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1
Content available remote Laws of large numbers for two tailed Pareto random variables
100%
Adler A.
Probability and Mathematical Statistics
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2008
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tom 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.
2
Content available remote Marcinkiewicz-type strong law of large numbers for pairwise independent random fields
100%
Czerebak-Mrozowicz E. B. , Klesov O. I. , Rychlik Z.
Probability and Mathematical Statistics
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2002
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tom Vol. 22, Fasc. 1
127--139
EN
We present the Marcinkiewicz-type strong law of large numbers for ran-dom fields{Xn, n ∈Zd+}of pairwise independent random variables, where Zd+, d≥1, is the set of positived-dimensional lattice points with coordinatewise partial ordering.
3
Content available remote On exact strong laws for sums of multidimensionally indexed random variables
100%
Adler A. , Qi Y.
Probability and Mathematical Statistics
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2003
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tom 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 → ∞.
4
Content available remote Complete exact laws
100%
Adler A.
Probability and Mathematical Statistics
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2000
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tom 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.
5
Content available remote Strong laws of large numbers for random permanents
100%
Rempała G. A. , Wesołowski J.
Probability and Mathematical Statistics
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2002
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tom Vol. 22, Fasc. 2
201--209
EN
The strong laws of large numbers for random permanents of increasing order are derived. The method of proofs relies on the martingale decomposition of a random permanent function similar to the one known for U-statistics.
6
Content available remote On stability of trimmed sums
100%
Hu T.-C. , Huang C.-Y. , Rosalsky A.
Probability and Mathematical Statistics
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2003
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tom 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.
7
Content available remote Limit theorems for a higher order time dependent Markov chain model
100%
Kokoszka P. , Kutta T. , Singh D. , Wang H.
Probability and Mathematical Statistics
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2023
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tom Vol. 43, Fasc. 1
121--139
EN
The paper establishes a strong law of large numbers and a central limit theorem for a sequence of dependent Bernoulli random variables modeled as a higher order Markov chain. The model under consideration is motivated by problems in quality control where acceptability of an item depends on the past k acceptability scores. Moreover, the model introduces dependence that may evolve over time and thus advances the theory for models with time invariant dependence. We establish explicit assumptions that incorporate this dynamic dependence and show how it enters into the limits describing long-term behavior of the system.
8
Content available remote On exact strong laws of large numbers under general dependence conditions
100%
Adler A. , Matuła P.
Probability and Mathematical Statistics
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2018
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tom 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.
9
Content available remote Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables
100%
Chang S. , Li D. , Rosalsky A.
Probability and Mathematical Statistics
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2019
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tom Vol. 39, Fasc. 1
19--38
EN
Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + … + Xn, n ≥ 1. Motivated by a theorem of Mikosch (1984), this note is devoted to establishing a strong law of large numbers for the sequence {max1 ≤ k ≤ n |Sk|; n ≥ 1}. More specifically, necessary and sufficient conditions are given for [wzór] a.s., where log x = loge max{e, x}, x ≥ 0.
10
Content available remote Exact strong laws of large numbers for independent random fields
100%
Kurasiński P. , Matuła P. , Adler A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2018
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tom 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.
11
Content available remote Strong law of large numbers for random variables with multidimensional indices
100%
Gdula A. M. , Krajka A.
Probability and Mathematical Statistics
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2017
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tom Vol. 37, Fasc. 1
185--199
EN
Let {Xn, n ϵ V ⸦ N2} be a two-dimensional random field of independent identically distributed random variables indexed by some subset V of lattice N2. For some sets V the strong law of large numbers [wzór] is equivalent to EX1 = μ and [wzór]. In this paper we characterize such sets V.
12
Content available remote On the strong laws of large numbers for two-dimensional arrays of blockwise independent and blockwise orthogonal random variables
88%
Quang N. V. , Thanh L. V.
Probability and Mathematical Statistics
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2005
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tom 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.
13
Content available remote On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in rademacher type p Banach spaces
88%
Rosalsky A. , Thanh L. V.
Probability and Mathematical Statistics
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2007
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tom 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.
14
Content available remote Exact laws for sums of ratios of order statistics from the Pareto distribution
75%
Adler A.
Open Mathematics
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2006
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tom 4
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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).
15
Content available remote Estimators of g-monotone dependence functions
75%
Krajka A.
Applicationes Mathematicae
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1998-1999
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tom 25
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nr 3
253-269
EN
The notion of g-monotone dependence function introduced in [4] generalizes the notions of the monotone dependence function and the quantile monotone dependence function defined in [2], [3] and [6]. In this paper we study the asymptotic behaviour of sample g-monotone dependence functions and their strong properties.
16
Content available remote Individual ergotic theorem for non-contractive normal operators
63%
Jajte R.
Probability and Mathematical Statistics
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2002
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tom 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.
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