier Series
*5-The Poisson Summation Formula
6-Positive Definite Sequences
7-L2 Theory
8-Stochastic Processes and Integralsc
9-Problems for Solution
ANSWERS TO PEOBLEMS
SOME BOOK ON COGNATE SUBJECTS
INDEX
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