7. Probability and Statistics Soviet Essays - Sheynin, Oscar

3) After the work of Borel (the strong law of large numbers) and Khinchin (the law of theiterated logarithm), the problems of estimating various probabilities connected with thebehavior of consecutive sums have ´been becoming ever more prominent (G&K, §§1.6 – 1.7).During the new period (1948 – 1957), the main attention was concentrated on studying thepaths of discrete Markov processes by the tools of the analytic theory of these processes withcontinuous time (G&K, §5).Limit theorems on the behavior of discrete Markov processes with a large number of smalljumps are derived by approximating these processes by continuous Markov processesobeying the Fokker – Planck equation. When the discrete process has a small number of largejumps, the approximation is achieved by a process with continuous time having paths withgaps of only the first kind and directed by integro-differential equations (Kolmogorov [77,§19] or by their various generalizations. Theoretically, the classical limit theorems and thepropositions of the second direction for independent summands are included into this patternas particular cases.We (G&K, end of §5) noted that up to 1947 the actual implementation of the second partof this program (application of processes continuous in time and having jumps) did not yetbegin if cases of the second direction only connected with processes with independentincrements are excluded.I postpone until the next section the review of the new contributions devoted tostrengthening the classical limit theorems on attraction to the Gauss distribution and on theissues of Item 2. Such a unification of the second direction with all the classical issues iscaused by the fact that, although this direction had ideologically grown out of the notions ofprocesses with continuous time, during its practical realization it followed along the linesunconnected with its origin.As to the studying of the behavior of consecutive sums, I should indicate first of all thatfrom among the difficult problems of the previous period Prokhorov [2; 3] essentiallyadvanced the determination of sufficient conditions for the applicability of the strong law oflarge numbers to sums of independent terms. His conditions are very close to necessaryrestrictions and they are both necessary and sufficient for the case of Gaussian terms. Anumber of works is devoted to the law of the iterated logarithm (Diveev [6], Sapogov [6; 15],Sarymsakov [30]) and to the strong law of large numbers for dependent variables (Bobrov[9]).Some findings in the contributions of Prokhorov and Skorokhod which are describedbelow are still formulated and proved only for sums of independent terms but in essence theirmethods lack anything inseparably linked with such a restriction.Gikhman, whose works [4 – 6] appeared in 1950 – 1951, studied processes (t) of a rathercomplicated nature which pass over in the limit into simpler Markov processes withcontinuous time *(t). Prokhorov [8; 16] and Skorokhod [4; 7; 8] investigated processes ofaccumulating sums of independent terms or discrete Markov processes, supplemented so asto apply their method, to those of continuous time. Their (and Gikhman’s) *(t) was aMarkov process with continuous time. The transition to *(t) was made because its analyticnature is simpler (probabilities connected with it admit of a strict analytical expression) 4 .Here is the general pattern of all such investigations that explicitly appeared in Americanwritings (Kac, Erdös et al): A process (t) is considered as being dependent on parameter n(the number of terms in the sum; or some indicator of their smallness; etc); it is required todetermine the conditions under which the functional F(x), defined in the space X of therealization of the processes (t) and *(t) (by some method the space is made common), obeythe relationEF() EF(*) as n . (1)