André Adler

Strong laws of large numbers for weighted sums of random elements in normed linear spaces.

Consider a sequence of independent random elements {Vn, n > in a real separable normed linear space (assumed to be a Banach space in most of the results), and sequences of con-

A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces

For weighted sums of the form Sn = [summation operator]j=1kn anj(Vnj - cnj) where {anj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]kn

A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces

For weighted sums [Sigma]j = 1najVj of independent random elements {Vn, n >= 1} in real separable, Rademacher type p (1 p 0 is established, where {vn, n >= 1} and bn --> [infinity] are suitable sequences. It is assumed that {Vn, n >= 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of V and the growth behaviors of the constants {an, n >= 1} and {bn, n >= 1}. No assumption is made concerning the existence of expected values or absolute moments of the {Vn, n >- 1}.

Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean

AbstractThe almost sure limiting behavior of weighted sums of independent and identically distributed random variables barely with or without finite mean are established. Results for these partial sums,

Stability of sums of independent random variables

\sum\limits_{k = 1}^n {k^\alpha X_k ,} \alpha \in R

Complete convergence for arrays of minimal order statistics

have been studied, but only when α=−1 or α=0. As it turns out, the two cases of major interest are α=−1 and α>−1. The purpose of this article is to examine the latter.

Laws of large numbers for ratios of uniform random variables

Consider independent and identically distributed random variables {Xnk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, Xn(i) ≤ Xn(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables Rij = Xn(j)/Xn(i).

Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

In this paper we establish a relationship between convergence in probability and almost surely for sums of independent random variables. It turns out that whenever there is a relatively stable weak law of large numbers, there is a corresponding strong law. Our goal is to explore whether or not there exist constants that asymptotically behave like our partial sums. Previous results seem to indicate, in the i.i.d. case, that whenever the tails of the distribution at hand are regularly varying with exponent minus one and P{X x}), then one can always find constants so that the weighted and normalized partial sums converge to one almost surely. However, a few extreme cases until now had offered evidence to the contrary. Herein, we show that even in those cases almost sure stability can be obtained.