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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.

Laws of large numbers for two tailed Pareto random variables

Adler A.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

121--128

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.

A substitute of the expected value

Urbanik K.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

285-297

To every probability measure /.t there corresponds a subspace X(u) generated by convex linear combinations of independent random variables X_1,X_2,... with the same probability distribution u and closed under the convergence in probability. The purpose of this paper is a construction of generalized expected values of u based on the space X (u).

Limit theorems for arrays of maximal order statistics

Adler A.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

211--220

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}.

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

Adler A.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

149--157

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.

Complete exact laws

Adler A.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

215--222

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.