Probability and moment inequalities for sums of weakly dependent random variables,with applications

Doukhan and Louhichi [P. Doukhan, S. Louhichi, A new weak dependence condition and application to moment inequalities, Stochastic Process. Appl. 84 (1999) 313–342] introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random variables. We employ such cumulant estimates to derive inequalities of Bernstein and Rosenthal type which both improve on previous results. Furthermore, we consider several classes of processes and show that they fulfill appropriate weak dependence conditions. We also sketch applications of our inequalities in probability and statistics.     

The distribution of product of independent beta random variables with application to multivariate analysis

Summary Schatzoff [9] obtained the forms of the probability density function (pdf) and the cumulative distribution function (cdf) of the product of independent beta random variables when their parameters had some special values. The forms, however, did not indicate the constants explicitly. In this paper his approach is modified so as to allow presentation of explicit expressions for the pdf and cdf of the product of independent beta random variables (without restriction to the values of the parameters) in neat forms. Applications in multivariate analysis are given for the central and the non-central cases. Research supported by the National Research Council of Canada, No. A-4060.     

A variance bound for a general function of independent noncommutative random variables

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab. 44(5) (2016), 3431–3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.     

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.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

On the moments of some first passage times for sums of dependent random variables

Let S_n,n = 1, 2, …, denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: S_n>ca(n)}, where c ≥ 0and a(y) is a positive continuous function on [0, ∞), such that a(y) = o(y)as y → ∞, are given. With the further assumption that a(y) = yP,0 ≤ p < 1, a law of large numbers and the asymptotic behaviour of the moments when c → ∞ are obtained. The corresponding stopped sums are also studied.     

Robustness of optimal designs for correlated random variables

Suppose that Y = (Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j), X = (X_ij) is an n × p matrix with X_ij ∈ {−1, 1}; this corresponds to a factorial design with −1, 1 representing low or high level respectively, or corresponds to a weighing design with −1, 1 representing an object j with weight b_j placed on the left and right of a chemical balance respectively. E-optimal designs Z are chosen that are robust in the sense that they remain E-optimal when the covariance of Y_i, Y_i′ is ρ > 0 for i ≠ i′. Within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the A- and D-criteria.     

On the transitivity of the comonotonic and countermonotonic comparison of random variables

A recently proposed method for the pairwise comparison of arbitrary independent random variables results in a probabilistic relation. When restricted to discrete random variables uniformly distributed on finite multisets of numbers, this probabilistic relation expresses the winning probabilities between pairs of hypothetical dice that carry these numbers and exhibits a particular type of transitivity called dice-transitivity. In case these multisets have equal cardinality, two alternative methods for statistically comparing the ordered lists of the numbers on the faces of the dice have been studied recently: the comonotonic method based upon the comparison of the numbers of the same rank when the lists are in increasing order, and the countermonotonic method, also based upon the comparison of only numbers of the same rank but with the lists in opposite order. In terms of the discrete random variables associated to these lists, these methods each turn out to be related to a particular copula that joins the marginal cumulative distribution functions into a bivariate cumulative distribution function. The transitivity of the generated probabilistic relation has been completely characterized. In this paper, the list comparison methods are generalized for the purpose of comparing arbitrary random variables. The transitivity properties derived in the case of discrete uniform random variables are shown to be generic. Additionally, it is shown that for a collection of normal random variables, both comparison methods lead to a probabilistic relation that is at least moderately stochastic transitive.     

Total positivity order and the normal distribution

Unlike the usual stochastic order, total positivity order is closed under conditioning. Here we provide a general formulation of the preservation properties of the order under conditioning; we study certain properties of the order including translation properties and the implications of having equality in the inequality defining the order. Specializing to the multivariate normal distribution, the study of total positivity order leads to new cones defined in terms of covariance M-matrices related to positive dependence, whose properties we study.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.     

Functions of elementary random variables and their application to system reliability

Summary LetX ₁,...,X _n be elementary random variables, i.e. random variables taking only finitely many values in . Then for an arbitray functionf(X ₁,...,X _n ) inX ₁,...,X _n a unique polynomial representation with the aid of Lagrange polynomials is given. This easily yields the moments as well as the distribution off(X ₁,...,X _n ) by terms of finitely many moments of the variablesX ₁,...,X _n . For n=1 a necessary and sufficient condition results thatm numbers are the firstm moments of a random variable takingm+1 different values. As an application of random functionsf(X ₁,...,X _n ) the reliability of technical systems with three states is treated.

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Zusammenfassung X ₁, ...,X _n seien elementare Zufallsvariable, d. h., Zufallsvariable, die nur endlich viele reelle Werte annehmen. Mit Hilfe von Lagrangepolynomen wird für eine beliebige Funktionf(X₁,...,X _n ) eine eindeutige polynomiale Darstellung angegeben. Daraus ergeben sich leicht die Momente wie auch die Verteilung von f(X₁,...,X _n ), ausgedrückt durch die Momente der VariablenX ₁,...,X _n . Fürn=1 erhält man eine notwendige und hinreichende Bedingung, daßm Zahlen die erstenm Momente einer Zufallsvariablen sind, diem+1 verschiedene Werte annimmt. Als Anwendung wird die Zuverlässigkeit eines technischen Systems mit drei Zuständen behandelt.

     

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.     

Effort associated with a class of random optimization methods

When differentiability is not assumed random procedures can be successfully used to estimate the extreme values of a given function. For a class of such algorithms we treat the problem of estimating the mean effort.Research partially supported by CNPq-Brazil.     

Almost sure Cesàro and Euler summability of sequences of dependent random variables

For a sequence of real random variables C _α-summability is shown under conditions on the variances of weighted sums, comprehending and sharpening strong laws of large numbers (SLLN) of Rademacher-Menchoff and Cramér-Leadbetter, respectively. Further an analogue of Kolmogorov’s criterion for the SLNN is established for E _α-summability under moment and multiplicativity conditions of 4th order, which allows one to weaken Chow’s independence assumption for identically distributed square integrable random variables. The simple tool is a composition of Cesàro-type and of Euler summability methods, respectively. Received: 12 June 2006, Revised: 14 May 2007     

Mixing normal approximations of vectors of sums and maximum sums

Summary The conditioned central limit theorem for the vector of maximum partial sums based on independent identically distributed random vectors is investigated and the rate of convergence is discussed. The conditioning is that of Rényi (1958,Acta Math. Acad. Sci. Hungar.,9, 215–228). Analogous results for the vector of partial sums are obtained. University of Petroleum and Minerals