On measuring the dependence of uncorrelated random variables

We discuss a way of measuring the strength of the dependence of uncorrelated random variables. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, PartI. Eger, Hungary, 1994.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

Representations for partially exchangeable arrays of random variables

Consider an array of random variables (X_i,j), 1 ≤ i,j < ∞, such that permutations of rows or of columns do not alter the distribution of the array. We show that such an array may be represented as functions f(α, ξ_i, η_j, λ_i,j) of underlying i.i.d, random variables. This result may be useful in characterizing arrays with additional structure. For example, we characterize random matrices whose distribution is invariant under orthogonal rotation, confirming a conjecture of Dawid.     

Representations of ordered random variables in the terms of the sums or products of independent variables

This is a survey of the available representations capable of expressing dependent ordered variables like order statistics and records as sums of independent summands or as products of independent random multipliers. Some new similar representations are obtained that in a similar manner express the common distributions of representatives of these two types of ordered random variables.     

A random CLT for dependent random variables

We introduce a new condition for {Y_τn} to have the same asymptotic distribution that {Y_n} has, where {Y_n} is a sequence of random elements of a metric space (S, d) and {τ_n} is a sequence of random indices. The condition on {Y_n} is that max_iDnd(Y_i, Y_an)→^p0 as n → ∞, where D_n = {i: |k_i−k_an| ≤ δ_ank_an} and {δ_n} is a nonincreasing sequence of positive numbers. The condition on {τ_n} is that P(|(k_τn/k_an)−1| > δ_an) → 0 as n → ∞. Under these conditions, we will show that d(Y_τn, Y_an) → ^P0 and apply this result to the CLT for a general class of sequences of dependent random variables.     

On identification of random polynomials of discrete random variables by their distributions

We consider the following problem: for what polynomials P and Q does the equality imply , the common distribution F of discrete random variables X₁ and X₂ being given. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger. Hungary, 1994     

Matricially free random variables

We show that the framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a concept which not only leads to a natural generalization of freeness, but also underlies other fundamental types of noncommutative independence, such as monotone independence and boolean independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws.     

Convex fuzzy random variables

A theory of fuzzy random variables is developed that applies to situations involving both randomness and fuzziness. The use of membership functions that are quasi-concave play an important role in the theory. The expectation of a fuzzy random variable is a fuzzy variable (fuzzy set). The usual linearity properties of probabilistic expectation carry over to fuzzy random variables. A special case of a fuzzy Law of Large Number is proven.     

Truncated random variables with application to random surfaces

There has been considerable interest in obtaining discrete results for random surfaces. Standard results have been published in journals of physics or engineering which have emphasised the applications. This paper gives a detailed background of the mathematical methods needed so that the central connection, namely truncated random variables, between these standard results can be understood. Distributions of discrete peak measures are obtained from the distributions of discrete profile measures of a random Gaussian surface by applying results for the distributions of truncated random variables. This enable the moments to be obtained from known results for the truncated distributions.     

On the random max-closure for heavy-tailed random variables

In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X ₁ ,X ₂ , . . . , which states that, given distributions \( {F}_{X_1} \) , \( {F}_{X_2} \) , . . . from some class of heavy-tailed distributions, the distribution of the random maximum X( _η) := max{0,X ₁ , . . . , X _η } or random maximum S _(η) := max{0, S ₁ , . . . , S _η } belongs to the same class of heavy-tailed distributions. Here, S _n = X ₁ + · · · + X _n , n ≥ 1, and η is a counting random variable, independent of {X ₁ ,X ₂ , . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window .     

On distributions of random variables chosen at random

The existence of typical distributions for random variables chosen at random from a finite-dimensional random variable vector space of high dimension is established. Possible typical distributions are described, and conditions for the typical distribution to be standard Gaussian are given. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 153–160. Translated by the author.     

Summation of independent random variables by the Riesz method

Under moment restrictions on X₁ exact sufficient conditions are established on the growth of a_n for the convergence almost surely of weighted Riesz means to MX₁, where (X_n) is a sequence of independent identically distributed random variables, (a _n) is a positive numerical sequence .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 641–647, May, 1992.     

Weak approximation of CIR equation by discrete random variables

For the CIR equation \( {\text{d}}{X_t} = \left( {\theta - k\,{X_t}} \right){\text{d}}t + \sigma \sqrt {{{X_t}}} {\text{d}}{B_t} \), we propose positive weak first- and second-order approximations that use, at each step, generation of discrete (respectively two- and three-valued) random variables (Theorems 3 and 4). The equation is split into deterministic part \( {\text{d}}{D_t} = \left( {\theta - k{D_t}} \right){\text{d}}t \), which is solved exactly, and stochastic part \( {\text{d}}{S_t} = \sigma \sqrt {{{S_t}}} {\text{d}}{B_t} \), which is actually approximated in distribution.     

Sequences of m-orthogonal random variables

A sequence of random variables {X_n} is said to be a sequence of m -orthogonal random variables if E X _n ² < for any n and E(X_kX_j) for k–j>m. Here m is a nonnegative integer. One proves a theorem on the law of the iterated logarithm for a sequence ofm-orthogonal random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 198–202, 1982.