Dependence with complete connections and its applications.
(Dependenţă cu legături complete şi aplicaţii.)

*(Romanian. English summary)*Zbl 0584.60064
Bucureşti: Editura Ştiinţifică şi Enciclopedică. 360 p. (1982).

This is the third monograph on dependence with complete connections (the first one was written by G. Ciucu and the reviewer [Chains with complete connections. (Romanian) Bucureşti: Editura Academiei Repub. Popul. Romîne (1960; Zbl 0095.12402)]; the second one by the second author and the reviewer [Random processes and learning. Berlin etc.: Springer-Verlag (1969; Zbl 0194.51101)]), since the introduction of the concept in 1935 by O. Onicescu and G. Mihoc [C. R. Acad. Sci., Paris 200, 511–512 (1935; Zbl 0010.40602)].

A quadruple \(\{(W,\mathcal W),(X,\mathcal X),p,u\}\) is called a random system with complete connections if (1) \((W,\mathcal W)\) and \((X,\mathcal X)\) are measurable spaces; (2) \(P\) is a stochastic kernel on \(W\times {\mathcal X}\); and (3) \(u\) is a measurable transformation of \(W\times X\) into \(W\). A general Markov chain on \(W\) is associated to this system whose transition probability function is \(Q(w,A)=P(w,\{x: u(w,x)\in A\})\).

Most of the material published since the appearance of the second monograph is reviewed and partly incorporated in the text, as well as new results obtained by the authors. Emphasis is laid on the associated chain, a Markov chain with very nice properties, on functional limit theorems for chains of infinite order and for the associated Markov chain, as well as on the study of certain special cases as OM-chains, representation algorithms for real numbers (continued fraction expansions, f-expansions, d-adic expansions), and chains of infinite order.

Research workers in stochastic processes will find the exposition and the coverage of this monograph extremely interesting and useful.

A quadruple \(\{(W,\mathcal W),(X,\mathcal X),p,u\}\) is called a random system with complete connections if (1) \((W,\mathcal W)\) and \((X,\mathcal X)\) are measurable spaces; (2) \(P\) is a stochastic kernel on \(W\times {\mathcal X}\); and (3) \(u\) is a measurable transformation of \(W\times X\) into \(W\). A general Markov chain on \(W\) is associated to this system whose transition probability function is \(Q(w,A)=P(w,\{x: u(w,x)\in A\})\).

Most of the material published since the appearance of the second monograph is reviewed and partly incorporated in the text, as well as new results obtained by the authors. Emphasis is laid on the associated chain, a Markov chain with very nice properties, on functional limit theorems for chains of infinite order and for the associated Markov chain, as well as on the study of certain special cases as OM-chains, representation algorithms for real numbers (continued fraction expansions, f-expansions, d-adic expansions), and chains of infinite order.

Research workers in stochastic processes will find the exposition and the coverage of this monograph extremely interesting and useful.

Reviewer: Radu Theodorescu (Quebec)