Asymptotic behavior for sums of non-identically distributed random variables

For any given positive integer m, let X_i, 1 ≤ i ≤ m be m independent random variables with distributions F_i, 1 ≤ i ≤ m. When all the summands are nonnegative and at least one of them is heavy-tailed, we prove that the lower limit of the ratio ■equals 1 as x →∞. When the summands are real-valued, we also obtain some asymptotic results for the tail probability of the sums. Besides, a local version as well as a density version of the above results is also presented.     

On the average number of records for sequences of not identically distributed random variables

The scheme of n series of independent random variables X ₁₁, X ₂₁, …, X _k1, X ₁₂, X ₂₂, …, X _k2, …, X _1n , X _2n , …, X _kn is considered. Each of these successive series X _1m , X _2m , …, X _km , m = 1, 2, …, n consists of k variables with continuous distribution functions F ₁, F ₂, …, F _k , which are the same for all series. Let N(nk) be the number of upper records of the given nk random variables, and EN(nk) be the corresponding expected value. For EN(nk) exact upper and lower estimates are obtained. Examples are given of the sets of distribution functions for which these estimates are attained.     

非同分布NA序列的完全收敛性

讨论了非同分布NA序列部分和与随机足标部分和的完全收敛性，推广了于浩在1989年得到的关于独立随机变量序列的一些结果。     

Strong approximation of empirical process with independent but non-identically distributed random variables

Let{Y_t,t=1,2,…} be independent random variables with continuous distribution functionsF_i(y).For any y,dencte s=F_t(y)=1/t sum from i=1 to t F_i(y).The empirical process is defind by t~(-1/2)R(s,t) whereR(s,t)=t(1/t sum from i=1 to t I_((?)_t(Y_i)≤s)-s)=sum from i=1 to t I_(?)-ts=sum from i=1 to t I_(?)-(?)_t(y)=sum from i=1 to t I_(Y_(?)≤y)-sum from i=1 to t F_i(y).The purpose of this paper is to investigate the asymptotic properties of the empirical processR(s,t).We shall prove that for some integer sequence {t_k},there is a (?)-process (?)(s,t) such that(?)|R(s,t_k)-(?)(s,t_k)|=O(t_k~(1/2)(log t_k)~(-1/4)(log log t_k)~(1/2))a.s.where (?)(s,t) is a two-parameter Gaussian process defined in §1.     

Some estimates of the normal approximation for independent non-identically distributed random variables

In this paper, we generalize some results of [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003] for independent identically distributed summands to to the case of independent non-identically distributed real summands. We derive the Edgeworth expansion with the first term only. Proofs are given following [V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003].     

An empirical process central limit theorem for dependent non-identically distributed random variables

This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.     

Weak convergence of products of sums of independent and non-identically distributed random variables

We study limit properties in the sense of weak convergence in the space D[0,1] of certain processes based on products of sums of independent and non-identically distributed random variables. The obtained results extend and generalize results known in the i.i.d. case.     

Recurrence relations for order statistics from n independent and non-identically distributed random variables

Some well-known reeurrence relations for order statistics in the i.i.d. case are generalized to the case when the variables are independent and non-identically distributed. These results could be employed in order to reduce the amount of direct computations involved in evaluating the moments of order statistics from an outlier model.     

A LIL for independent non-identically distributed random variables in Banach space and its applications

In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes.     

A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables

The bounded-dual-Lipschitz and Prohorov distances from the ‘empirical measure’ to the ‘average measure’ of independent random variables converges to zero almost surely if the sequence of average measures is tight. Three examples are also given.     

On gaps and unoccupied urns in sequences of geometrically distributed random variables

This paper continues the study of gaps in sequences of n geometrically distributed random variables, as started by Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239], who concentrated on sequences which were gap-free. Now we allow gaps, and count some related parameters.Our terminology of gaps just means empty “urns” (within the range of occupied urns), if we think about an urn model. This might be called weak gaps, as opposed to maximal gaps, as in Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239]. If one considers only “gap-free” sequences, both notions coincide asymptotically, as n→∞.First, the probability p_n(r) that a sequence of length n has a fixed number r of empty urns is studied; this probability is asymptotically given by a constant p^*(r) (depending on r) plus some small oscillations. When , everything simplifies drastically; there are no oscillations.Then, the random variable ‘number of empty urns’ is studied; all moments are evaluated asymptotically. Furthermore, samples that have r empty urns, in particular the random variable ‘largest non-empty urn’ are studied. All moments of this distribution are evaluated asymptotically.The behavior of the quantities obtained in our asymptotic formulæ is also studied for p→0 resp. p→1, through a variety of analytic techniques.The last section discusses the concept called ‘super-gap-free.’ A sample is super-gap-free, if r=0 and each non-empty urn contains at least 2 items (and d-super-gap-free, if they contain ?d items). For the instance , we sketch how the asymptotic probability (apart from small oscillations) that a sample is d-super-gap-free can be computed.     

The number of records in a sequence of nonidentically distributed random variables

One obtains limit theorems for the number of records and for the times of the attainment of the record values in a sequence of independent random variables.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 373–388, 1986.     

Recurrence relations among moments of order statistics from two related sets of independent and non-identically distributed random variables

Some recurrence relations among moments of order statistics from two related sets of variables are quite well-known in the i.i.d. case and are due to Govindarajulu (1963a, Technometrics, 5, 514–518 and 1966, J. Amer. Statist. Assoc., 61, 248–258). In this paper, we generalize these results to the case when the order statistics arise from two related sets of independent and non-identically distributed random variables. These relations can be employed to simplify the evaluation of the moments of order statistics in an outlier model for symmetrically distributed random variables.     

Record and interrecord times for sequences of nonidentically distributed random variables

Assume that the independent random variables X₁,X₂,... have the distribution functions , ..., respectively, where F is an arbitrary continuous distribution function, while _i are positive constants. In this situation, one obtains some theorems for the record moments and interrecord times.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 109–118, 1985.     

Comparison of numbers of records in the sequences of discrete and continuous random variables

Different record achievements are fixed in many domains of human activities. This process very often happens with some rate of digitization (up to seconds, meters, or thousands of individuals) of the observed results. By the examples of exponential and geometrical distributions, it is shown how such a type of the transitions from continuous to discrete distributions may vary the numbers of the record values in the corresponding sequences of the random variables.     

Maximal inequality and complete convergences of non-identically distributed negatively associated sequences

A maximal inequality for the partial sum of NA sequence is constructed.By using this inequality the complete convergence rates in the strong laws for a class of dependent random variables for weighted sums are discussed.The results obtained extend the results of Liang(1999, 2000).     

On the uniform distribution of sequences of random variables

Summary This paper deals with the almost sure uniform distribution (modulo 1) of sequences of random variables. In the case where the law of the increments X _n+h –X _n of the sequence X ₀, X ₁, does not depend on n, sufficient conditions are given to assure the uniform distribution (modulo 1) with probability one. As an illustrative example the partial sums of a sequence of independent, identically distributed variables is considered.     

On the minimum correlation between symmetrically distributed random variables

Using linear programming, we show that families of symmetrically distributed Bernoulli random variables have a maximal negative correlation that almost always is strictly above the general lower limit.