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Weighted Laws of Large Numbers for A Class of Independent Summands

Pakes Anthony G.

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

55--75

This paper obtains a necessary and sufficient condition for a weak law of large numbers for weighted averages of positive-valued independent random variables whose distributions belong to a class which includes the Fα-scheme of record theory. Additional general conditions are found under which the weak law extends to a strong law with the same norming. Examples show these conditions can be fulfilled, and that if they are not, then the weighted averages exhibit multiple growth rates.

Small deviation probabilities of weighted sums with fast decreasing weights

Rozovsky L. V.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

161--178

We examine small deviation probabilities of weighted sums of i.i.d. positive random variables whose distribution function is regularly varying at zero provided that weights are decreasing fast enough.

Large deviation theorems for weighted compound Poisson sums

Bonin O.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

357--368

In this paper we present two large deviation results for weighted compound sums ∑Ni = 1 ai Xi, where Xi’s are i.i.d. (possibly lattice) random variables, ai’s are non-negative real numbers, and N is a Poisson variable. These results are generalizations of approximations for non-weighted compound sums and for non-compound weighted sums.