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1

Limit theorems for a higher order time dependent Markov chain model

Kokoszka Piotr, Kutta Tim, Singh Deepak, Wang Haonan

Probability and Mathematical Statistics

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2023

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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.

2

Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

Chang Shuhua, Li Deli, Rosalsky Andrew

Probability and Mathematical Statistics

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2019

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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.

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

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.

5

Strong law of large numbers for random variables with multidimensional indices

Gdula A. M., Krajka A.

Probability and Mathematical Statistics

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2017

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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.

6

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.

7

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.

8

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.

9

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.

10

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 → ∞.

11

Strong laws of large numbers for random permanents

Rempała G. A., Wesołowski J.

Probability and Mathematical Statistics

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2002

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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.

12

Marcinkiewicz-type strong law of large numbers for pairwise independent random fields

Czerebak-Mrozowicz E. B., Klesov O. I., Rychlik Z.

Probability and Mathematical Statistics

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2002

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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.

13

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.

14

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.