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On exact strong laws of large numbers under general dependence conditions

Adler A., Matuła P.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 1

103--121

EN

We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.

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Exact strong laws of large numbers for independent random fields

Kurasiński P., Matuła P., Adler A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2018

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Vol. 66, no. 2

179--188

EN

Let {Xṉ, ṉ ϵ Nd} be a family of independent random variables with multidimensional indices (a random field) with the same distribution as the r.v. X. A necessary and sufficient condition for the strong law of large numbers in this setting is E|X| logd-1+|X| < ∞. Our goal is to study the almost sure convergence of normalized or weighted sums in the case when this moment condition is not satisfied.

3

Laws of large numbers for two tailed Pareto random variables

Adler A.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 1

121--128

EN

We sample m random variables from a two tailed Pareto distribution. A two tailed Pareto distribution is a random variable whose right tail is px−2 and whose left tail is qx−2, where p + q = 1. Next, we look at the largest of these random variables and establish various Weak and Strong Laws that can be obtained with weighted sums of these random variables. The case of m = 1 is completely different from m > 1.

4

On exact strong laws for sums of multidimensionally indexed random variables

Adler A., Qi Y.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

305--313

EN

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 → ∞.

5

Limit theorems for arrays of maximal order statistics

Adler A.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

211--220

EN

Let {X, Xnj,1≤j≤mn, n≥1} be independent and identically distributed random variables with the Pareto distribution. Let Xn(k) be the k-th largestorder statistic from then-th row of our array. This paper establishes unusual limit theorems involving weighted sums for the sequence {Xn(k), n≥1}.

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A generalized Law of the Iterated Logarithm for the largest observation of a triangular array

Adler A.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

149--157

EN

Consider independent and identically distributed random variables {X, Xkj1≤j≤k, k≥1} from a particular distribution with EX=∞.Weshow that there exists an unusual generalized Law of the Iterated Logarithm involving max 1≤j≤kXkj.

7

Complete exact laws

Adler A.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 1

215--222

EN

Consider independent and identically distributed random variables {X,Xn, n ≥ 1} with xP{X > x} ~α(log x)α, where α > −1 and P{X < −x} = o(P{X > x}). Even though the mean does not exist, we establish Laws of Large Numbers of the form [formula].. for all ε > 0 and a particular nonsummable sequence {cn, n ≥ 1}, where L ≠ 0.