7. Probability and Statistics Soviet Essays - Sheynin, Oscar

and the proof of their convergence as n to the respective moments of the normaldistribution.In a number of cases Markov surmounted great calculational difficulties. On principle,even more important was that he substantiated new limit propositions, prototypes of theso-called ergodic theorems. For the Markov chains, the distribution of n as n increasesever less depends on the value taken by 1 : a remote state of the system ever less dependson its initial state.The next direction of the theory of probability developed by Markov and otherresearchers before the Great October Socialist Revolution {before the Nov. 7, 1917, newstyle, Bolshevist coup} is connected with the construction of the theory of errors.Astronomers paid much attention to this subject, and their contribution was not restrictedto methodologically improving the exposition of already known results 3 .During the 19 th , and the beginning of the 20 th century, Buniakovsky {1846},Tikhomandritsky {1898}, Ermakov {1878}, Markov (1900) and Bernstein {1911}compiled textbooks on probability theory on the level corresponding to the contemporarystate of that science. Markov’s textbook played a considerable part in developingprobability theory in our country. He explicated a number of findings in sufficient detail,and, in the same time, in an elementary way 4 which fostered the readers’ interest not onlyin passive learning, but in active reasoning as well. Already in its first edition, Bernstein’sbook, distinguished by many peculiar traits, for a long time exerted considerable influence.Then, Slutsky (1912) acquainted his Russian readers with the new issues in mathematicalstatistics that had originated in England in the first decade of the 20 th century.The works of the two mathematicians, Bernstein and Slutsky, who played an importantpart in building up new directions of research in probability theory and mathematicalstatistics in our country 5 , began to appear in the years immediately preceding theRevolution. During the first period of his work, Bernstein examined such important issuesas the refinement of the {De Moivre –} Laplace theorem, the logical justification ofprobability theory, and the transfer of its peculiar methods to problems in the theory offunctions. It was in this very period that he was able to discover a remarkable proof of theWeierstrass theorem (1912). {At the time,} Slutsky studied problems in mathematicalstatistics chiefly connected with correlation theory.[3] Thus, already before the Revolution, scientific pre-requisites for the development ofprobability theory were created in our country. And the establishment, after the Revolution,of a vast network of academic and research institutes and of academies of sciences in theUnion Republics 6 fostered the growth of scientific investigations in many cities as well asthe creation of considerable mathematical bodies and the initiation of new directions ofresearch.In the then young Central Asian University {Tashkent} Romanovsky established aprominent school of mathematical statistics and the theory of Markov chains. In Moscow,in the nation’s oldest university, the well-known school of the theory of probability wascreated on the basis of the school of the theory of functions of a real variable. It is difficultto overestimate its influence on the development of probability theory during the latestdecades. The construction of the foundation of the theory; a vast development of theclassical issues concerning limit theorems for sums of independent variables; the conceptof stochastic processes (without aftereffect; stationary and with stationary increments,branching processes); the development of methods of statistical physics; of queuing,reliability and information theories; and many other issues are the subject of research doneby Moscow specialists. The beginning of stochastic investigations in Moscow wasconnected with two outstanding mathematicians, Khinchin and Kolmogorov.