Asymptotic normal distribution of multidimensional statistics of dependent random variables

A central limit theorem for multidimensional processes in the sense of [9], [10] is proved. In particular the asymptotic normal distribution of a sum of dependent random functions of m variables defined on the positive part of the integral lattice is established by the method of moments. The results obtained can be used, for example, in proving the asymptotic normality of different statistics of n₀-dependent random variables as well as to determine the asymptotic behaviour of the resultant of reflected waves of telluric type.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

Gaps in samples of geometric random variables

In this note we continue the study of gaps in samples of geometric random variables originated in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math. 294 (2005) 225-239] and continued in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at 〈http://www.ulb.ac.be/di/mcs/louchard/〉 (number 81 on the list) or at 〈http://math.sun.ac.za/∼ prodinger/pdffiles/gapsAPRIL27.pdf.〉] In particular, since the notion of a gap differs in these two papers, we derive some of the results obtained in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at 〈http://www.ulb.ac.be/di/mcs/louchard/〉 (number 81 on the list) or at 〈http://math.sun.ac.za/∼prodinger/pdffiles/gapsAPRIL27.pdf.〉] for gaps as defined in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math. 294 (2005) 225-239].     

Convergence of series of dependent φ-subgaussian random variables

The almost sure convergence of weighted sums of φ-subgaussian m-acceptable random variables is investigated. As corollaries, the main results are applied to the case of negatively dependent and m-dependent subgaussian random variables. Finally, an application to random Fourier series is presented.     

Distributional Properties of Exceedance Statistics

Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold is investigated. Three statistics connected with exceeding the threshold are introduced, their exact and asymptotic distributions are derived. Also distribution-free properties, leading to some common and some new discrete distributions, are considered. Identification of equidistribution of observations and the threshold are discussed. In this context relations between the exponential and gamma distributions are studied and a new derivation of the celebrated Laplace expansion for the standard normal distribution function is given.     

Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability

From the classical notion of uniform integrability of a sequence of random variables, a new concept of integrability (called h-integrability) is introduced for an array of random variables, concerning an array of constants. We prove that this concept is weaker than other previous related notions of integrability, such as Cesàro uniform integrability [Chandra, Sankhyā Ser. A 51 (1989) 309-317], uniform integrability concerning the weights [Ordóñez Cabrera, Collect. Math. 45 (1994) 121-132] and Cesàro α-integrability [Chandra and Goswami, J. Theoret. Probab. 16 (2003) 655-669].Under this condition of integrability and appropriate conditions on the array of weights, mean convergence theorems and weak laws of large numbers for weighted sums of an array of random variables are obtained when the random variables are subject to some special kinds of dependence: (a) rowwise pairwise negative dependence, (b) rowwise pairwise non-positive correlation, (c) when the sequence of random variables in every row is φ-mixing. Finally, we consider the general weak law of large numbers in the sense of Gut [Statist. Probab. Lett. 14 (1992) 49-52] under this new condition of integrability for a Banach space setting.     

Conditional waiting time distributions of runs and patterns and their applications

In this paper, a simple and general method based on the finite Markov chain imbedding technique is proposed to determine the exact conditional distributions of runs and patterns in a sequence of Bernoulli trials given the total number of successes. The idea is that given the total number of successes, the Bernoulli trials are viewed as random permutations. Then, we extend the result to multistate trials. The conditional distributions studied here lead to runs and patterns-type distribution-free tests whose applications are widespread. Two applications are considered. First, a distribution-free test for randomness is applied to rainfall data at Oxford from 1858 to 1952. The second application is to develop runs and patterns-type distribution-free control charts which can be used as Phase I and/or Phase II control charts. Numerical results for two commonly used runs-type statistics, the longest run and scan statistics, are also given.

