7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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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
L = Ek( k| ξ − Eξ|kvarξ)kk3/ 23(4)ensures the closeness of F n (x) = P( n < x) to the class G of normal distributions g in the sensethat, in a uniform metric 1 (F; g) = sup |F(x) – g(x)|, (5)the estimation(F n ; G) ≤ CL (6)takes place. Here, C is an absolute constant whose best possible value is not yet determined.We (G&K, §1.3) have spoken about Linnik’s works adjoining this problem.If 1 , 2 , … , n , … is a sequence of independent identically distributed summands, then,owing to the abovementioned Khinchin’s theorem, the distribution of the variables k =ξnk− ABkkwhich correspond to any subsequence of n k can only converge to an infinitely divisibledistribution. In the general case this fact is, however, hardly interesting since the initialdistribution of each of the n ’s can be such that, whichever subsequence and values of A k andB k be chosen, the limiting distribution (naturally, in the sense of weak convergence) will onlybe degenerate. Prokhorov [11], however, proved that under these conditions and without anynorming a relation (F n ; D) 0 as n with D being the class of all the infinitely divisibledistributions took place for a uniform metric (5) and distribution F n of the sum (3) whereasKolmogorov [152] showed that (F n ; D) ≤ Cn –1/5 where C was an absolute constant. It isunknown whether the order n –1/5 is definitive.It is natural to formulate the problem of deriving approximations g to the distributions F nof sums (3) in as wide as possible boundaries and being uniform in the sense of some metricin the space of distributions. Dobrushin’s work [3] provides an example of solving a rathercomplicated problem where this requirement of uniformity of the estimation is met. Heprovided a system G of limiting distributions approximating F n in the metric with respect tovariation 2 (F; g) = var (F – g) so that3 / 2C ln n 2 (F n ; G) ≤1/ 13nwhere C was an absolute constant. F n was the distribution of the number of the occurrencesof the separate states during n steps for a homogeneous Markov chain with two states.The third general desire, yet rarely accomplished, is the derivation of best estimates of theapproximation of distributions F n by distributions from some class G under various naturalconditions with respect to the construction of the sum n , i.e., the precise or asymptoticcalculation of expressionsE(F; G) = sup inf (F n ; g), F n ;∈ F, g ∈ G,
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[3] I bear in mind the well-known p
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successes of physical statistics. B
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classes of independent facts whose
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distribution is a corollary of the
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examine in the first place the curv
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12. According to Bortkiewicz’ ter
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generality, the similarities taking
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one on another, as well as the corr
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is inapplicable because the right s
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Instead, Slutsky introduced new not
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abandoned in August 1936, but it is
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last decades, mathematicians more o
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charged with making the leading ple
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motion and a number of others) are
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phenomena. It is self-evident that
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Such new demands were formulated in
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The addition of independent random
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automatic lathes, etc. Here, the ma
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11. Kolmogorov, A.N. Grundbegriffe
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period 1 and remained, until the ap
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of the analytical tool rather than
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with probability approaching unity,
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logic. The ensuing vagueness in his
Page 50 and 51: 2. Gnedenko, B.V. (1949), On Lobach
Page 52 and 53: will be sufficient, although not ne
Page 54 and 55: nlimk = 1P(| k (n) - m k (n) | > H
Page 56 and 57: favorite classical issue as the gam
Page 58 and 59: and some quite definite (not depend
Page 60 and 61: influenced by a construction that a
Page 62 and 63: P ij (1) = p ij (1) , P ij (t) =kP
Page 64 and 65: 4.2d. Bebutov [1; 2] as well as Kry
Page 66 and 67: are yet no limit theorems correspon
Page 68 and 69: described, from the viewpoint that
Page 70 and 71: conditional variance and determined
Page 72 and 73: Romanovsky [45] and Kolmogorov [46]
Page 74 and 75: Let S be the general population wit
Page 76 and 77: Part 1. Russian/Soviet AuthorsAmbar
Page 78 and 79: 2. On necessary and sufficient cond
Page 80 and 81: Gnedenko, B.V., Groshev, A.V. 1. On
Page 82 and 83: 52. ( (Mathematical Principl
Page 84 and 85: Kozuliaev, P.A. 1. Sur la répartit
Page 86 and 87: Obukhov, A.M. 1. Normal correlation
Page 88 and 89: 30. Généralisations d’un théor
Page 90 and 91: 22. Alcune applicazioni dei coeffic
Page 92 and 93: 10. A.N. Kolmogorov. The Theory of
Page 94 and 95: Kuznetsov, Stratonovich & Tikhonov
Page 96 and 97: In the homogeneous case H s t = H t
Page 98 and 99: to such a generalization. He only s
Page 102 and 103: as it is usual for the modern theor
Page 104 and 105: 1. {The second reference to Pugache
Page 106 and 107: Smirnov, Romanovsky and others made
Page 108 and 109: determined the precise asymptotic c
Page 110 and 111: for finite values of N, M and R 2 .
Page 112 and 113: Mikhalevich’s findings by far exc
Page 114 and 115: Uch. Zap. = Uchenye ZapiskiUkr = Uk
Page 116 and 117: Khinchin, A.Ya. 43. Math. Ann. 101,
Page 118 and 119: 7. DAN 115, 1957, 49 - 52.Pinsker,
Page 120 and 121: Anderson, T.W., Darling, D.A. (1952
Page 122 and 123: Statistical problems in radio engin
Page 124 and 125: observations for its power with reg
Page 126 and 127: securing against mistakes (A.N. Kry
Page 128 and 129: of the others, then its distributio
Page 130 and 131: In Kiev, in the 1930s, N.M. Krylov
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7. Probability and Statistics Soviet Essays - Sheynin, Oscar

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