7. Probability and Statistics Soviet Essays - Sheynin, Oscar

with probability approaching unity, becomes arbitrarily small as compared with the sums.The necessary and sufficient conditions under which such convergence takes place were alsodiscovered. It turned out, that, in addition to the normal law appropriate for the classical limittheorems, all the other stable laws, found by the French mathematician Lévy, can also appear,whereas the entire class of admissible laws coincides with the so-called infinitely divisiblelaws whose study was begun by the Italian de Finetti.A large part of the further work done by the Moscow school was connected with theconcept of random processes (for the time being, in its classical, non-quantumunderstanding). Two large fields were here studied:4) The theory of processes without aftereffect. Being a direct generalization of Markovchains, they are therefore called Markov processes. For them, the probabilities of transitionF(x; E; s; t) from state x at moment s to one of the states belonging to set E at moment t areconnected by the so-called equation of Smoluchowski or Chapman 12 .5) The theory of stationary random processes with their spectral theory.Kolmogorov originated the first of these directions. He discovered that, under some wideconditions and given transition probabilities, the non-linear Smoluchowski integral equationinvariably leads to some linear partial differential equation called after Fokker and Planck 13 .Still wider are the conditions under which a variable Markov process depending on aparameter asymptotically approaches an ideal Markov process obeying the Fokker – Planckequations. In such relations we perceive now the common root of all the limit theorems of theLaplacean and Liapunov type. Only from this point of view the fact that the classical functionof the normal density of probability(x; D) = (1/2 πD) exp(– x 2 /2D)is the solution of the equation of heat conduction∂ϕ∂=∂ ϕ ∂2 2/ D 2 / xceased to appear accidental.Mathematicians of the Moscow school (including, in particular, Petrovsky) and Bernsteinare studying this new vast field, termed theory of stochastic differential equations, thatopened up here. Most works of the Moscow school assume that the considered states of thesystem are represented by points of some compact part of space whereas Bernstein examinedwith special attention those new facts that appear when this restriction is abandoned. Hisgeneralization is all the more natural, since the special case of the classical limit theoremsleading to the normal probability distribution should indeed be considered on the entirenumber axis, i.e., on a non-compact set 14 .I am unable to dwell as minutely on the spectral and ergodic theories of stationary randomprocesses created (as a general mathematical theory) by Khinchin. This direction of researchoccupies a prominent place in the work of other representatives of the Moscow school aswell and many foreign mathematicians cultivated it. The remarkable investigations onstatistical periodography independently started by Slutsky have also joined Khinchin’sdirection of work. In the field of statistics, the importance of all this research is widelyrecognized abroad. One of Slutsky’s main contributions was reprinted {translated intoEnglish} in England on the initiative of the English statisticians. Wold’s book on stationarytime series published in Sweden was entirely based on the works of Khinchin, etc. For somereason the appreciation of the importance of the stochastic, statistical concept of oscillationswith a continuous spectrum for physics and mechanics, as insisted on by Wiener in asomewhat different form even before the appearance of the Moscow works, is established toa lesser degree. Here, the contributions of the Moscow school sometimes become known