On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables

Lithuanian Mathematical Journal - We present upper bounds of the integral $$ {int}_{-infty}^{infty }{left|xright|}^lleft|mathrm{P}left{{Z}_N0left({S}_N{X}_1+dots +{X}_Nright) $$ of...     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

On the global central limit theorem forM-dependent random variables

Published in Lietuvos Matematikos Rinkinys, Vol. 34, No. 2, pp. 259–265, April–June, 1994.     

On the global central limit theorem for ϕ-mixing random variables

The estimate of the remainder term is obtained in the global central limit theorem for π-mixing r.v.s. As a consequence of Theorem 1 the convergence rate of absolute moments for sums of π-mixing r.v.s. to corresponding absolute moments of the normal r.v. is found. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 233–247, April–June, 1995.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

On the rate of convergence in the central limit theorem for sequences of weakly dependent Hilbert-valued random variables

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 194, pp. 174–175, 1992.     

Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.     

On a lower bound of the rate of convergence in the central limit theorem form-dependent random variables

We derive a lower bound of the rate of convergence in the central limit theorem form-dependent random variables under the finiteness of the fifth absolute moments of summands. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 388–398, July–September, 1997.     

A limit theorem for multiple sums of identically distributed independent random variables

One investigates the asymptotic behavior (with respect to a small parameter) of series of probabilities of large deviations for sums of random variables with multidimensional indices.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 265–277, 1986.     

A limit theorem for sums of independent,nonidentically distributed random variables

A generalization and refinement of Chen's theorem related to a strong law of large numbers for sums of independent, nonidentically distributed random variables are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta in im. V. A. Steklova AN SSSR, Vol. 85, pp. 188–192, 1979.     

A limit theorem for a certain transform of sums of independent random variables

Summary A certain transformation of distribution functions (d.f.'s) of positive random variables (r.v.'s) has been studied by the author and Harkness in [2] and [3]. In this paper, a limit theorem concerning such a transform of convolutions of d.f.'s is proved.     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.     

Vague convergence of sums of independent random variables

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, a_n → ∞, 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.