7. Probability and Statistics Soviet Essays - Sheynin, Oscar

of the analytical tool rather than the essence of the problem led him to demand the existenceof finite moments of any order for the random variables under his consideration.Liapunov offered a formulation of this limit theorem free from the restriction justmentioned. His methods, and the generality of his result created such a great impression, thateven to our day the proposition in his wording is often called the main, or the central limittheorem of probability theory 7 .From among the other directions of research followed by the Petersburg school the socalledMarkov chains should be especially cited. This peculiar term screens one of the mostgeneral and fruitful patterns of natural processes. One of the main concepts for the entiremodern natural science is the notion of phase space of the possible states of a studied system.A change of an isolated system is supposed to be deterministic and free from the so-calledaftereffect if, during time interval , the system certainly transfers from state to state =F(; ) where F is some definite single-valued function of the initial state and the interval oftime. For random processes without aftereffect, given and , we only have, instead of thefunction F(; ), a definite probability distribution depending on and for the state toreplace the state after units of time.Markov considered the simplest case of such processes in which the phase space onlyconsisted of a finite number of states 1 , 2 , …, n . In addition, he restricted his attention toconsidering processes, as we say, with discrete time, i.e., only observed after = 1, 2, 3, …Under these conditions, the studied probability distributions are given by transitionprobabilities p ij( ) for the time interval from state i to state j with the recurring formulasp ij( +1) =kp ik( ) p kj(1)allowing the calculation of p ik( ) for any = 1, 2, 3, … given the matrix (p ij ) of the transitionprobabilities p ij = p ij (1) for an elementary unit time interval.This simple pattern nevertheless allows to study all the main general properties ofprocesses without aftereffect. In particular, Markov ascertained the first rigorously provedergodic theorem: if all p ij are positive, then p ij ( ) p j as where p j do not depend on i.I have not in the least exhausted even the main achievements of the Petersburg school. Ihave mainly dwelt on its essentially new ideas, and sometimes explicated them in asomewhat modernized form so as to show more clearly their influence on the furtherdevelopment of the theory of probability, and their importance for mathematical naturalscience. It would have been more difficult to offer, in a popular article, a notion of thetechnical skill, elegance and wit of the school’s exceptionally eminent analytical methods.Only with considerable delay, in the 1920s or even the 1930s, the importance of the worksof Chebyshev, Markov and Liapunov was quite appreciated in Western Europe. Nowadaysthey are everywhere perceived as the point of departure for the entire further development ofthe theory of probability. In particular, the main Liapunov limit theorem 8 and the theory ofMarkov chains were exactly what was most of all needed for a reliable substantiation of thedeveloping statistical physics. That the West had slowly adopted the ideas of the Petersburgschool may perhaps be indeed partly explained by the fact that the school was very remotefrom statistical physics, so that Markov only illustrated the application of his theory of trialsconnected into a chain (the application of Markov chains) by considering the distribution ofvowels and consonants in the text of {Pushkin’s} ! (Eugene Onegin) 9 .Hopefully, my last remark will not lead to an impression that the works of the Petersburgschool lacked an animated feeling of connection with the requirements of mathematicalnatural science. A keen sense of reality in formulating mathematical problems was especiallycharacteristic of Chebyshev. Issuing from comparatively special elementary, and sometimesrather old-fashioned applied problems, he elicited from them with exceptional insight such