     

Runs,scans and URN model distributions: A unified Markov chain approach

This paper presents a unified approach for the study of the exact distribution (probability mass function, mean, generating functions) of three types of random variables: (a) variables related to success runs in a sequence of Bernoulli trials (b) scan statistics, i.e. variables enumerating the moving windows in a linearly ordered sequence of binary outcomes (success or failure) which contain prescribed number of successes and (c) success run statistics related to several well known urn models. Our approach is based on a Markov chain imbedding which permits the construction of probability vectors satisfying triangular recurrence relations. The results presented here cover not only the case of identical and independently distributed Bernoulli variables, but the non-identical case as well. An extension to models exhibiting Markov dependence among the successive trials is also discussed in brief.     

Limit theorems for runs based on ‘small spacings’

A new concept of runs was proposed in the work of Eryilmaz and Stepanov (2008). A sequence of spacings forms a run if the lengths of these spacings do not exceed ε>0. In that paper, asymptotic properties of such spacings were investigated and statistical criteria proposed. In our present study, we maintain research on runs associated with these spacings. We derive limit theorems for the total number of runs, longest run and propose a statistical criterion.     

Gap-free compositions and gap-free samples of geometric random variables

We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free.     

具有不同分布NA随机变量列的强大数律及应用

给出了具有不同分布的NA随机变量列满足的若干强大数律;作为应用,不仅将独立随机变量的一类强极限定理完整的推广到NA随机变量情形,而且关于NA随机变量的一些已有结果可以作为推论得出.     

Long repetitive patterns in random sequences

Summary Appearances of long repetitive sequences such as 00...0 or 1010...101 in random sequences are studied. The expected length of the longest repetitive run of any specified type in a random binary sequence of length n is shown to tend to the binary logarithm of n plus a periodic function of log n. Necessary and sufficient conditions are derived to ensure that with probability 1 an infinite random sequence should contain repetitive runs of specified lengths in given initial segments. Finally, the number of long repetitive runs of a specified kind that occur in a random sequence is studied. These results are derived from simple expressions for the generating functions for the probabilities of occurrences of various repetitive runs. These generating functions are rational, and lead to sharp asymptotic estimates for the probabilities.     

The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m,k), i=1,…,n, be generalized order statistics based on F. For fixed r∈N, and a suitable counting process N(t), t>0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n)−r+1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237-244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.     

Comportement asymptotique des fonctions de répartition perturbées pour des processus non stationnaires et absolument réguliers

The object is to study the asymptotic normality of the statistics associated to the perturbed empirical distribution function via the slow convergence of multivariate U-statistic. We extend the results of Sun (1993) from the case of identically distributed absolutely regular random variables to the case of nonstationary absolutely regular random vectors. To cite this article: M. Harel, E. Elharfaoui, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On Success Runs of Length Exceeded a Threshold

Consider a sequence of n two state (success-failure) trials with outcomes arranged on a line or on a circle. The elements of the sequence are independent (identical or non identical distributed), exchangeable or first-order Markov dependent (homogeneous or non homogeneous) random variables. The statistic denoting the number of success runs of length at least equal to a specific length (a threshold) is considered. Exact formulae, lower/upper bounds and approximations are obtained for its probability distribution. The mean value and the variance of it are derived in an exact form. The distributions and the means of an associated waiting time and the length of the longest success run are provided. The reliability function of certain general consecutive systems is deduced using specific probabilities of the studied statistic. Detailed application case studies, covering a wide variety of fields, are combined with extensive numerical experimentation to illustrate further the theoretical results.     

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.     

Maximal pattern complexity of higher dimensional words

This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1-3) (2006) 15-27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof.Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.     

$\rho^*$-混合随机变量列加权和的完全收敛性

In this paper, the complete convergence of weighted sums for ρ*-mixing sequence of random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete convergence of weighted sums for ρ*-mixing sequence of random variables are established. We not only promote and improve the results of Li et al. (J. Theoret. Probab., 1995, 8(1): 49-76) from i.i.d. to ρ*-mixing setting but also obtain their necessities and relax their conditions.     

渐近正态随机变量函数的极限分布

基于渐近正态随机变量,导出随机变量函数极限分布的两个一般性理论结果.作为应用,证明了渐近正态随机变量一系列具体函数的极限分布,其中包括泊松随机变量平方根的渐近正态性,以及随机变量部分和在正则化常数是随机变量情况下的渐近正态性.