Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. II

Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 1, pp. 94–106, January–March, 1994.     

Boundary-value problems for sums of random variables,defined on a countably valued Markov chain. I

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 139–146, February, 1989.     

Limit theorems for sums of independent random variables defined on a nonrecurrent random walk

Let {X_i} _i= -, {_i} _i=1 be two independent sequences of randoms variables, where the _i are identically distributed and assume integer values. LetIn the paper the question of the asymptotic behavior as n of the quantity is considered. It is shown that the distribution of W_n converges to the distribution of the normal law and that the estimate of the rate of convergence has the same order as the classical estimate of Berry-Esseen.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 17–29, 1979.The author is grateful to I. A. Ibragimov for his attention to the present work and for useful discussions.     

Limit distributions of sums of m-dependent Bernoulli random variables

Summary The set of limit distributions of row sums of a triangular array of Bernoulli random variables which is strictly stationary and m-dependent in each row is characterized. Necessary and sufficient conditions for the convergence of the row sums to a given limit distribution are found. The case of convergence to a Poisson distribution is given special attention.     

Boundary-value problems for sums of random variables,defined on a countably-valued markov chain. II

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 590–597, May, 1989.     

Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

We study the fractal properties of distributions of random variables digits of polybasic Q-representations (a generalization of n-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.     

Limit behavior of sums of a random number of independent random variables

Lithuanian Mathematical Journal -     

Limit theorems for sums of dependent random variables

Lithuanian Mathematical Journal -     

Limit theorems for sums of dependent random variables

Summary In Lai and Stout [7] the upper half of the law of the iterated logarithm (LIL) is established for sums of strongly dependent stationary Gaussian random variables. Herein, the upper half of the LIL is established for strongly dependent random variables {X _i} which are however not necessarily Gaussian. Applications are made to multiplicative random variables and to f(Z _i ) where the Z _i are strongly dependent Gaussian. A maximal inequality and a Marcinkiewicz-Zygmund type strong law are established for sums of strongly dependent random variables X _i satisfying a moment condition of the form E¦S _a,n ¦^pg(n), where , generalizing the work of Serfling [13, 14].Research supported by the National Science Foundation under grant NSF-MCS-78-09179Research supported by the National Science Foundation under grant NSF-MCS-78-04014