Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

Vizcarra, Andrew B.; Viens, Frederi

doi:10.1007/978-3-7643-8458-6_20

Cited by 5 publications

(18 citation statements)

References 22 publications

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“…While the proof in [17] completely generalized the standard Borell-Sudakov inequality to the sub-Gaussian and sub-2nd-chaos cases, it ran into inefficiencies in the case of higher order chaos.…”

Section: Introductionmentioning

confidence: 96%

“…Also in [17], using a new concise Malliavin-derivative-based proof, a Borell-Sudakov 2 -type concentration inequality for such processes was proved, which shows that the supremum of a sub-nth chaos process is again a sub-nth chaos random variable with a well-controlled scale. While the proof in [17] completely generalized the standard Borell-Sudakov inequality to the sub-Gaussian and sub-2nd-chaos cases, it ran into inefficiencies in the case of higher order chaos.…”

Section: Introductionmentioning

confidence: 97%

“…The question remains of how many such results are true for other classes of processes. While the majorizing measure conditions of Talagrand show that supremum estimates can be achieved for processes with tails decaying no slower than exponentially (in the nomenclature of [8], these are processes with increments in the Orlicz space relative to convex Young functions of the type exp(z q ) − 1, with q 1), the authors of this note provided in [17] an extension of Dudley's entropy upper bound to all q = 2/n with n an integer.…”

Section: Introductionmentioning

confidence: 98%

“…In this article we correct these inefficiencies by combining a new use of iterated Malliavin derivatives, via a fractional exponential Poincaré lemma, with the innovative relation between Malliavin derivatives and suprema of processes discovered in [17]. This technique provides a new usage of the versatile Malliavin calculus, whose many applications are discussed in [9]; we summarize the Malliavin derivative's properties that we use herein, making our treatment essentially self-contained.…”

Section: Introductionmentioning

confidence: 98%

“…Such a process (random field) X on an arbitrary index set I is essentially required to have increments X(t) − X(s) with tails that decay no slower than exp(−c|z| 2/n /δ(s, t)) where δ is some pseudo-metric on I . This definition was motivated by results including: (i) the fundamental observation [17,Lemma 3.3], that if the Malliavin derivative of a random variable X is almost surely bounded, then X must be sub-Gaussian; and (ii) the discovery in that same article of conditions on the nth Malliavin derivatives extending this observation to the sub-nth chaos case.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Viens¹,

Vizcarra²

2007

Journal of Functional Analysis

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This article provides a detailed analysis of the behavior of suprema and moduli of continuity for a large class of random fields which generalize Gaussian processes, sub-Gaussian processes, and random fields that are in the nth chaos of a Wiener process. An upper bound of Dudley type on the tail of the random field's supremum is derived using a generic chaining argument; it implies similar results for the expected supremum, and for the field's modulus of continuity. We also utilize a sharp and convenient condition using iterated Malliavin derivatives, to arrive at similar conclusions for suprema, via a different proof, which does not require full knowledge of the covariance structure.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Viens¹,

Vizcarra²

2007

Journal of Functional Analysis

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IPAD: Stable Interpretable Forecasting with Knockoffs Inference

Fan

Sharifvaghefi

et al. 2019

Journal of the American Statistical Association

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Interpretability and stability are two important features that are desired in many contemporary big data applications arising in economics and finance. While the former is enjoyed to some extent by many existing forecasting approaches, the latter in the sense of controlling the fraction of wrongly discovered features which can enhance greatly the interpretability is still largely underdeveloped in the econometric settings. To this end, in this paper we exploit the general framework of model-X knockoffs introduced recently in Candès, Fan, Janson and Lv (2018), which is nonconventional for reproducible large-scale inference in that the framework is completely free of the use of p-values for significance testing, and suggest a new method of intertwined probabilistic factors decoupling (IPAD) for stable interpretable forecasting with knockoffs inference in high-dimensional models. The recipe of the method is constructing the knockoff variables by assuming a latent factor model that is exploited widely in economics and finance for the association structure of covariates. Our method and work are distinct from the existing literature in that we estimate the covariate distribution from data instead of assuming that it is known when constructing the knockoff variables, our procedure does not require any sample splitting, we provide theoretical justifications on the asymptotic false discovery rate control, and the theory for the power analysis is also established. Several simulation examples and the * Yingying Fan is Dean's Associate Professor in Business Administration,

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Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data

Koike¹

2019

Ann. Statist.

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This paper establishes an upper bound for the Kolmogorov distance between the maximum of a highdimensional vector of smooth Wiener functionals and the maximum of a Gaussian random vector. As a special case, we show that the maximum of multiple Wiener-Itô integrals with common orders is well-approximated by its Gaussian analog in terms of the Kolmogorov distance if their covariance matrices are close to each other and the maximum of the fourth cumulants of the multiple Wiener-Itô integrals is close to zero. This may be viewed as a new kind of fourth moment phenomenon, which has attracted considerable attention in the recent studies of probability. This type of Gaussian approximation result has many potential applications to statistics.To illustrate this point, we present two statistical applications in high-frequency financial econometrics: One is the hypothesis testing problem for the absence of lead-lag effects and the other is the construction of uniform confidence bands for spot volatility.Let m * be an integer such that |ρ m * |Σ(θ m * ) = max 1≤m≤M |ρ m |Σ(θ m ). By assumption [A4], for each n ∈ N there is a number ϑ n ∈ G n such that |ϑ n − θ m * | ≤ υ n . Now noting (B.5), we havefor sufficiently large n. Let us denote by ⊖ the symmetric difference between two sets. Then we have

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Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

Cited by 5 publications

References 22 publications

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

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

IPAD: Stable Interpretable Forecasting with Knockoffs Inference

Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data

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