7. Probability and Statistics Soviet Essays - Sheynin, Oscar

In the particular case of a characteristic functionalF() = 1 if ∈A and = 0 otherwiseof the set A ⊆ X the problem is reduced to that of the convergence of probabilitiesP( ∈A) P(* A). (2)By definition of weak convergence P P * (see §1) of distribution to distribution * inX, this is the convergence (1) for all bounded continuous functionals F which indeedGikhman, Prokhorov and Skorokhod proved under sufficiently wide assumptions. If thefunctional F is not only continuous but also smooth, then, in some cases, even goodquantitative estimations of the rapidity of the convergence can be derived (Gikhman [8]).Regrettably, in the most interesting problem of convergence (2) the functional F isdiscontinuous. Nevertheless, weak convergence (1) leads to (2) if the boundary of A hasprobability zero in the limiting distribution P * . But good quantitative estimates of theconvergence are difficult to make. Prokhorov [16] had by far overcome his predecessors byobtaining in some simple problems a remainder term of the order of n –1/8 when it wasexpected to be of the order n –1/2 .Only in such simple problems as the estimation of the probability P( ≤ a) for themaximum = maxµ − npnNp( 1−p), 0 ≤ n ≤ Nin the Bernoulli trials Koroliuk [10] was able to arrive at asymptotic expressions with aremainder term of the order of 1/N. With similar refinements Smirnov [12] and Koroliuk [5]solved some problems on the probability of passing beyond assigned boundaries forwanderings connected with issues in mathematical statistics 5 . Gikhman [16; 19] had alsoapplied methods explicated in this section to problems in the same discipline. Restricting myattention to this very general description of the pertinent writings, I note that much dependsthere on a successful choice of a topology of the space of realizations X (see §1).5. Distributions of Sums of Independent and Weakly DependentTerms and Infinitely Divisible Distributions5.1. Owing to the insignificant success in estimating the remainder terms in limit theoremsof the general type (§4), the study of the limiting behavior of the distributions of the sums ofa large number of independent and weakly dependent terms remains important in itself.Many works of the last decade belong here. Kolmogorov, in his essay [136], attempted tosystematize the trends in the works done during the latest period and to list the problems tobe solved next in this already greatly studied sphere. The convergence to a limitingdistribution of adequately normed and centered sums usually is not the real aim ofinvestigation. It is usually required to have a good approximation to the distribution F n of n = 1 + 2 + … + n (3)in the form of a distribution g belonging to some class of distributions G. Thus, forindependent terms k the smallness of the Liapunov ratio