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.     

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.     

A local limit theorem for independent random variables

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.     

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.     

Asymptotics for dependent Bernoulli random variables

This paper considers a sequence of Bernoulli random variables which are dependent in a way that the success probability of a trial conditional on the previous trials depends on the total number of successes achieved prior to the trial. The paper investigates almost sure behaviors for the sequence and proves the strong law of large numbers under weak conditions. For linear probability functions, the paper also obtains the strong law of large numbers, the central limit theorems and the law of the iterated logarithm, extending the results by James et al. (2008).     

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。     

A strong approximation for logarithmic averages of partial sums of random variables

LetS _n be the partial sums of -mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S _n / _n ) converges almost surely to _– f(x)d(x). We also obtain strong approximation forH(n)= _k=1 ⁿ k ^–1 f(S _k /k)=logn _– f(x)d(x) which will imply the asymptotic normality ofH(n)/log^1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for -mixing random variables.     

The mixing advantage for bounded random variables

Corresponding to n independent non-negative random variables X₁,…,X_n concentrated on a bounded interval set are values M₁,…,M_n, where each M_i is the expected value of the maximum of n independent copies of X_i. We obtain a sharp upper bound for the expected value of the maximum of X₁,…,X_n in terms of M₁,…,M_n. This inequality is sharp. A similar result is demonstrated for minima.     

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].     

Laplace approximations for sums of independent random vectors

Let X _i , i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping B→R. Under a central limit theorem assumption, an asymptotic evaluation of Z _n = E (exp (n Φ (∑ _{i =1} ⁿ X _i /n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without the central limit theorem assumption. Received: 19 September 1997 / Revised version:22 April 1999     

Entropy conditions for subsequences of random variables with applications to empirical processes

We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under the assumption that the entropy function of is not too large, many strong limit theorems will continue to hold uniformly over all sequences in . We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly over all sequences in . We prove a similar result for sequences (n _k ω) mod 1 where the sequence (n _k ) of real numbers satisfies a Hadamard gap condition. Authors’ addresses: István Berkes, Department of Statistics, Technical University Graz, Steyrergasse 17/IV, A-8010 Graz, Austria; Walter Philipp, Department of Statistics, University of Illinois, 725 S. Wright Street, Champaign, IL 61820, USA; Robert F. Tichy, Department of Analysis and Computational Number Theory, Technical University Graz, Steyrergasse 30, A-8010 Graz, Austria     

On the Bahadur representation of sample quantiles for sequences of φ-mixing random variables

The object of the present investigation is to show that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of φ-mixing random variables. Two different orders of the remainder term, under different φ-mixing conditions, are obtained and used for proving two functional central limit theorems for sample quantiles. It is also shown that the law of iterated logarithm holds for quantiles in stationary φ-mixing processes.     

A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers

We consider a new family of convex weakly compact valued integrable random sets which is called an adapted array of convex weakly compact valued integrable random variables of type p (1?p?2). By this concept, more general laws of large numbers will be established. Some illustrative examples are provided.     

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.     

Inequalities for sums of independent geometrical random variables

Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the functionF(p ₁,...,p_n;t)=P(X₁+...+X_n≤t) is Schur-concave with respect to (p ₁,...,p_n) for every realt, whereX _i are independent geometric random variables with parametersp _i. A generalization to negative binomial random variables is also presented.     

Maxima of branching random walks vs. independent random walks

In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random walks. A comparison of our results with the known results of branching random walk then identifies the limit behaviors which are due to the number of particles and those which are determined by the branching structure.     

An empirical central limit theorem for dependent sequences

We prove a central limit theorem for the d$d$-dimensional distribution function of a class of stationary sequences. The conditions are expressed in terms of some coefficients which measure the dependence between a given σ$\sigma$-algebra and indicators of quadrants. These coefficients are weaker than the corresponding mixing coefficients, and can be computed in many situations. In particular, we show that they are well adapted to functions of mixing sequences, iterated random functions, and a class of dynamical systems.     

Asymptotic results for the empirical process of stationary sequences

We prove a strong invariance principle for the two-parameter empirical process of stationary sequences under a new weak dependence assumption. We give several applications of our results.     

Fluctuations of shapes of large areas under paths of random walks

Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.