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A remark on the exact laws of large numbers for ratios of independent random variables

Matuła Przemysław

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 2

219--225

Let (Xn)n∈N and (Yn)n∈N be two sequences of i.i.d. random variable ξ which are independent of each other and all have the distribution of a positive random variable ξ with density fξ . We study weighted strong laws of large numbers for the ratios of the form [wzór]1 in the cases when IEξ = ∞ or limx→0+ fξ (x) = 0 or fξ is unbounded. This research complements some results known so far.

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.