相关论文: Three series theorem for independent random variab…
The paper deals with the order statistics and empirical mathematical expectation (which is also called the estimate of mathematical expectation in the literature) in the case of infinitely increasing random variables. The Kolmogorov concept…
Let $\{X_{i}, i\in J\}$ be a family of locally dependent non-negative integer-valued random variables with finite expectations and variances. We consider the sum $W=\sum_{i\in J}X_i$ and use Stein's method to establish general upper error…
The aim of this paper is to establish the Marcinkiewicz-Zygmund (MZ) type law of large numbers for the randomly weighted sums with weights chosen randomly, uniformly over the unit sphere in $\mathbb{R}^n$. We also establish a theorem that…
The law of large numbers extends to random sets by employing Minkowski addition. Above that, a central limit theorem is available for set-valued random variables. The existing results use abstract isometries to describe convergence of the…
We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of…
We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions…
We prove a general transfer theorem for multivariate random sequences with independent random indexes in the double array limit setting. We also prove its partial inverse providing necessary and sufficient conditions for the convergence of…
Abstract. The purpose of this paper is twofold. We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work arXiv:1910.08492, to study the propagation of randomness under…
We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. In the case of positive variables, the main theorem extends a result proved earlier for arrays with identically…
Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time.For the normalised counting measure of the number of…
In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance…
We provide an improved version of the Darling-Erd\"os theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance…
We develop the information-theoretical concepts required to study the statistical dependencies among three variables. Some of such dependencies are pure triple interactions, in the sense that they cannot be explained in terms of a…
Statistical independence is a notion ubiquitous in various fields such as in statistics, probability, number theory and physics. We establish the stability of independence for any pair of random variables by their corresponding Brockwell…
In this paper, we present three remarkable properties of the normal distribution: first that if two independent variables's sum is normally distributed, then each random variable follows a normal distribution (which is referred to as the…
Central limit theorems are established for the sum, over a spatial region, of observations from a linear process on a $d$-dimensional lattice. This region need not be rectangular, but can be irregularly-shaped. Separate results are…
The paper that is commented by Touchette contains a computational study which opens the door to a desirable generalization of the standard large deviation theory (applicable to a set of $N$ nearly independent random variables) to systems…
Non-linear renewal theory is extended to include random walks perturbed by both a slowly changing sequence and a stationary one. Main results include a version of the Key Renewal Theorem, a derivation of the limiting distribution of the…
We derive concentration inequalities for sums of independent and identically distributed random variables that yield non-asymptotic generalizations of several strong laws of large numbers including some of those due to Kolmogorov [1930],…
We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting…