7. Probability and Statistics Soviet Essays - Sheynin, Oscar

automatic lathes, etc. Here, the matter concerns stochastic processes in which chance entersas the random distribution of the number of calls in a telephone network or of the interruptionof the work of a lathe owing to some breakdown; or as the form of the law of distribution forthe duration of telephone calls; or of the time needed for repairing the lathes. Only undergreatly simplifying assumptions such processes conform to the pattern of processes withoutaftereffect having a discrete set of possible states and are {then} easily included in thegeneral theory. Without these simplifications such a subordination of the processes withoutaftereffect to the general theory is {also} possible but too complicated for direct application.In a number of works Khinchin developed special methods for examining the phenomena ofqueuing. He himself and N.V. Smirnov used these methods for solving the problems raisedby the Central Institute of Communications and pertaining to automatic telephony. Gnedenkoapplied the same methods to problems of the textile industry.Peculiar problems of spatial queuing originate when studying the crystallization of metalsand metallic alloys. Kolmogorov solved some of them as formulated by the Institute of Steel.M.A. Leontovich developed a statistical theory of bimolecular reactions which is a somewhatspecialized and complicated version of the theory of stochastic processes without aftereffecthaving a finite number of states. He issued from differential equations corresponding to thepattern of continuous time. Smirnov andV.I. Glivenko showed that a similar specializationand complication of classical Markov chains (with a discrete sequence of trials) should be anessential tool of the theory of heredity.We note finally that the extraordinary extension of the sphere of stochastic researchoutlined above would have been exceptionally difficult without a reconstruction of thelogical foundation of the theory of probability, – of its axiomatics and the system of its mainnotions. From among the various directions in justifying the theory only one is yet developedto an extent that enables it to provide a formally irreproachable system of the main notionscovering all the ramifications of the theory caused by the various requirements of physics andtechnology. This is the direction developing the axiomatics of probability theory by issuingfrom the definition of probability as an additive function of sets given on an adequate systemof sets of elementary events 11 . In particular, the modern theory of probability cannot besatisfied only by considering finite-dimensional laws of distribution. The study of continuousstochastic processes and of a number of other physical problems inevitably leads to theconsideration of random functions, or, in other words, of laws of distríbution in functionalspaces. Slutsky and Kolmogorov engaged in a systematic construction of the theory of thesefunctions.3. From among the general problems of mathematical statistics, Soviet mathematiciansdeveloped with maximal success, first, those concerned with the statistical detection of latentperiodicities and the establishment of the formulas of forecasting; and, second, problems ofthe statistical determination of distribution functions.The classical theory of periodograms enables to analyze a series of random variables if weassume that it is made up by superimposing several periodic oscillations and additionalperturbations, independent from one trial to another one. The last-mentioned hypothesis is,however, usually arbitrary and Slutsky and Romanovsky deeply studied all the circumstancesoccurring when we assume dependences between the random perturbations. Slutsky, in hisconcrete contributions, accomplished at the Geophysical Institute, offered many simple andelegant methods for analyzing series with a supposed latent periodicity.Meteorologists were very interested in establishing formulas of forecasts by statisticalmeans (we especially mention the works of V.Yu. Weese {Vise?}. The same problemespecially concerned the hydrometeorological service; incidentally, Slutsky worked on itsinstructions. A number of investigations was devoted to the composition of formulas fordetermining the prospects for the harvest by issuing from meteorological data (Obukhov).