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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 uniform dimension result for two-dimensional fractional multiplicative processes

Xiong Jin (2014)

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

Given a two-dimensional fractional multiplicative process ( F t ) t ∈ [ 0 , 1 ] determined by two Hurst exponents H 1 and H 2 , we show that there is an associated uniform Hausdorff dimension result for the images of subsets of [ 0 , 1 ] by F if and only if H 1 = H 2 .

Asymptotic behavior of the empirical process for gaussian data presenting seasonal long-memory

Mohamedou Ould Haye (2002)

ESAIM: Probability and Statistics

We study the asymptotic behavior of the empirical process when the underlying data are gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and U -Statistics.

Asymptotic behavior of the Empirical Process for Gaussian data presenting seasonal long-memory

Mohamedou Ould Haye (2010)

ESAIM: Probability and Statistics

We study the asymptotic behavior of the empirical process when the underlying data are Gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and U-Statistics.

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...

Cluster continuous time random walks

Agnieszka Jurlewicz, Mark M. Meerschaert, Hans-Peter Scheffler (2011)

Studia Mathematica

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW model is useful in physics, to model diffusing particles. Its scaling limit is a time-changed process, whose densities solve an anomalous diffusion equation. This paper develops limit theory and governing equations for cluster CTRW, in which a random number of jumps cluster together into a single jump. The clustering introduces a dependence between the waiting times and jumps that significantly affects...

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...

Elliptic self-similar stochastic processes.

Albert Benassi, Daniel Roux (2003)

Revista Matemática Iberoamericana

Let M be a random measure and L be an elliptic pseudo-differential operator on Rd. We study the solution of the stochastic problem LX = M, X(O) = O when some homogeneity and integrability conditions are assumed. If M is a Gaussian measure the process X belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of M is not necessarily Gaussian. We characterize the solutions X which are self-similar and with stationary increments in terms of the driving mcasure...

Estimation in models driven by fractional brownian motion

Corinne Berzin, José R. León (2008)

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

Let {bH(t), t∈ℝ} be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type X(t)=c+∫0tσ(X(u)) dbH(u)+∫0tμ(X(u)) du. In different particular models where σ(x)=σ or σ(x)=σ  x and μ(x)=μ or μ(x)=μ  x, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(⋅)...

Excursions of the integral of the brownian motion

Emmanuel Jacob (2010)

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

The integrated brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been...

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