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.

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.

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.

Operator semistable probability measures on $R^{N}$

A. Łuczak (1981)

Colloquium Mathematicae

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The right tail exponent of the Tracy–Widom $β$ distribution

Laure Dumaz, Bálint Virág (2013)

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

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The Tracy–Widom $β$ distribution is the large dimensional limit of the top eigenvalue of $β$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a \to \infty$ the tail of the Tracy–Widom distribution satisfies $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

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We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$ . Restrict it to some large cube $Λ$ . Consider now $I_{Λ}$ , a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Λ$ grows to infinity. We prove that, with probability one in the...

On the computation of moments of the partial non-central $χ$ -square distribution function

Gil, Amparo, Segura, Javier, Temme, Nico C.

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Properties satisfied by the moments of the partial non-central $χ$ -square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.

Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

Rafał M. Łochowski (2006)

Banach Center Publications

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Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S = \sum a_{i ₁, . . ., i_{d}} X_{i ₁}^{(1)} \dots X_{i_{d}}^{(d)}$ generated by symmetric random variables $X_{i ₁}^{(1)}, . . ., X_{i_{d}}^{(d)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments...

Universal stability of Banach spaces for ε -isometries

Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou (2014)

Studia Mathematica

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Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T : L (f) \equiv \bar{s p a n} f (X) \to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally...