Some notions of conditionally dominated random variables are introduced and characterized, Under rather minimal assumptions on random variables {X,Xn, n ≥ 1}, some limit theorems of Jamison’s type of weighted sums of random variables are obtained. Mathematics Subject Classification: 60F15

Sufficient conditions are given under which a sequence of independent random elements taking values in a Banach space satisfy the Hsu and Robbins law of large numbers. The complete convergence of random indexed sums of random elements is also considered.

We consider sums of functions of subtrees of a random binary search tree, and obtain general laws of large numbers and central limit theorems. These sums correspond to random recurrences of the quicksort type, Xn L = XIn+X ′ n−1−In+Yn, n ≥ 1, where In is uniformly distributed on {0, 1, . . . , n− 1}, Yn is a given random variable, Xk L = X ′ k for all k, and given In, XIn and X ′ n−1−In are ind...

In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distr...

We prove equidistribution results for certain exponential sums that arise in the work of Kurlberg-Rudnick on "cat maps". We show (Theorems 1 and 2) that suitable normalizations of these sums behave like the traces of random matrices in SU(2). We also show that as a suitable parameter varies, the corresponding sums are statistically independent (Theorems 3 and 4). The main tools are Deligne's Eq...