共查询到20条相似文献,搜索用时 15 毫秒
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V. V. Petrov 《Journal of Mathematical Sciences》1992,61(1):1905-1906
Upper bounds are obtained for the absolute moments of order p>1 of sums of independent random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 120–121, 1989. 相似文献
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Let {Xk} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s
Fn(x) of sums
is studied, where 0<α≤2, ank>0, 1≤k≤mn, and, as n→∞, bothmax
1≤k≤mna
nk→0 and
. It is shown that such convergence, with suitably chosen An's and necessarily stable limit laws, holds for all such arrays {αnk} provided it holds for the special case αnk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence
of the moments of the sequence {Fn(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums
, where an≠0, bn>0, andmax
1≤k≤n ak/bn→0. The strong law is said to hold if there are constants An for which Sn→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which bn/|an|<x if x>0. The main result is as follows. If the strong law holds,EN (|X1|)<∞. If
for some 0<p≤2, then the strong law holds with
if 1≤p≤2 and An=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various
conditions imposed on F(x), the coefficients an and bn, and the function N(x).
Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993. 相似文献
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Large deviations of sums of independent random variables 总被引:3,自引:0,他引:3
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Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the
functionF(p
1,...,pn;t)=P(X1+...+Xn≤t) is Schur-concave with respect to (p
1,...,pn) for every realt, whereX
i are independent geometric random variables with parametersp
i. A generalization to negative binomial random variables is also presented. 相似文献
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A sequence (μ
n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ
n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence.
Let ‖μ‖ denote the total mass ofμ andδ
0 denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in
the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible
probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague
convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e
−1] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x
1 + ... +x
n)/a
n, an → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ
0. If furthermore the ratiosa
n+1/a
n are bounded above and below by positive numbers, thenL(x)=P[|X
1|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea
n → ∞ so that the limit measure=βδ
0.
To the memory of Shlomo Horowitz
This research was partially supported by the National Science Foundation. 相似文献
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L. V. Rozovsky 《Journal of Mathematical Sciences》2009,159(3):341-349
Let Sn = X1 + · · · + X
n
, n ≥ 1, and S
0 = 0, where X
1, X
2, . . . are independent, identically distributed random variables such that the distribution of S
n/B
n converges weakly to a nondeoenerate distribution F
α
as n → ∞ for some positive B
n
. We study asymptotic behavior of sums of the form
where
a function d(t) is continuous at [0,1] and has power decay at zero,
Bibliography: 13 titles.
Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122. 相似文献
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V. V. Petrov 《Journal of Mathematical Sciences》2005,127(1):1763-1766
Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 158–164.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by V. V. Petrov. 相似文献
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Professor S. D. Chatterji 《Probability Theory and Related Fields》1969,13(3-4):338-341
Summary We prove that the only continuous functions representable as a series of Rademacher functions on the unit interval are the linear functions. A generalization using general k-expansions of real numbers is also proved. 相似文献
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In 1952 Darling proved the limit theorem for the sums of independent identically distributed random variables without power moments under the functional normalization. This paper contains an alternative proof of Darling’s theorem, using the Laplace transform. Moreover, the asymptotic behavior of probabilities of large deviations is studied in the pattern under consideration. 相似文献
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O. P. Vinogradov 《Journal of Mathematical Sciences》1995,76(1):2208-2213
Many papers exist dealing with the distribution of the maximum of partial sums of independent random variables and studying
the relationship between the sums and the sample extremes (see [L. Takacs,Combinator Methods in the Theory of Stochastic Processes, Wiley, New York (1967)], [J. Calambos,The Asymptotic Theory of Extreme Order Statistics, Wiley, New York (1978)]). The present paper deals with the effect the sample extremes have on the distribution of the maximum
of sums of independent random variables.
Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993 相似文献