On the convergence in law of almost all sums of independent random variables

The paper deals with sums of independent and identically distributed random variables defined on some probability space which are multiplied by random coefficients. These coefficients are the values of independent random variables defined on another probability space. We obtain conditions for the weak convergence of weighted sums, for almost all coefficients, to some infinitely divisible distribution. The limit distribution for these sums is found. © 1995 Plenum Publishing Corporation

Convergence of Insurance Payout Stochastic Processes to Generalized Poisson Process

© 2015, Springer Science+Business Media New York. We consider stochastic processes describing the size of a company’s insurance payouts in the case of a growing number of clients. Convergence of such processes in Skorokhod space is proved. As a result, a functional limit theorem for risk processes is obtained, which allows us to use well-known formulas for estimating an insurance company’s ruin probability in the considered case

The probability of successful allocation of particles in cells (the general case)

© 2015, Springer Science+Business Media New York. Let pnN be the probability of successful allocation of n groups of particles in N cells with the following assumptions: (a) each group contains m particles and has allocation as a general allocation scheme; (b) each cell contains at most r particles from the same group; (c) events connected with different groups are independent. We obtain an asymptotically exact bound of pnN as n,N →∞ such that n/N is bounded. Applications to problems in error-correcting coding are considered

An exponential inequality and strong limit theorems for conditional expectations

An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for the nonuniformly distributed allocation scheme. © 2010 Akadémiai Kiadó, Budapest, Hungary

The probability of correcting errors by an antinoise coding method when the number of errors belongs to a random set

We consider n messages of N blocks each, where each block is encoded by some antinoise coding method. The method can correct no more than one error. We assume that the number of errors in the ith message belongs to some finite random subset of nonnegative integer numbers. Let A stand for the event that all errors are corrected; we study the probability P(A) and calculate it in terms of conditional probabilities. We prove that under certain moment conditions probabilities P(A) converge almost sure as n and N tend to infinity so that the value n/N has a finite limit. We calculate this limit explicitly. © 2010 Allerton Press, Inc

Almost sure limit theorems for random allocations

Almost sure limit theorems are presented for random allocations. A general almost sure limit theorem is proved for arrays of random variables. It is applied to obtain almost sure versions of the central limit theorem for the number of empty boxes when the parameters are in the central domain. Almost sure versions of the Poisson limit theorem in the left domain are also proved. © 2005 Akadémiai Kiadó, Budapest

Strong laws of large numbers for random forests

Random forests are studied. A moment inequality and a strong law of large numbers are obtained for the number of trees having a fixed number of nonroot vertices. © Akadémiai Kiadó, Budapest, Hungary, 2009

Asymptotic normality of kernel type density estimators for random fields

Kernel type density estimators are studied for random fields. It is proved that the estimators are asymptotically normal if the set of locations of observations become more and more dense in an increasing sequence of domains. It turns out that in our setting the covariance structure of the limiting normal distribution can be a combination of those of the continuous parameter and the discrete parameter cases. The proof is based on a new central limit theorem for α-mixing random fields. Simulation results support our theorems. © Springer 2006

Convergence of random step lines to Ornstein-Uhlenbeck-type processes

The paper deals with random step-line processes defined by sums of independent identically distributed random variables multiplied by independent indicators. These processes describe some models in which random variables are replaced with other ones. We prove the convergence in distribution of such processes to the weighted Ornstein-Uhlenbeck process. ©1998 Plenum Publishing Corporation

Almost sure versions of some analogues of the invariance principle

Almost sure versions of some functional limit theorems for random step lines and random broken lines defined by sums of independent identically distributed random variables with replacements are obtained