The asymptotic distribution of randomly weighted sums and self-normalized sums

The asymptotic distribution of randomly weighted sums and self - normalized sums ∗

We consider the self-normalized sums Tn = ∑n i=1 XiYi/ ∑n i=1 Yi, where {Yi : i ≥ 1} are non-negative i.i.d. random variables, and {Xi : i ≥ 1} are i.i.d. random variables, independent of {Yi : i ≥ 1}. The main result of the paper is that each subsequential limit law of Tn is continuous for any non-degenerate X1 with finite expectation, if and only if Y1 is in the centered Feller class.

A note on the normal approximation error for randomly weighted self-normalized sums

Let X = {Xn}n≥1 and Y = {Yn}n≥1 be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums ψn(X,Y) = n ∑ i=1 XiYi/Vn, Vn = √ Y 2 1 + · · ·+ Y 2 n . These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student’s t-statistic or the empirical correlation coefficient.

Asymptotic Behavior of Weighted Sums of Weakly Negative Dependent Random Variables

Let be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let be other sequence of positive random variables independent of and for some and for all . In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy...

The Asymptotic Distribution of Exponential Sums, I

On the Asymptotic Distribution of Certain Sums