7. Probability and Statistics Soviet Essays - Sheynin, Oscar

P ij (1) = p ij (1) , P ij (t) =kP ij (t–1) p kj (t) , t > 1. (4.1.1)Especially important is the time-homogeneous case p ij (t) = p ij . Here, the matrix (P ij (t) ) is equal tomatrix (p ij ) raised to the power of t.A natural question of the limiting behavior of the probabilities P ij (t) as t ¤ ¥presents itself. In case of a finite number of states and time-homogeneity Markov proved that whenall the p ij were positive there existed limitslim P ij (t) = P j as t ¤ ¥ (4.1.2)which did not depend on the initial state E 1 . Beginning with 1929, this result (Markov’s ergodictheorem) became the point of departure for a long cycle of works done by Romanovsky and a numberof foreign authors (Mises, Hadamard, Fréchet, et al). Romanovsky made use of algebraic methodsdating back not only to Markov but also to Frobenius whereas Mises initiated direct probabilitytheoreticmethods. Romanovsky, in his fundamental memoir [35], fully studied by algebraic methodsthe limiting behavior of the transition probabilities P ij (t) under conditions of time-homogeneity andfiniteness of the number of states. An exhaustive exposition of the solution by means of a directmethod can be found in Bernstein’s treatise [41]. Kolmogorov [29] was able completely to ascertainby direct methods the limiting behavior of these probabilities under time-homogeneity for a countablenumber of states.Already Markov had studied non-homogeneous chains. For them, a natural generalization of therelation (4.1.2) islim |P ik (t) – P jk (t) | = 0 as t ¤ ¥ (4.1.3)which expresses the vanishing of the dependence of state E k after t steps (as t ¤ ¥ ) from the initialstate (E i or E j ). On related problems see Kolmogorov [22; 32] and Bernstein [41].Markov’s typical problem was the study of the sums of a sequence of random variables 1 , 2 , …, t ,… such that t takes value a j if, after t steps, the system, having initially been in state E i , finds itself instate E j . The sums t = 1 + 2 + … + tcan obviously be written down as t =ja j µ ij(t)where µ ij (t) is the number of times that the system finds itself in state E j during the first t steps giventhat its initial state was E i . Therefore, from the modern point of view, the main natural problem is thestudy of the limiting behavior of the random variables µ ij (t) as t ¤ ¥ .For the case of time-homogeneity, all the problems connected with the usual, and the strong law oflarge numbers are easily solved. Sarymsakov devoted his article [9] to the law of the iteratedlogarithm. The problem of the limiting laws of distribution are deeper. For a finite number n of statesthe main problem consists in studying the limiting behavior as t ¤ ¥ of probabilitiesQ i (t) (m 1 ; m 2 ; …; m n ) = P(µ i1 (t) = m 1 ; µ i2 (t) = m 2 ; …; µ in (t) = m n ).A number of authors have quite prepared a full solution of this problem; more precisely, theydiscovered the necessary and sufficient conditions for the applicability of the appropriate local limittheorem concerning the (n – 1)-dimensional Gauss law (indeed, m 1 + m 2 + …+ m n = n) and even for