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A Ciesielski–Taylor type identity for positive self-similar Markov processes

A. E. Kyprianou, P. Patie (2011)

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

The aim of this note is to give a straightforward proof of a general version of the Ciesielski–Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski–Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Lévy processes into itself. Secondly, some classical features...

A remarkable σ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano (2013)

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

The σ -finite measure 𝒫 sup which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s h -transform processes with respect to these functions are utilized for the construction of 𝒫 sup .

A simple approach to functional inequalities for non-local Dirichlet forms

Jian Wang (2014)

ESAIM: Probability and Statistics

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for Lp(p> 1) settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in [P.T. Gressman,...

Alpha-stable branching and beta-coalescents.

Birkner, Matthias, Blath, Jochen, Capaldo, Marcella, Etheridge, Alison M., Möhle, Martin, Schweinsberg, Jason, Wakolbinger, Anton (2005)

Electronic Journal of Probability [electronic only]

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

Z. Michna (2007)

Studia Mathematica

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 that Y x can...

Behavior near the extinction time in self-similar fragmentations I : the stable case

Christina Goldschmidt, Bénédicte Haas (2010)

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

The stable fragmentation with index of self-similarity α∈[−1/2, 0) is derived by looking at the masses of the subtrees formed by discarding the parts of a (1+α)−1–stable continuum random tree below height t, for t≥0. We give a detailed limiting description of the distribution of such a fragmentation, (F(t), t≥0), as it approaches its time of extinction, ζ. In particular, we show that t1/αF((ζ−t)+) converges in distribution as t→0 to a non-trivial limit. In order to prove this, we go further and...

Convex rearrangements of Lévy processes

Youri Davydov, Emmanuel Thilly (2007)

ESAIM: Probability and Statistics

In this paper we study asymptotic behavior of convex rearrangements of Lévy processes. In particular we obtain Glivenko-Cantelli-type strong limit theorems for the convexifications when the corresponding Lévy measure is regularly varying at + with exponent α ∈ (1,2).

Dynamical attraction to stable processes

Albert M. Fisher, Marina Talet (2012)

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

We apply dynamical ideas within probability theory, proving an almost-sure invariance principle in log density for stable processes. The familiar scaling property (self-similarity) of the stable process has a stronger expression, that the scaling flow on Skorokhod path space is a Bernoulli flow. We prove that typical paths of a random walk with i.i.d. increments in the domain of attraction of a stable law can be paired with paths of a stable process so that, after applying a non-random regularly...

Exponential ergodicity of semilinear equations driven by Lévy processes in Hilbert spaces

Anna Chojnowska-Michalik, Beniamin Goldys (2015)

Banach Center Publications

We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric α-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance...

Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series

M. Magdziarz, A. Weron (2007)

Studia Mathematica

We introduce a fractional Langevin equation with α-stable noise and show that its solution Y κ ( t ) , t 0 is the stationary α-stable Ornstein-Uhlenbeck-type process recently studied by Taqqu and Wolpert. We examine the asymptotic dependence structure of Y κ ( t ) via the measure of its codependence r(θ₁,θ₂,t). We prove that Y κ ( t ) is not a long-memory process in the sense of r(θ₁,θ₂,t). However, we find two natural continuous-time analogues of fractional ARIMA time series with long memory in the framework of the Langevin...

Generalized tempered stable processes

Jan Rosiński, Jennifer L. Sinclair (2010)

Banach Center Publications

This work introduces the class of generalized tempered stable processes which encompass variations on tempered stable processes that have been introduced in the field, including "modified tempered stable processes", "layered stable processes", and "Lamperti stable processes". Short and long time behavior of GTS Lévy processes is characterized and the absolute continuity of GTS processes with respect to the underlying stable processes is established. Series representations of GTS Lévy processes are...

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

Tomasz J. Kozubowski (2010)

Banach Center Publications

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 stable....

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