7. Probability and Statistics Soviet Essays - Sheynin, Oscar

motion and a number of others) are connected with this picture. Separate problems of thiskind have been comparatively long ago solved by the theory of probability. Nevertheless, nosatisfactory general method existed, and this led to the need for introducing very restrictiveassumptions as well as for inventing a special method for each problem. In 1932 Petrovskydiscovered a remarkable method connecting the most general problem of random walks withproblems in the theory of differential equations and thus providing a possibility of a commonapproach to all such problems. At the same time, his method eliminates the need for almostall the restrictive assumptions so that the problems can be formulated in their naturalgenerality.Since the De Moivre limit theorem; the Liapunov theorem; and all their later extensionsincluding those considering many-dimensional cases, are particular instances of the generalproblem of random walks, the Petrovsky method provides, in particular, a new, and, for thatmatter, a remarkable because of its generality proof of all these propositions. Moreover, thepower of this analytic method is so great that in a number of cases (as, for example, in theabovementioned law of the iterated logarithm) it enabled to solve also such problems that didnot yield to any other known method. The application of the Petrovsky method by otherworkers of the Moscow collective (Kolmogorov, Khinchin) had since already led to thesolution of a large number of problems of both theoretical and directly applied nature. In anad hoc monograph Khinchin showed that this method covered most various stochasticproblems.When speaking about the theoretical investigations of the Moscow school, it is alsonecessary to touch on its considerable part in the logical justification of the doctrine ofprobabilities. The construction of a robust logical foundation for the edifice of the theorybecame possible comparatively recently, after the main features of the building weresufficiently clearly outlined. And, although a large number of scientists from all quarters ofthe globe participated in constructing a modern, already clearly determined logical base forthe theory, it is nevertheless necessary to note that it was Kolmogorov who first achieved thisgoal having done it completely and systematically.[5] However, considering the development of a stochastic theory as its main aim, theMoscow school may nevertheless be reproached for not sufficiently or not altogethersystematically studying the issues of practical statistics. The main problem next in turn, towhich the Moscow collective is certainly equal, consists in revising and systematizing thestatistical methods that still include many primitive and archaic elements. This subject ispermanently included in the plans of the Moscow school and is just as invariably postponed.However, the collective has been accomplishing a number of separate studies of considerableworth in mathematical statistics. Here, statisticians have even created a special directionarousing ever more interest in the scientific world, viz, the systematic study of theconnections and interrelations between the theoretical laws of distribution and their empiricalrealizations. If a long series of trials is made on a random variable obeying a given law ofdistribution, and a graph of the empirical distribution obtained is constructed, we shall see aline unboundedly approaching the graph of the given law as the number of trials increases.The investigation of the rapidity and other characteristics of this approach is a natural andimportant problem of mathematical statistics. The closeness, and, in general, the mutuallocation of these two lines can be described and estimated by most various systems ofparameters; the limiting behavior of each of these parameters provides a special stochasticproblem. Its solution often entails very considerable mathematical difficulties, and, as a rule,is obtained as some limit theorem. During the last years, Moscow mathematicians achieved anumber of interesting results in this direction and Smirnov should be named here first andforemost. He had discovered a number of strikingly elegant and whole limiting relations andGlivenko and Kolmogorov followed suit. These works were met with lively response also by