7. Probability and Statistics Soviet Essays - Sheynin, Oscar

of the others, then its distribution should be close to normal. I do not know any {rigorous}mathematical findings made in this direction before Chebyshev. And, although his proof ofthe {central limit} theorem has logical flaws and the formulation of his theorem lacks thenecessary restrictions, Chebyshev’s merit in solving this issue is everlasting. It consists inthat he was able, first, to develop a method of proof (the method of moments) 2 ; second, toformulate the problem of establishing the rapidity of approximation and to discoverasymptotic expansions; and third, to stress the importance of the theorem.Incidentally, I note that soon after Chebyshev’s work had appeared, Markov publishedtwo memoirs where he rigorously proved more general propositions. He applied the samemethod of investigation, – the method of moments. After Liapunov had made public tworemarkable writings on the same subject, this method apparently lost its importance;indeed, whereas Chebyshev and Markov demanded that the terms {of the studied sum}possessed finite moments of all orders, Liapunov was able to establish such conditions ofthe theorem that only required a restricted number of moments (up to the third order andeven somewhat lower). Liapunov’s method was actually a prototype of the modern methodof characteristic functions. Not only had he managed to prove the sufficiency of hisconditions for the convergence of the distribution functions of the appropriately normedand centered sums of independent variables to the normal law; he also estimated therapidity of the convergence.Markov exerted great efforts to restore the honor of the method of moments. Hesucceeded by employing a very clever trick which is not infrequently used nowadays aswell. Its essence consists in that, instead of a sequence of given random variables 1 , 2 , …,we consider curtailed variables* n = n , if | n | ≤ N n , and = 0 otherwise.The number N n remains at our disposal, and, when it is sufficiently large, the equality* n = n holds with an overwhelming probability. Unlike the initial variables, the new onespossess moments of all orders, and Markov’s previous results are applicable to them. Anappropriate choice of the numbers N n ensures that the sums of * n and of the initialvariables have approaching distribution functions. Markov was thus able to show that themethod of moments allowed to derive all the Liapunov findings.[2] In 1906 Markov initiated a cycle of investigations and thus opened up a new objectof research in probability and its applications to natural sciences and technology. He beganconsidering sequences of peculiarly dependent random variables (or trials) n . Thedependence was such that the distribution of n , given the value taken by n–1 , does notchange once the values of k , k < n – 1, become known.Markov only illustrated the idea of these chainwise dependences, which in our timeenjoy various applications, by examples of the interchange of the vowels and consonantsin long extracts from Russian poetry (Pushkin) and prose (S.T. Aksakov). In the newcontext of random variables connected in chain dependences he encountered the problemsformulated by Chebyshev for sums of independent terms. The extension of the law of largenumbers to such dependent variables proved not excessively difficult, but the justificationof the central limit theorem was much more troublesome. The method of moments thatMarkov employed required the calculation of the central moments of all the integral ordersfor the sumsn(1/B n )k = 1( k – E k ), B n 2 = varnk = 1 k