Asymptotic Expansions of Large Deviations for Sums of Nonidentically Distributed Random Variables

The work is designated for obtaining asymptotic expansions and determination of structures of the remainder terms that take into consideration large deviations both in Cramer zones and Linnik power zones for the distribution function of sums of independent nonidentically distributed random variables (r.v.). In this scheme of summation of r.v., the results are obtained first by mainly using the general lemma on large deviations considering asymptotic expansions for an arbitrary r.v. with regular behaviour of its cumulants [11]. Asymptotic expansions in the Cramer zone for the distribution function of sums of identically distributed r.v. were investigated in the works [1,2]. Note that asymptotic expansions for large deviations were first obtained in the probability theory by J. Kubilius [3].     

On the Rate of Multivariate Poisson Convergence

The distribution of the sum of independent nonidentically distributed Bernoulli random vectors inR^kis approximated by a multivariate Poisson distribution. By using a multivariate adaption of Kerstan's (1964,Z. Wahrsch. verw. Gebiete2, 173–179) method, we prove a conjecture of Barbour (1988,J. Appl. Probab.25A, 175–184) on removing a log-term in the upper bound of the total variation distance. Second-order approximations are included.     

On random power series with nonidentically distributed coefficients

The distribution of the radius of convergence r(ω) of a random power series with nonidentically distributed coefficients is considered. The results obtained here extend the well-known work in the identically distributed case.     

A general estimate in the invariance principle

We obtain estimates for the accuracy with which a random broken line constructed from sums of independent nonidentically distributed random variables can be approximated by a Wiener process. All estimates depend explicitly on the moments of the random variables; meanwhile, these moments can be of a rather general form. In the case of identically distributed random variables we succeed for the first time in constructing an estimate depending explicitly on the common distribution of the summands and directly implying all results of the famous articles by Komlós, Major, and Tusnády which are devoted to estimates in the invariance principle.     

A limit theorem for sums of independent,nonidentically distributed random variables

A generalization and refinement of Chen's theorem related to a strong law of large numbers for sums of independent, nonidentically distributed random variables are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta in im. V. A. Steklova AN SSSR, Vol. 85, pp. 188–192, 1979.     

On the L~p-Consistency of Wavelet Estimators

This paper deals with the L~p-consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L~p-consistency of wavelet estimators for independent and identically distributed random vectors in R~d. Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function.     

A Laplace transform method for order statistics from nonidentical random variables and its application in Phase-type distribution

In this paper, we derive a method for obtaining the Laplace transform of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). A survey of the most important properties, applications and the o.s. of a Phase-type (PH) distribution are also presented. Two illustrative examples are provided.     

Strong law of large numbers for weighted sums

The law of large numbers for weighted sums of independent identically distributed random variables due to Chow and Lai is generalized to nonidentically distributed or pairwise independent summands. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic, Models, Moscow, Russia, 1996, Part. II.     

Records in the Fα-scheme. II. Limit laws

Limit theorems for the number of records in a sequence of independent nonidentically distributed random variables are obtained. A generalization of the so-called F^α-scheme is given. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 42–51. Translated by A. Sudakov.     

Multivariate order statistics based on dependent and nonidentically distributed random variables

In this paper some identities and inequalities which involve the joint distribution of order statistics in a set of dependent and nonidentically distributed random variables are derived. These identities and inequalities provide a unified way to handle the joint distribution of order statistics in a set of univariate or bivariate observations.     

Limit theorems for extremal flows of events

We study the max-generalized Cox process describing a model for an inhomogeneous flow of extremal events. The results obtained describe the asymptotic behavior of this process if it is generated by dependent and, in general, nonidentically distributed random variables.     

On Asymptotics of Large Increments of Sums in Non-i.i.d. Case

The almost sure asymptotic behavior of large increments of partial sums is investigated in case of independent nonidentically distributed random variables. One-sided variants of the Csörg–Révész strong approximation laws are obtained in non-i.i.d. case.     

Almost sure approximation theorems for the multivariate empirical process

Summary Let R(s,t) be the empirical process of a sequence of independent random vectors with common but arbitrary distribution function. In this paper we give an almost sure approximation of R(s,t) by a Kiefer process. The result continues to hold for stationary sequences of random vectors with continuous distribution function and satisfying a strong mixing condition.Supported in part by an NSF grant     

On the convolution of Mittag–Leffler distributions and its applications to fractional point processes

We obtain the distribution of the sum of independent Mittag–Leffler (ML) random variables which are not necessarily identically distributed. Firstly we discuss the corresponding known result for independent and identically distributed ML random variables which follows as a special case of our result. Some applications of the obtained result to fractional point processes are also discussed.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

Estimates in the invariance principle in terms of truncated power moments

We obtain estimates for the distributions of errors which arise in approximation of a random polygonal line by a Wiener process on the same probability space. The polygonal line is constructed on the whole axis for sums of independent nonidentically distributed random variables and the distance between it and the Wiener process is taken to be the uniform distance with an increasing weight. All estimates depend explicitly on truncated power moments of the random variables which is an advantage over the earlier estimates of Komlos, Major, and Tusnady where this dependence was implicit.     

Properties of asymptotic expansions

The distribution function of a normalized sum of identically distributed independent random vectors is considered and some properties of its asymptotic expansions are studied. With the help of the results obtained here a theorem due to I. A. Ibragimov is strengthened.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 907–914, December, 1977.The author thanks V. V. Petrov and L. V. Osipov for taking interest in this article.     

Computing the moments of order statistics from nonidentically distributed phase-type random variables

In this paper, we derive a recurrence relation for the single moments of order statistics (o.s.) arising from n independent nonidentically distributed phase-type (PH) random variables (r.v.’s). This recurrence relation will enable one to compute all single moments of all o.s. in a simple recursive manner.     

Sample Covariance Matrix for Random Vectors with Heavy Tails

We compute the asymptotic distribution of the sample covariance matrix for independent and identically distributed random vectors with regularly varying tails. If the tails of the random vectors are sufficiently heavy so that the fourth moments do not exist, then the sample covariance matrix is asymptotically operator stable as a random element of the vector space of symmetric matrices.     

Extreme Values and the Multivariate Compact Law of the Iterated Logarithm

For a sequence of independent identically distributed Euclidean random vectors, we prove a compact Law of the iterated logarithm when finitely many maximal terms are omitted from the partial sum. With probability one, the limiting cluster set of the appropriately operator normed partial sums is the closed unit Euclidean ball. The result is proved under the hypotheses that the random vectors belong to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The integrability condition characterizes how many maximal terms must be omitted from the partial sum sequence.