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1

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

Matuła Przemysław

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 2

219--225

EN

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.

2

Unusual limit theorems for the two tailed Pareto distribution

Adler André, Matuła Przemysław

Probability and Mathematical Statistics

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2021

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Vol. 41, Fasc. 2

267--281

EN

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.

3

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.

4

Kernel conditional quantile estimator under left truncation for functional regressors

Helal N., Said E. O.

Opuscula Mathematica

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2016

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Vol. 36, no. 1

25--48

EN

Let Y be a random real response which is subject to left-truncation by another random variable T. In this paper, we study the kernel conditional quantile estimation when the covariable X takes values in an infinite-dimensional space. A kernel conditional quantile estimator is given under some regularity conditions, among which in the small-ball probability, its strong uniform almost sure convergence rate is established. Some special cases have been studied to show how our work extends some results given in the literature. Simulations are drawn to lend further support to our theoretical results and assess the behavior of the estimator for finite samples with different rates of truncation and sizes.

5

Some convergence properties of the sum of Gaussian functionals

Wałachowska A.

Demonstratio Mathematica

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2016

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Vol. 49, nr 4

453--462

EN

In the paper, some aspects of the convergence of series of dependent Gaussian sequences problem are solved. The necessary and sufficient conditions for the convergence of series of centered dependent indicators are obtained. Some strong convergence results for weighted sums of Gaussian functionals are discussed.

6

Weighted averages of random variables and exact weights

Matuła P., Seweryn M.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

377-384

EN

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 (…)

7

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.

8

On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in rademacher type p Banach spaces

Rosalsky A., Thanh L. V.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 2

205--222

EN

For a sequence of random elements {Vn, n ≥1} taking values in a real separable Rademacher type p (1 ≤p ≤2) Banach space and positive constants bn↑∞, conditions are provided for the strong law of large numbers ∑ni=1Vi/bn→0 almost surely. We treat the following cases: (i) {Vnn ≥1} is blockwise independent with EVn=0, n≥1, and (ii) {Vn, n≥1} is blockwise p-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of Móricz [12], Móricz et al. [13], and Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative examples or counterexamples are provided.

9

On the strong laws of large numbers for two-dimensional arrays of blockwise independent and blockwise orthogonal random variables

Quang N. V., Thanh L. V.

Probability and Mathematical Statistics

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2005

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Vol. 25, Fasc. 2

385--391

EN

In this paper we obtain the conditions of the strong law of large numbers for two-dimensional arrays of random variables which are blockwise independent and blockwise orthogonal. Some well-known results on the strong laws of large numbers for two-dimensional arrays of random variables are extended.

10

On multivalued amarts

Dudek D., Zięba W.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 1

93-99

EN

In recent years, convergence results for multivalued functions have been developed and used in several areas of applied mathematics: mathematical economics, optimal control, mechanics, etc. The aim of this note is to give a criterion of almost sure convergence for multivalued asymptotic martingales (amarts). For every separable Banach space B the fact that every L^1-bounded B- valued martingale converges a.s. in norm to an integrable B-valued random variable (r.v.) is equivalent to the Radon-Nikodym property [6]. In this paper we solve the problem of a.s. convergence of multivalued amarts by giving a topological characterization.

11

On stability of trimmed sums

Hu T.-C., Huang C.-Y., Rosalsky A.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 1

153--172

EN

Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables and let {an, n ≥ 1} and {bn, n ≥ 1} be sequences of constants where 0 < bn ↑ ∞. Let Xn(1), Xn(2),…, Xn(n) be a rearrangement of X1,…, Xn such that |Xn(1)| ≥ |Xn(2)| ≥ … ≥ |Xn(n)|. Consider the sequence of weighted sums Tn = Σni=1 ai Xi, n ≥ 1, and, for fixed r ≥ 1, set T(r)n = Σni=1 ai Xi I(|Xi| ≤ |X(r+1)n|), n ≥ r + 1; i.e., T(r)n is the sum Tn minus the sum of the X(k)n’s multiplied by their corresponding coefficients for k = 1,…, r. The main results provide sufficient and, separately, necessary conditions for b−1n T(r)n − kn → 0 almost surely for some sequence of centering constans {kn, n ≥1}. The current work extends that of Mori [14], [15] wherein an ≡ 1.

12

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 → ∞.

13

Almost sure convergence of the distributional limit theorem for order statistics

Peng L., Qi Y.

Probability and Mathematical Statistics

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2003

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Vol. 23, Fasc. 2

217--228

EN

Let Xn, n ≥ 1, be a sequence of independent and identically distributed random variables and Xn,1 ≤ Xn,2 ≤...≤ Xn,n denote the order statistics of X1,…, Xn. For any sequence of integers {kn} with 1 ≤ kn ≤ n and limn→∞min {kn, n − kn + 1} = ∞, if there exist constants an > 0, bn ∈ R and some non-degenerate distribution function G such that (Xn,kn − bn)/ an converges in distribution to G, then with probability one [wzór] = G(x) for all x ∈ C (G), where C (G) is the set of continuity points of G.

14

On the sequences whose conditional expectations can approximate any random variable

Kaniowski K.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

115--126

EN

Let(Ω,F, P)be a non-atomic probability space. For a given sequence (Xn)of random variables we indicate a number of conditions whichimply that for anyrandom variable Y there exists a sequence(Un) of σ-fields satisfying E(Xn|Un)→Ya.s. In particular, we formulate a sufficient condition using the distributions of Xn’sonly.

15

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}.

16

Individual ergotic theorem for non-contractive normal operators

Jajte R.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

85--99

EN

A condition implying the strong law of large numbers for trajectories of a normal non-contractive operator is given. The condition has been described in terms of a spectral measure, in the spirit of the well-known theorem of V. F. Gaposhkin. To embrace the non-contractive operators we pass from the classical arithmetic (Cesáro) means to the Borel methods of summability.

17

On the approximation of a random variable by a conditioning of a given sequence

Kaniowski K.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 2

321--328

EN

Let (Ω,F, P) be a non-atomic probability space. If (Xn) is a sequence of r.v.’s satisfying Xn → 0 a.s. (respectively, in probability) as n→∞ and EX+n→∞, EX-n→ ∞, as n → ∞, then for any r.v. Y there exists a sequence (Un) of σ-fields such that E(Xn|Un|)→Ya.s. (respectively, in probability) as n→∞.