Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.     

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.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

Convergence of series of independent random variables

Sufficient conditions are given for the convergence almost surely of a series composed of independent symmetric random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mateamticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 158–160, 1987.     

Moments of sums of independent random variables

Upper bounds are obtained for the absolute moments of order p>1 of sums of independent random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 120–121, 1989.     

Weighted lattice polynomials of independent random variables

We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems.     

Self-normalized moderate deviations for independent random variables

Let X1,X2,...be a sequence of independent random variables(r.v.s) belonging to the domain of attraction of a normal or stable law.In this paper,we study moderate deviations for the self-normalized sum ∑ni=1 Xi/Vn,p,where Vn,p =(∑ni=1 |Xi|p)1/p(p>1).Applications to the self-normalized law of the iterated logarithm,Studentized increments of partial sums,t-statistic,and weighted sum of independent and identically distributed(i.i.d.) r.v.s are considered.     

Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.     

Convergence of distributions of extremal independent random variables

Kaunas Polytechnic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 2, pp. 219–232, April–June, 1990.     

Convergence of sequences of pairwise independent random variables

In spite of the fact that the tail -algebra of a sequence of pairwise independent random variables may not be trivial, we have discovered that if such a sequence converges in probability or almost everywhere, then the limit has to be a constant. This enables us to provide necessary and sufficient conditions for its convergence, in terms of its marginal distribution functions.