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

Tomasz J. Kozubowski

Displaying similar documents to “Geometric infinite divisibility, stability, and self-similarity: an overview”

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...

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....

Classes of measures closed under mixing and convolution. Weak stability

Jolanta K. Misiewicz, Krzysztof Oleszkiewicz, Kazimierz Urbanik (2005)

Studia Mathematica

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For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = μ∘λ: λ ∈ , which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for which ℳ(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ₁, Θ₂ independent of X, X’ there exists a random variable Θ independent...

α-stable random walk has massive thorns

Alexander Bendikov, Wojciech Cygan (2015)

Colloquium Mathematicae

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We introduce and study a class of random walks defined on the integer lattice $ℤ^{d}$ -a discrete space and time counterpart of the symmetric α-stable process in $ℝ^{d}$ . When 0 < α <2 any coordinate axis in $ℤ^{d}$ , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.

Premium evaluation for different loss distributions using utility theory

Harman Preet Singh Kapoor, Kanchan Jain (2011)

Discussiones Mathematicae Probability and Statistics

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For any insurance contract to be mutually advantageous to the insurer and the insured, premium setting is an important task for an actuary. The maximum premium ( $P_{m a x})$ that an insured is willing to pay can be determined using utility theory. The main focus of this paper is to determine $P_{m a x}$ by considering different forms of the utility function. The loss random variable is assumed to follow different Statistical distributions viz Gamma, Beta, Exponential, Pareto, Weibull, Lognormal and Burr....

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...

Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders

Z. Michna (2007)

Studia Mathematica

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In this paper we consider a symmetric α-stable Lévy process Z. We use a series representation of Z to condition it on the largest jump. Under this condition, Z can be presented as a sum of two independent processes. One of them is a Lévy process $Y_{x}$ parametrized by x > 0 which has finite moments of all orders. We show that $Y_{x}$ converges to Z uniformly on compact sets with probability one as x↓ 0. The first term in the cumulant expansion of $Y_{x}$ corresponds to a Brownian motion which implies...

On the limit distribution of the well-distribution measure of random binary sequences

Christoph Aistleitner (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove the existence of a limit distribution of the normalized well-distribution measure $W (E_{N}) / \sqrt{N}$ (as $N \to \infty$ ) for random binary sequences $E_{N}$ , by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

On two fragmentation schemes with algebraic splitting probability

M. Ghorbel, T. Huillet (2006)

Applicationes Mathematicae

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Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass x₀ ∈ (0,1) undergoes splitting into b > 1 fragments of random sizes with some size-dependent probability p(x₀). With probability 1-p(x₀), this particle is left unchanged forever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable and unstable case with $p (x ₀) = x ₀^{a}$ and $p (x ₀) = 1 - x ₀^{a}$ respectively, for some a > 0. In the first (resp. second) case,...

Stability of scheduling with random processing times on one machine

Paweł Rajba, Mieczysław Wodecki (2012)

Applicationes Mathematicae

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We consider a strong NP-hard single-machine scheduling problem with deadlines and minimizing the total weight of late jobs on a single machine ( $1 | | \sum w_{i} U_{i}$ ). Processing times are deterministic values or random variables having Erlang distributions. For this problem we study the tolerance to random parameter changes for solutions constructed according to tabu search metaheuristics. We also present a measure (called stability) that allows an evaluation of the algorithm based on its resistance to...

Asymptotic stability of a system of randomly connected transformations on Polish spaces

Katarzyna Horbacz (2001)

Annales Polonici Mathematici

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We give sufficient conditions for the existence of a matrix of probabilities ${[p_{i k}]}_{i, k = 1}^{N}$ such that a system of randomly chosen transformations $Π_{k}$ , k = 1,...,N, with probabilities $p_{i k}$ is asymptotically stable.

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

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A generalized form of the usual Lognormal distribution, denoted with $_{γ}$ , is introduced through the γ-order Normal distribution $_{γ}$ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of $_{γ}$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

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...

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.

Uniformly convex spiral functions and uniformly spirallike functions associated with Pascal distribution series

Gangadharan Murugusundaramoorthy, Basem Aref Frasin, Tariq Al-Hawary (2022)

Mathematica Bohemica

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The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes ${𝒮𝒫}_{p} (α, β)$ and ${𝒰𝒞𝒱}_{p} (α, β)$ of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

On asymmetric distributions of copula related random variables which includes the skew-normal ones

Ayyub Sheikhi, Fereshteh Arad, Radko Mesiar (2022)

Kybernetika

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Assuming that $C_{X, Y}$ is the copula function of $X$ and $Y$ with marginal distribution functions $F_{X} (x)$ and $F_{Y} (y)$ , in this work we study the selection distribution $Z \overset{d}{=} (X | Y \in T)$ . We present some special cases of our proposed distribution, among them, skew-normal distribution as well as normal distribution. Some properties such as moments and moment generating function are investigated. Also, some numerical analysis is presented for illustration.

Stability of characterizations of distribution functions using failure rate functions

Maia Koicheva, Edward Omey (1990)

Aplikace matematiky

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Let $λ$ denote the failure rate function of the $d, f$ . $F$ and let $λ_{1}$ denote the failure rate function of the mean residual life distribution. In this paper we characterize the distribution functions $F$ for which $λ_{1} = c λ$ and we estimate $F$ when it is only known that $λ_{1} / λ$ or $λ_{1} - c λ$ is bounded.