On the invariance principle for sums of independent identically distributed random variables

The paper deals with the invariance principle for sums of independent identically distributed random variables. First it compares the different possibilities of posing the problem. The sharpest results of this theory are presented with a sketch of their proofs. At the end of the paper some unsolved problems are given.     

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.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

On r-quick limit sets for empirical and related processes based on mixing random variables

The r-quick limit points of normalized sample paths and empirical distribution functions of mixing processes are characterized. An r-quick version of Bahadur-Kiefer-type representation for sample quantiles is established, which yields the r-quick limit points of quantile processes. These results are applied to linear functions of order statistics. Some results on r-quick convergence of certain Gaussian processes are also established.     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

Weak convergence of empirical and quantile processes in sup-norm metrics via kmt-constructions

There has been much interest recently in the specially constructed empirical processes of Komlós, Major and Tusnády [2]; as one would guess, much of the application has come from the Hungarian school.In this note we contribute to the unifying effect this profound work has had by showing how the major theorem of O'Reilly [4] follows in rather elementary fashion from this powerful construction. We also take this opportunity to restate O'Reilly's criterion in an elementary form that is far more intelligible.     

On the discounted global CLT for some weakly dependent random variables

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑ _r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ _n1:= ∫ _−∞ ^∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 584–597, October–December, 2006.     

Weak convergence of random polygonal lines to a Gaussian process

We obtain a limit theorem of convergence in distribution for random polygonal lines defined by sums of independent random variables with replacements. In a particular case, the limit is the Gaussian Ornstein-Uhlenbeck process.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 33–44, January–March, 2005.Translated by V. Mackeviius     

H可积下的相依随机变量和的完全收敛性质

在满足H可积的条件下,利用随机变量的截尾方法,以及相关引理,给出了行内两两NQD序列以及p混合条件的随机组列部分和的完全收敛定理以及强大数定理．     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

On the asymptotic power of some k-sample statistics based on the multivariate empirical process

Some k-sample Kolmogorov-Smirnov and Cramér-von Mises-type statistics, based on the multivariate empirical process, are studied. Expressions for their asymptotic power are obtained against various classes of alternative distribution functions.     

The limit theorems for maxima of stationary Gaussian processes with random Index

Let {X(t), t ≥ 0} be a standard(zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(TT) exists under some additional conditions related to the correlation function r(·).     

Bootstrapping the empirical distribution of a linear process

The validity of the moving block bootstrap for the empirical distribution of a short memory causal linear process is established under simple conditions that do not involve mixing or association. Sufficient conditions can be expressed in terms of the existence of moments of the innovations and summability of the coefficients of the linear model. Applications to one and two sample tests are discussed.     

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little??s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.     

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.     

On convergence of the maximum of dependent random variables

Kaunas University of Technology, Studentu 50, 3028 Kaunas, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 1, pp. 3–6, January–March, 1992.     

Strong approximations of the Hoeffding,Blum, Kiefer,Rosenblatt multivariate empirical process

Hoeffding (Ann. Math. Statist. 1948) and Blum, Kiefer and Rosenblatt (Ann. Math. Statist. 1961) constructed distribution free tests of independence based on a multivariate empirical process. We establish strong invariance principles for the latter and also for appropriate functionals of it.     

On some multivariate density estimates and empirical Bayes problems

This paper consider estimates of multidimensional density functions and their derivatives. The asymptotic unbiasedness and the convergence properties of these estimates are established.Some applications to empirical Bayes problems are considered.     

Asymptotic results for empirical means of independent geometric distributed random variables

We consider independent geometric distributed random variables which satisfy suitable hypotheses. We study large and moderate deviations for their empirical means, and we illustrate applications of the large deviation results for the weak record values of i.i.d. discrete random variables.