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  1. 3 Open Mathematics

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  1. 3 Adler A.

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  1. 1 2017

  2. 1 2015

  3. 1 2006

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Exact laws for sums of ratios of order statistics from the Pareto distribution

100%
Adler A.
Open Mathematics
|
2006
|
tom 4
|
nr 1
1-4
EN
Consider independent and identically distributed random variables {X nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X n(i) ≤ X n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R ij = X n(j)/X n(i).
2
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One sided strong laws for random variables with infinite mean

100%
Adler A.
Open Mathematics
|
2017
|
tom 15
|
nr 1
828-832
EN
This paper establishes conditions that secure the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables. These random variables do not possess a finite first moment so these results are not typical. These mild conditions allow us to show that the almost sure upper limit is infinity while the almost sure lower bound is one.
3
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Laws of large numbers for ratios of uniform random variables

100%
Adler A.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
Let {Xnn n ≥ 1} and {Yn, n ≥ 1} be two sequences of uniform random variables. We obtain various strong and weak laws of large numbers for the ratio of these two sequences. Even though these are uniform and naturally bounded random variables the ratios are not bounded and have an unusual behaviour creating Exact Strong Laws.
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