Uncorrelatedness sets of bounded random variables

An uncorrelatedness set of two random variables shows which powers of random variables are uncorrelated. These sets provide a measure of independence: the wider an uncorrelatedness set is, the more independent random variables are. Conditions for a subset of to be an uncorrelatedness set of bounded random variables are studied. Applications to the theory of copulas are given.     

Uncorrelatedness sets for random variables with given distributions

Let and be random variables having finite moments of all orders. The set

is said to be an uncorrelatedness set of and It is known that in general, an uncorrelatedness set can be arbitrary. Simple examples show that this is not true for random variables with given distributions. In this paper we present a wide class of probability distributions such that there exist random variables with given distributions from the class having a prescribed uncorrelatedness set. Besides, we discuss the sharpness of the obtained result.

     

Periodically correlated multivariate second order random distribution fields and stationary cross correlatedness

In this paper we consider multivariate second order operator periodically correlated random distribution fields for which we complete and extend the results from [5].     

Diophantine approximation with prime variables and mixed powers

The Ramanujan Journal - Suppose that $$\lambda _1, \lambda _2, \lambda _3, \lambda _4$$ are positive real numbers and $$\frac{\lambda _1}{\lambda _2}$$ is irrational and algebraic. Let $$\mathcal...     

Continuous random walks and fractional powers of operators

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes. This short paper represents a bridge between probabilists and researchers working in PDE?s.     

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.     

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.     

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.     

Transforms of pseudo-Boolean random variables

As in earlier works, we consider {0,1}ⁿ as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions.     

Uniform nonintegrability of random variables

Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116: 27–37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability (W-UNI) of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W-UNI, one of which is a W-UNI analogue of the celebrated de La Vallée Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.     

Strategic disclosure of random variables

We consider a game G_n played by two players. There are n independent random variables Z₁, … , Z_n, each of which is uniformly distributed on [0,1]. Both players know n, the independence and the distribution of these random variables, but only player 1 knows the vector of realizations z ? (z₁, … , z_n) of them. Player 1 begins by choosing an order _zk1,…,_zkn$z_{k_1}\text{,}\dots \text{,}{z}_{k_n}$ of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns _zk1$z_{k_1}$. If player 2 accepts, then player 1 pays _zk1$z_{k_1}$ euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering _zk2$z_{k_2}$ euros to player 2. Play continues until player 2 accepts an offer. If player 2 has rejected n − 1 times, player 2 has to accept the last offer at period n. This model extends Moser’s (1956) problem, which assumes a non-strategic player 1.     

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.     

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler’s randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction:

1. The randomisation of a stable structure is stable.
2. The randomisation of a simple unstable structure is not simple.

We also prove that in the randomised structure, every type is a Lascar type.     

Weak association of random variables

The dependence concept of weak association is introduced and is shown to be equivalent to positive quadrant dependence. Furthermore, a characterization of independence in the class of positive quadrant dependent random variables by means of moment conditions is proved. Both results generalize some theorems proved by Lehmann and Jogdeo for the two- and three-dimensional case.