Stable convergence of random sequences with random indices

In this paper, we give a complete solution to Rényi's conjecture about stable convergence under a general metric space setup, and provide new Anscombe-type theorems concerning the stable convergence of random sequences with random indices. In particular, we present a random version of the famous arcsine law. Supported by the Postdoctoral Foundation of China. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Convergence of series of probabilities of large deviations for sums of random variables with multidimensional indices

One finds necessary and sufficient conditions for the convergence of series of weighted probabilities of large deviations for sums of independent random variables with multidimensional indices. These conditions are expressed in terms of the initial distribution.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 114–131, 1986.     

Asymptotic distribution of sequences with random indices

A general form is determined for the limit distribution function of a sequence of random vectors with random indices (S₁(N _n ⁽¹⁾ ) ..., Sr(N_n ^(r)) in the case when sequences (S₁(n), ..., S_r(n)) and (N _n ⁽¹⁾ , ..., N _n ^(r) ) for appropriate normalization have a nonsingular joint limit distribution.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 705–712, December, 1969.     

Convergence of weighted sums of independent random variables

Let {X_k} 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 F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=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 {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<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 A_n=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 a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Convergence of distributions of extremal independent random variables

Kaunas Polytechnic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 2, pp. 219–232, April–June, 1990.     

Moments of the maximum of normed partial sums of random variables with multidimensional indices

Summary For a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm.     

On limit distributions of randomly indexed multidimensional random sequences with an operator normalization

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 85–100, 1991.     

The asymptotics of randomly indexed random sequences: Independent indices

Translated from:Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, 1989, pp. 60–72.     

Asymptotics of randomly indexed infinite-dimensional random sequences: Independent indices

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 38–44, 1990.     

Convergence and relative compactness in distribution of sequences of random hypermeasures

Let Φ be a compact set in a vector space equipped with a convergence which is metrizable in Φ but not certainly in the whole space. We endow the space of continuous on Φ linear functionals on span Φ with the norm \( {\left\| u \right\|_\Phi } = \sup \varphi \in \Phi \left| {u\varphi } \right| \) and call the elements of the completion of Φ hypermeasures. We prove theorems on the convergence in probability or in distribution and relative compactness in distribution of a sequence of random hypermeasures.     

Convergence of two functionals of asymptotically normal sums of independent random variables

We consider two functionals of sums of independent random variables and demonstrate that the validity of the central limit theorem for the sums of independent random variables that enter the arguments of those functionals is a sufficient condition for one of the functionals and a necessary and sufficient condition for the other one to have a weak limit. Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

A remark on the law of the logarithm for weighted sums of random variables with multidimensional indices

In this work, a sharp upper bound on the law of the logarithm for the weighted sums of random variables with multidimensional indices is obtained. The main result improves the result in [Li, Rao and Wang, 1995. On strong law of large numbers and the law of the logarithm for weighted sums of independent random variables with multidimensional indices. J. Multivariate Anal. 52, 181–198], partly.