7. Probability and Statistics Soviet Essays - Sheynin, Oscar

last decades, mathematicians more often than not somewhat distrust the rigor andirrevocability of its conclusions. During this very period there appeared Czuber’s celebratedcourse which serves as a lively embodiment of the most ugly period in the life of the theoryof probability and which was a model for many translations including Russian ones{?}.[2] But then, in Russia, at the beginning of the second half of the 19 th century, Chebyshevbegan destroying, one after another, the obstacles that for almost half a century had beenarresting the development of the theory of probability. At first he discovered his majesticallysimple solution of the problem of extending the law of large numbers. The history of thisproblem is indeed remarkable and provides almost the only example in its way known in theentire evolution of the mathematical sciences. Until Chebyshev, the law was considered as avery complicated theorem; to prove those particular cases that Jakob Bernoulli and Poissonhad established, transcendental and complicated methods of mathematical analysis wereusually applied.Chebyshev, however, proved his celebrated theorem, that extended the law of largenumbers to any independent random variables with bounded variances, by the mostelementary algebraic methods, – just like it could have been proved before the invention ofthe analysis of infinitesimals. And it certainly contained as its simplest particular cases theresults of Bernoulli and Poisson. His proof is so simple that it can be explained during alecture in 15 or 20 minutes. Its underlying idea is concentrated in the so-called Chebyshevlemma that allows to estimate the probabilities of large values of a random variable by meansof its expectation 4 whose calculation or estimation is in most cases considerably simpler.Formally speaking, this lemma is so simple as to be trivial; however, neither Chebyshev’spredecessors or contemporaries, nor his immediate followers were able to appreciate properlythe extreme power and flexibility of its underlying idea. This power only fully manifesteditself in the 20 th century, and, moreover, mostly not in probability theory but in analysis andthe theory of functions.Chebyshev, however, did not restrict his attention to establishing the general form of thelaw of large numbers. Until the end of his life he continued to work also on the more difficultproblem, on extending just as widely the De Moivre – Laplace limit theorem. He developedin detail for that purpose the remarkable method of moments which {Bienaymé and} he hadcreated, and which still remains one of the most powerful tools of probability theory and isessentially important for other mathematical sciences. Chebyshev was unable to carry out hisinvestigations to a complete proof of the general form of the limit theorem. However, he hadcorrectly chosen the trail which he blazed to that goal, and his follower, Markov, completedChebyshev’s studies, although only at the beginning of this {the 20 th } century, after thelatter’s death.However, another follower of Chebyshev, Liapunov, published the first proof of thegeneral form of the limit theorem somewhat earlier, in 1901. He based it on a completelydifferent method, the method of Fourier transforms, not less powerful and nowadaysdeveloped into a vigorous theory of the so-called characteristic functions which are one ofthe most important tools of the modern probability theory. When applying the modern formof the theoryof these functions, the Liapunov theorem is proved in a few lines, but at that time, when thattheory was not yet developed, the proof was extremely cumbersome.In a few years after Liapunov had published his results, Markov, as indicated above,showed that the limit theorem can be proved under the same conditions by the moreelementary method created by {Bienaymé and} Chebyshev. And we ought to note that thelater development of this issue confirmed that the conditions introduced by Liapunov andMarkov for proving the limit theorem were very close to their natural boundaries: thosediscovered recently were only insignificantly wider.