Results 1  10
of
129
Weak convergence of positive self similar Markov processes and overshoots of Lévy processes
, 2004
"... We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy pr ..."
Abstract

Cited by 34 (2 self)
 Add to MetaCart
We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying Levy process # in the Lamperti's representation converges weakly at infinity and E T1 where T 1 = inf{t : # t 1}. Under these conditions, we give a pathwise construction of the limit law.
2006): Analysis of the Rosenblatt process
 Preprint, SAMOS, Université de Paris 1
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract

Cited by 32 (13 self)
 Add to MetaCart
(Show Context)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. cc sd
A law of iterated logarithm for increasing selfsimilar Markov processes
, 2002
"... We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[. ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
(Show Context)
We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[.
Small deviations for fractional stable processes
, 2003
"... Let {Rt, 0 ≤ t ≤ 1} be a symmetric αstable RiemannLiouville process with Hurst parameter H> 0. Consider a translation invariant, βselfsimilar, and ppseudoadditive functional seminorm .. We show that if H> β + 1/p and γ = (H − β − 1/p) −1, then lim ε ε↓0 γ log P [R  ≤ ε] = −K ∈ ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
Let {Rt, 0 ≤ t ≤ 1} be a symmetric αstable RiemannLiouville process with Hurst parameter H> 0. Consider a translation invariant, βselfsimilar, and ppseudoadditive functional seminorm .. We show that if H> β + 1/p and γ = (H − β − 1/p) −1, then lim ε ε↓0 γ log P [R  ≤ ε] = −K ∈ [−∞, 0), with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H> β + 1/p + 1/α. We also show that under the above assumptions, lim ε ε↓0 γ log P [X  ≤ ε] = −K ∈ (−∞, 0), where X is the linear αstable fractional motion with Hurst parameter H ∈ (0, 1) (if α = 2, then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and NonGaussian frameworks.
Variations and estimators for the selfsimilarity order through Malliavin calculus
, 2007
"... Using multiple stochastic integrals, we analyze the asymptotic behavior of quadratic variations for Gaussian and nonGaussian selfsimilar process. We apply our results to the study of statistical estimators for the selfsimilarity index. ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
(Show Context)
Using multiple stochastic integrals, we analyze the asymptotic behavior of quadratic variations for Gaussian and nonGaussian selfsimilar process. We apply our results to the study of statistical estimators for the selfsimilarity index.
Random rewards, fractional Brownian local times and stable selfsimilar processes
 ANN. APPL. PROBAB
, 2006
"... We describe a new class of selfsimilar symmetric αstable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards or incurring random costs. The resulting models are different from the ones studied earlier both in their mem ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
(Show Context)
We describe a new class of selfsimilar symmetric αstable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards or incurring random costs. The resulting models are different from the ones studied earlier both in their memory properties and smoothness of the sample paths.
A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the selfsimilarity parameter
, 2008
"... ..."
LONG RANGE DEPENDENCE
"... Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced pr ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced processes, heavy tails and light tails, limit theorems and large deviations. 1.
Functional Quantization Rate and mean regularity of processes with an application to Lévy Processes, preprint LPMA1048
, 2006
"... We investigate the connections between the mean pathwise regularity of stochastic processes and their L r (P)functional quantization rates as random variables taking values in some L p ([0,T],dt)spaces (0 < p≤r). Our main tool is the Haar basis. We then emphasize that the derived functional qua ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
(Show Context)
We investigate the connections between the mean pathwise regularity of stochastic processes and their L r (P)functional quantization rates as random variables taking values in some L p ([0,T],dt)spaces (0 < p≤r). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (e.g., for Brownian motion or symmetric stable processes) so that the rate is optimal as a universal upper bound. As a first application, we establish the O((log N) −1/2) upper bound for general Itô processes which include multidimensional diffusions. Then, we focus on the specific family of Lévy processes for which we derive a general quantization rate based on the regular variation properties of its Lévy measure at 0. The case of compound Poisson processes, which appear as degenerate in the former approach, is studied specifically: we observe some rates which are between the finitedimensional and infinitedimensional “usual ” rates.
A FergusonKlassLePage series representation of . . .
, 2010
"... The study of nonstationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In [9], a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, wh ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
The study of nonstationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In [9], a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, which were built using sums over Poisson processes. We present here a different construction of these processes, based on the Ferguson Klass LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse OrnsteinUhlenbeck process, logfractional multistable motion and linear multistable multifractional motion. We also show that the processes defined here have the same finite dimensional distributions as the corresponding processes constructed in [9]. Finally, we display numerical experiments showing graphs of synthesized paths of such processes.