Supremum concentration inequality and modulus of continuity for sub-nth chaos processes

Viens, Frederi; Vizcarra, Andrew B.

doi:10.1016/j.jfa.2007.03.019

Cited by 29 publications

(40 citation statements)

References 12 publications

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“…This is similar to the upper bound portion of our work (Section 4.1), and closer yet to the rst-chaos portion of the work in [16]; they do not, however, address lower bound issues.…”

Section: Concentration Inequalitiessupporting

confidence: 70%

Density Formula and Concentration Inequalities with Malliavin Calculus

Nourdin¹,

Viens

2009

Electron. J. Probab.

108

206

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We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and dierentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator (dual of the Malliavin derivative). In particular, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process, thus extending a formula recently used by Chatterjee [4]. We also explain how to derive concentration inequalities for Z in our framework.

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“…This is similar to the upper bound portion of our work (Section 4.1), and closer yet to the rst-chaos portion of the work in [16]; they do not, however, address lower bound issues.…”

Section: Concentration Inequalitiessupporting

confidence: 70%

Density Formula and Concentration Inequalities with Malliavin Calculus

Nourdin¹,

Viens

2009

Electron. J. Probab.

108

206

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“…(1.4) The modulus of continuity for X H,N (ω) obtained thanks to [33] allows to derive, for almost all ω ∈ Ω, that lim sup…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Lower bound for local oscillations of Hermite processes

Ayache

2020

Stochastic Processes and their Applications

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The most known example of a class of non-Gaussian stochastic processes which belongs to the homogenous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N . They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound of the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N , which, among other things, shows that these paths are nowhere differentiable functions.Running Title: On the oscillations of Hermite processes.

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“…The reader will check that the first of the following two corollaries is trivial to prove using the tools in this article. The second corollary requires techniques in [12], and can also be proved directly by using sub-Gaussian concentration results (see [27]). We do not give any details of its proof, for the sake of conciseness.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Sharp Asymptotics for the Partition Function of Some Continuous-time Directed Polymers

Cadel

Tindel²,

Viens³

2008

Potential Anal

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This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is white-noise in time and homogeneous in space. The second is a continuous-time discrete-space Markov process on Z d , in a random environment with similar properties as in continuous space, albeit defined only on R + × Z d . The case of a space-time white noise environment can be achieved in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder regime of the objects under consideration.Keywords Polymer model · Random medium · Gaussian field · Free energy · Lyapunov exponent Mathematics Subject Classifications (2000) 82D60 · 60K37 · 60G15 Frederi Viens' research partially supported by NSF grants no.: DMS 0204999 and DMS 0606615.A. Cadel · S. Tindel (B)

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Supremum concentration inequality and modulus of continuity for sub-nth chaos processes

Cited by 29 publications

References 12 publications

Density Formula and Concentration Inequalities with Malliavin Calculus

Density Formula and Concentration Inequalities with Malliavin Calculus

Lower bound for local oscillations of Hermite processes

Sharp Asymptotics for the Partition Function of Some Continuous-time Directed Polymers

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