Classes of measures closed under mixing and convolution. Weak stability

Jolanta K. Misiewicz; Krzysztof Oleszkiewicz; Kazimierz Urbanik

Displaying similar documents to “Classes of measures closed under mixing and convolution. Weak stability”

Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Jonathon Peterson, Gennady Samorodnitsky (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $κ g t; 0$ that determines the fluctuations of the process. When $0 l t; κ l t; 2$ , the averaged distributions of the hitting times of the random walk converge to a $κ$ -stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true...

On some limit distributions for geometric random sums

Marek T. Malinowski (2008)

Discussiones Mathematicae Probability and Statistics

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We define and give the various characterizations of a new subclass of geometrically infinitely divisible random variables. This subclass, called geometrically semistable, is given as the set of all these random variables which are the limits in distribution of geometric, weighted and shifted random sums. Introduced class is the extension of, considered until now, classes of geometrically stable [5] and geometrically strictly semistable random variables [10]. All the results can be straightforward...

A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski (2007)

Discussiones Mathematicae Probability and Statistics

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A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_{p}$ such that $X \overset{d}{=} \sum_{k = 1}^{T (p)} X_{p, k}$ , where $X_{p, k}$ ’s are i.i.d. copies of $X_{p}$ , and random variable T(p) independent of $X_{p, 1}, X_{p, 2}, . . .$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions....

Further results on laws of large numbers for uncertain random variables

Feng Hu, Xiaoting Fu, Ziyi Qu, Zhaojun Zong (2023)

Kybernetika

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The uncertainty theory was founded by Baoding Liu to characterize uncertainty information represented by humans. Basing on uncertainty theory, Yuhan Liu created chance theory to describe the complex phenomenon, in which human uncertainty and random phenomenon coexist. In this paper, our aim is to derive some laws of large numbers (LLNs) for uncertain random variables. The first theorem proved the Etemadi type LLN for uncertain random variables being functions of pairwise independent...

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

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We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

(2012)

Colloquium Mathematicae

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The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra $_{C}^{'} (ℝ ⁿ; M_{m \times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G \in_{C}^{'} (ℝ ⁿ; M_{m \times m})$ is the generating distribution of an i.d.c.s. if and only if the operator $\partial_{t} \otimes_{m \times m} - G *$ on $ℝ^{1 + n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Geometric infinite divisibility, stability, and self-similarity: an overview

Tomasz J. Kozubowski (2010)

Banach Center Publications

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The concepts of geometric infinite divisibility and stability extend the classical properties of infinite divisibility and stability to geometric convolutions. In this setting, a random variable X is geometrically infinitely divisible if it can be expressed as a random sum of $N_{p}$ components for each p ∈ (0,1), where $N_{p}$ is a geometric random variable with mean 1/p, independent of the components. If the components have the same distribution as that of a rescaled X, then X is (strictly) geometric...

A characterization of the space $D_{F}^{'}$

S. Pilipović (1982)

Matematički Vesnik

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On the limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

On the existence and asymptotic behavior of the random solutions of the random integral equation with advancing argument

Henryk Gacki (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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1. Introduction Random Integral Equations play a significant role in characterizing of many biological and engineering problems [4,5,6,7]. We present here new existence theorems for a class of integral equations with advancing argument. Our method is based on the notion of a measure of noncompactness in Banach spaces and the fixed point theorem of Darbo type. We shall deal with random integral equation with advancing argument $x (t, ω) = h (t, ω) + \int^{t + δ (t)} ₀ k (t, τ, ω) f (τ, x_{τ} (ω)) d τ$ , (t,ω) ∈ R⁺ × Ω, (1) where (i) (Ω,A,P) is a complete probability...

Universality for random tensors

Razvan Gurau (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We prove two universality results for random tensors of arbitrary rank $D$ . We first prove that a random tensor whose entries are $N^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability...

Tail and moment estimates for sums of independent random variables with logarithmically concave tails

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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For random variables $S = \sum_{i = 1}^{\infty} α_{i} ξ_{i}$ , where $(ξ_{i})$ is a sequence of symmetric, independent, identically distributed random variables such that $l n P (| ξ_{i} | \geq t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

Slowdown estimates and central limit theorem for random walks in random environment

Alain-Sol Sznitman (2000)

Journal of the European Mathematical Society

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This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on $ℤ^{d}$ , when $d > 2$ . We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important...

Excited against the tide: a random walk with competing drifts

Mark Holmes (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study excited random walks in i.i.d. random cookie environments in high dimensions, where the $k$ th cookie at a site determines the transition probabilities (to the left and right) for the $k$ th departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that...

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Faranak Goodarzi, Mohammad Amini, Gholam Reza Mohtashami Borzadaran (2017)

Applications of Mathematics

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Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via $w (\cdot)$ -function defined by Cacoullos...

Random differential inclusions with convex right hand sides

Krystyna Grytczuk, Emilia Rotkiewicz (1991)

Annales Polonici Mathematici

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Abstract. The main result of the present paper deals with the existence of solutions of random functional-differential inclusions of the form ẋ(t, ω) ∈ G(t, ω, x(·, ω), ẋ(·, ω)) with G taking as its values nonempty compact and convex subsets of n-dimensional Euclidean space $R^{n}$ .