Random Processes and Learning

Random Processes and Learning

by Marius Iosifescu, Radu Theodorescu

Paperback(Softcover reprint of the original 1st ed. 1969)

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Product Details

ISBN-13: 9783642461866
Publisher: Springer Berlin Heidelberg
Publication date: 02/25/2012
Series: Grundlehren der mathematischen Wissenschaften Series , #150
Edition description: Softcover reprint of the original 1st ed. 1969
Pages: 308
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1 A study of random sequences via the dependence coefficient.- 1.1. The general case.- 1.1.1. The dependence coefficient.- 1.1.1.1. Borel-Cantelli type properties.- 1.1.1.2. The 0–1 law.- 1.1.1.3. Two auxiliary results.- 1.1.2. Generalizations of Bienaymé’s equality.- 1.1.2.1. Some inequalities concerning the covariance.- 1.1.2.2. Applications to the variance of sums.- 1.1.3. Convergence of series.- 1.1.3.1. The a.s. convergence.- 1.1.3.2. The strong law of large numbers.- 1.1.4. The central limit theorem.- 1.1.4.1. The variance of sums.- 1.1.4.2. Different variants.- 1.1.5. The law of the iterated logarithm.- 1.1.5.1. Two auxiliary results.- 1.1.5.2. The main theorem.- 1.2. The Markovian case.- 1.2.1. The coefficient of ergodicity.- 1.2.1.1. Introductory definitions.- 1.2.1.2. Properties.- 1.2.1.3. The relationship to the independence coefficient.- 1.2.2. A lower bound for the variance.- 1.2.2.1. The main theorem.- 1.2.2.2. Auxiliary results.- 1.2.3. Asymptotic properties.- 1.2.3.1. Borel-Cantelli lemma and the 0–1 law.- 1.2.3.2. Generalizations of Bienaymé’s equality.- 1.2.3.3. Convergence of series.- 1.2.3.4. The strong law of large numbers.- 1.2.3.5. The central limit theorem and the law of the iterated logarithm.- 2 Random systems with complete connections.- 2.1. Ergodicity.- 2.1.1. Basic definitions.- 2.1.1.1. The concept of random system with complete connections.- 2.1.1.2. The associated Markov system.- 2.1.1.3. The associated operators.- 2.1.2. Different types of ergodicity.- 2.1.2.1. Definitions and auxiliary results.- 2.1.2.2. Uniform ergodicity in the weak sense.- 2.1.2.3. Uniform ergodicity for the homogeneous case.- 2.1.2.4. Uniform ergodicity in the strong sense.- 2.1.2.5. Application to multiple Markov chains.- 2.1.2.6. Application to the associated Markov system.- 2.1.3. An operator-theoretical approach.- 2.1.3.1. Mean and uniform ergodic theorems.- 2.1.3.2. Ergodic theorems for a special class of operators.- 2.1.3.3. Application to the associated Markov system.- 2.1.3.4. Application to the ergodicity of homogeneous random systems with complete connections.- 2.2. Asymptotic behaviour.- 2.2.1. Properties not supposing the ergodicity.- 2.2.1.1. Borel-Cantelli lemma and the 0–1 law.- 2.2.1.2. Convergence and the strong law of large numbers.- 2.2.2. Properties supposing the ergodicity.- 2.2.2.1. Basic results.- 2.2.2.2. The strong law of large numbers.- 2.2.2.3. The weak law of large numbers.- 2.2.2.4. The central limit theorem.- 2.2.2.5. The law of the iterated logarithm.- 2.2.2.6. Some nonparametric statistics.- 2.3. Special random systems with complete connections.- 2.3.1. OM-chains.- 2.3.1.1. Basic definitions.- 2.3.1.2. Examples.- 2.3.1.3. Ergodic theorems.- 2.3.1.4. The Monte Carlo simulation.- 2.3.1.5. The case of an arbitrary set of states.- 2.3.2. Chains of infinite order.- 2.3.2.1. Definition and several special cases.- 2.3.2.2. An existence theorem.- 2.3.2.3. The case of a finite set of states.- 2.3.3. Other examples.- 2.3.3.1. Partially observable sequences.- 2.3.3.2. Miscellanea.- 3 Learning.- 3.1. Basic models.- 3.1.1. Introductory definitions and notions.- 3.1.1.1. Description of models.- 3.1.1.2. The simulation of models.- 3.1.2. Distance diminishing models.- 3.1.2.1. Description of the model.- 3.1.2.2. Theorems concerning states.- 3.1.2.3. Theorems concerning events.- 3.1.3. Finite state models.- 3.1.3.1. Introductory comments.- 3.1.3.2. Properties.- 3.2. Linear models.- 3.2.1. The (t + 1)-operator model.- 3.2.1.1. Description of the model.- 3.2.1.2. The (m + 1)2-operator model with reinforcement.- 3.2.1.3. The limiting distribution function.- 3.2.2. Experimenter-, subject- and experimenter-subject-controlled events.- 3.2.2.1. Experimenter-controlled events.- 3.2.2.2. Subject-controlled events.- 3.2.2.3. Experimenter-subject-controlled events.- 3.3. Nonlinear models.- 3.3.1. The beta model.- 3.3.1.1. Description of the model.- 3.3.1.2. Some auxiliary results.- 3.3.1.3. Properties.- 3.3.2. The simultaneous discrimination learning model.- 3.3.2.1. Description of the model.- 3.3.2.2. Properties.- 3.3.3. The fixed sample size model.- 3.3.3.1. Description of the model.- 3.3.3.2. Properties.- Notation index.- Author and subject index.

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