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Unusual limit theorems for the two tailed Pareto distribution

Adler André, Matuła Przemysław

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 2

267--281

We examine order statistics from a two-sided Pareto distribution. It turns out that the smallest two order statistics and the largest two order statistics have very unusual limits. We obtain strong and weak exact laws for the smallest and the largest order statistics. For such statistics we also study the generalized law of the iterated logarithm. For the second smallest and second largest order statistics we prove the central limit theorem even though their second moment is infinite.

On exact strong laws for sums of multidimensionally indexed random variables

Adler A., Qi Y.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

305--313

Let {X, Xn, n ∈ Zd+} be independent and identically distributed random variables satisfying xP (|X| > x) ≈ L(x) with either EX = 0 or E|X| = ∞, where L(x) is slowly varying at infinity. This paper proves that there always exist sequencesof constants {an} and {BN} such that an Exact Strong Law holds, that is [wzór] an Xn/BN → 1 almost surely as N → ∞.