Weak convergence of random sums and maximum random sums under nonrandom norming

Necessary and sufficient conditions are obtained for the weak convergence of random sums and maximum random sums of independent identically distributed random variables. Limit distributions for these sums are described. The indices are not assumed to be independent of the summands. Supported in part by Grants NFW000 and NFW300 from the International Science Foundation and the Russian Government and by Grant No. 93-011-1446 from the Russian Foundation for Fundamental Research. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Some limit theorems for independent random processes at random points in time and random elements defined by them

The paper deals with random variables which are the values of independent identically distributed stochastic processes at random points in time. We obtain conditions for the weak convergence of their sums, at almost all points in time, to the same infinitely divisible distribution and describe the limit distribution for these sums. Also we obtain an analog of the Donsker theorem and limit theorems for empirical processes for such random variables. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

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. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-16099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Invariance principle for nonhomogeneous random walks on the grid ℤ1

A nonhomogeneous random walk on the grid ℤ¹ with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables and Brownian motion. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 459–472, September, 1999.     

The convergence of the random lines defined by observations of a stochastic process

We deal with independent random variables which are the values of a stochastic process taken at random points in time. So we have random variables depending upon a random parameter. We obtain the conditions providing the weak convergence of random lines defined by sums or maxima or bilinear forms of these random variables for almost all values of the parameter, to one and the same stochastic process. These limit stochastic processes are described. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

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. Supported by the Hungarian Foundation for Scientific Research (grant No. OTKA-T016933-1996) and by the Hungarian Ministry of Culture and Education (grant No. 179-1995). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Remarks on estimates in the total-variation metric

For sums of independent lattice random variables, the limiting normal approximation is trivial if the total-variation distance is considered. In this paper, we show that the normal distribution can be replaced by a suitably chosen lattice distribution. Mixtures of distributions are approximated by a convolution of the normal and Poisson laws. Construction of an asymptotic expansion is discussed. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 1–16, January–March, 2000. Translated by V. Čekanavičius     

Indistinguishability of particles or independence of the random variables?

After establishing the formula of information, we turn to two cases: to the independent case and to the dependent case. If the characteristics of the particles (e.g., the energy) as random variables are independent, we come in the discrete case to the Boltzmann distribution, while in the continuous case we come to the Gibbs formula-irrespectively of whether the particles are distinguishable or not. Our effort is nimed at clarity of the ideas, not at their mathematical completeness. Supported by the Hungarian National Foundation for Scientific Research (grant No. T 016384 and No. 4007-016237) Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part III.     

Distribution of the lower boundary functional of the step process of semi-Markov random walk with delaying screen at zero

On the basis of a given sequence of independent identically distributed pairs of random variables, we construct the step process of semi-Markov random walk that is later delayed by a screen at zero. For this process, we obtain the Laplace transform of the distribution of the time of the first hit of the level zero. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 912–919, July, 2007.     

Sums of independent poisson subordinators and their connection with strictly α-stable processes of Ornstein-Uhlenbeck type

We describe a construction in which the discrete time of a sequence of independent, identically distributed random variables changes with the Poisson time. The Poisson time is independent of this sequence. The defined process with continuous time is called a random index process. We establish several properties of random index processes. We study asymptotics of sums of independent, identically distributed random index processes in the case where elements of the initial sequence have strictly α-stable distribution. By calculating characteristic functions we establish relationships of these sums with strictly α-stable processes of the Ornstein- Uhlenbeck type. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 361, 2008, pp. 123–137.     

A criterion of convergence of nonrandomly centered random sums of independent identically distributed random variables

Necessary and sufficient conditions are presented for the weak convergence of random sums of independent identically distributed random variables in the double array scheme. As corollaries, two criteria of the normal convergence of random sums are given. Supported by the Russian Foundation for Fundamental Research (grant No. 96-011-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

On the law of the iterated logarithm for increments of sums of independent random variables

We obtain the law of the iterated logarithm for increments of sums of independent random variables. Our results generalize the Kolmogorov theorem and the Hartman—Wintner theorem on the law of the iterated logarithm. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 174–186.     

An analog of the Baum-Katz theorem for weakly dependent random variables

In this paper we study the limiting behavior of sums of dependent random variables under a strong mixing condition. We obtain conditions for which an analog of the Baum-Katz theorem holds and cite an example showing their optimality. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 360–368, March, 2000.     

Limit theorems for random processes constructed from sums of independent identically distributed random variables

There are proved limit theorems for random processes constructed from sums of independent identically distributed random variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 141–145, January, 1991.     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

On distributions of random variables chosen at random

The existence of typical distributions for random variables chosen at random from a finite-dimensional random variable vector space of high dimension is established. Possible typical distributions are described, and conditions for the typical distribution to be standard Gaussian are given. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 153–160. Translated by the author.     

Integrals of certain random functions with respect to general random measures

For random functions that are sums of random functional series, we determine an integral over a general random measure and prove limit theorems for this integral. We consider the solution of an integral equation with respect to an unknown random measure. National University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1087–1095, August, 1999.     

On the functional law of the iterated logarithm for trimmed sums

New conditions for the functional law of the iterated logarithm for trimmed sums of symmetric independent identically distributed random variables are obtained. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 119–125. Translated by N. B. Lebedinskaya.     

On some limit theorems for the maximum of sums of independent random processes

We investigate conditions for the weak convergence of the maximum of sums of independent random processes in the space L _p and present several applications to the asymptotic analysis of certain ω ²-type statistics. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1664–1674, December, 2008.