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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.

Weighted averages of random variables and exact weights

Matuła P., Seweryn M.

Demonstratio Mathematica

2012

Vol. 45, nr 2

377-384

In the present note we discuss the conditions under which for a sequence of nonnegative i.i.d. random variables (…) with infinite mean, there exist sequences of nonnegative real numbers (…) and (…) such that (…)

A sharp correlation inequality with application to almost sure local limit theorem

Weber M.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

79--98

We prove a new sharp correlation inequality for sums of i.i.d. square integrable lattice distributed random variables. We also apply it to establish an almost sure version of the local limit theorem for i.i.d. square integrable random variables taking values in an arbitrary lattice. This extends a recent similar result jointly obtained with Giuliano-Antonini under a slightly stronger absolute moment assumption (of order 2+u with u > 0). The approach used to treat the case u > 0 breaks down when u = 0. Mac- Donald’s concept of the Bernoulli part of a random variable is used in a crucial way to remedy this.