Potential theory for finitely additive Markov chains

In this paper potential theory is developed for finitely additive Markov chains and this is used to obtain various characterization theorems for discrete time Markov chains with an arbitrary state space, with finitely additive stationary transition probabilities and a finitely additive initial distribution.     

On the use of purely finitely additive multipliers in mathematical programming

In this paper, duality results are obtained for the problem of finding =inf{f(x):g(x)0,g(x)Y,xC}, where the constrainst spaceY is someL -space, by using the norm topology ofY. The corresponding multiplier is the sum of a countably additive part and a purely finitely additive part. Conditions are given such that the purely finitely additive part may be discarded.     

A functional central limit theorem for a generalized occupancy problem

A functional central limit theorem is proved for a class of finitely exchangeable random variables which are based on an occupancy scheme.     

The Edgeworth's conjecture in finitely additive production economies

This paper presents the core-Walras equivalence theorem in the very general setting of finitely additive coalition production economies. In the case of infinitely many commodities, a version of the equivalence result is proved without necessarily assuming that the positive cone has non-empty interior or that preferences satisfy cone conditions.     

Functional limit theorems for U-statistics in the degenerate case

Sen (1974) established multidimensional functional limit theorems for generalized U-statistics for estimable parameters which are stationary of order zero (the nondegenerate case). Similar univariate and bivariate functional limit theorems are derived for U-statistics in the degenerate case. The latter are close connected with one- or two-sample Cramér-von Mises statistics.     

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.⁽⁷⁾ Then we use the chaining argument with stratification adapted from that of Ossiander.⁽¹¹⁾ The results of this paper generalize those of Levental⁽¹⁰⁾ and Ossiander.⁽¹¹⁾     

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. Partially supported by Lithuanian Foundation of Studies and Science     

A general central limit theorem for strong mixing sequences

A central limit theorem for strong mixing sequences is given that applies to both non-stationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by Politis, Romano and Wolf in 1997.     

Joint discrete limit theorems in the space of analytic functions for general Dirichlet series

We prove joint discrete limit theorems in the sense of weak convergence of probability measures in the space of analytic functions for general Dirichlet series.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 104–116, January–March, 2005.Translated by R. Macaitien     

A local limit theorem for the probability of ruin

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs to S(v) with v≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.     

α-Stable limit theorems for sums of dependent random vectors

Several α-stable limit theorems for sums of dependent random vectors are proved via point processes theory; p-mixing, m-dependence, and the type of mixing treated within the extreme value theory are considered.     

Some estimates in theL p metric in the central limit theorem for additive functions

The convergence of the value distributions of a normalized sequence of strongly additive arithmetic functions to the standard normal law in theL _p metric is considered. An asymptotic formula for the variance is obtained. In both cases, the remainders are expressed in terms of the third absolute moment of the additive function. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 74–80, January–March, 1999. Translated by A. Mačiulis     

The limit absorption principle and homogenization procedure for periodic elliptic operators

For a periodic matrix elliptic operator \(A_\varepsilon \) with (x ?-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent 〈x〉^?1/2?· (\(A_\varepsilon \) ? (η ± i? ^σ )I)^?1〈x〉^?1/2?· has a limit in the operator norm in L ₂ as ? → 0 provided that η > 0, · > 0, and 0 < σ < 1/2.     

Three limit theorems for vacancy in multivariate coverage problems

Three limit theorems describing asymptotic distribution of vacancy in general multivariate coverage problems are proved, in which nk-dimensional spheres are distributed within a k-dimensional unit cube according to a density f. The first result (a central limit theorem) describes the case where the proportion of vacancy converges to a fixed constant lying between 0 and 1. The last two results treat the case where the proportion of vacancy tends to 1 as n → ∞. Results of this nature have hitherto been available only for restricted k and/or for f equal to the uniform density.