Direct calculation of probabilities of sums of independent lattice random variables

Kaunas Academy of Agriculture, Noreikiškes, 4324 Kaunas, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 1, pp. 52–62, January–March, 1994.     

Superlarge deviation probabilities for sums of independent lattice random variables with exponential decreasing tails

In the note we study large and superlarge deviation probabilities of sum of i.i.d. lattice random variables, whose distribution function has an exponentially decreasing tail at infinity.     

Complete convergence for randomly indexed sums of random variables

The complete convergence of normed sums of independent identically distributed random variables with random indices is studied. Some applications for subsequences and sequences with multidimensional indices are given. Supported by the Hungarian Foundation for Scientific Research (grant OTKA-1650/1991). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Two identities for sums of lattice random variables

For the sums S_n of independent, identically distributed random variables X_k, assuming nonnegative integer values, if EX₁ < 1/m,=">

Small deviation probabilities for sums of independent positive random variables

In this note, we give estimates of small deviation probabilities of the sum ∑_j≥1 λ_j X_j, where {λ_j} are nonnegative numbers and {X_j} are i.i.d. positive random variables that satisfy mild assumptions at zero and infinity. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 151–167.     

On probabilities of moderate deviations of sums for independent random variables

We investigate asymptotics of probabilities of moderate deviations and their logarithms for an array of row-wise independent random variables with finite variations and finite one-sided moments of order p > 2. The range of the zone of normal convergence is calculated in terms of Lyapunov ratios constructed from the positive parts of the random variables. Bounds for probabilities of moderate deviations are also derived in the case where the normal convergence fails. Bibliography: 16 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 200–215.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by A. N. Frolov.     

Tail asymptotics of randomly weighted sums of dependent strong subexponential random variables

Lithuanian Mathematical Journal - Following [C. Yu, Y. Wang, and D. Cheng, Tail behavior of the sums of dependent and heavy-tailed random variables, J. Korean Stat. Soc., 44(1):12–27, 2015],...     

Lower estimations of the deviation probabilities for sums of Bernoulli random variables

We obtain lower bilateral and unilateral estimations of the deviation probability of the arithmetic mean of independent Bernoulli random variables from mathematical expectation.     

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d.standard uniform random variables.This extends a corresponding result of Korshunov.As an application,we generalize a result of Tang,the uniform asymptotic estimate for the finite-time ruin probability,to the whole strongly subexponential class.     

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].     

Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let {_i} _i=1 be a sequence of independent identically distributed nonnegative random variables, S _n = ξ₁ + ? +ξ_n. Let Δ = (0, T] and x + Δ = (x, x + T]. We study the ratios of the probabilities P(S _n ε x + Δ)/P(ξ ₁ ε x + Δ) for all n and x. The estimates uniform in x for these ratios are known for the so-called Δ-subexponential distributions. Here we improve these estimates for two subclasses of Δ-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T ≤ ∞. Also, a characterization of the class LC is given.     

Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables

The series is studied, where S_n are the sums of independent equally distributed random variables, _n is a sequence of nonnegative numbers, >0, and >0 is an arbitrary positive number. For a broad class of sequences _n, the necessary and sufficient conditions are established for the convergence of this series for any >0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 770–784, June, 1993.     

Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.