Random Processes and Learning

The aim of" the present monograph is two-fold: (a) to give a short account of the main results concerning the theory of random systems with complete connections, and (b) to describe the general learning model by means of random systems with complete connections. The notion of chain with complete connections has been introduced in probability theory by ONICESCU and MIHOC (1935a). These authors have set themselves the aim to define a very broad type of dependence which takes into account the whole history of the evolution and thus includes as a special case the Markovian one. In a sequel of papers of the period 1935-1937, ONICESCU and MIHOC developed the theory of these chains for the homogeneous case with a finite set of states from differ­ ent points of view: ergodic behaviour, associated chain, limit laws. These results led to a chapter devoted to these chains, inserted by ONI­ CESCU and MIHOC in their monograph published in 1937. Important contributions to the theory of chains with complete connections are due to DOEBLIN and FORTET and refer to the period 1937-1940. They consist in the approach of chains with an infinite history (the so-called chains of infinite order) and in the use of methods from functional analysis.

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.