Some applications of the Malliavin calculus to sub-Gaussian and non-sub-Gaussian random elds

We introduce a boundedness condition on the Malliavin derivative of a random variable to study subGaussian and other non-Gaussian properties of functionals of random …elds, with particular attention to the estimation of suprema. We relate the boundedness of nth Malliavin derivatives to a new class of “sub-nth Gaussian chaos” processes. An expected supremum estimation, extending the DudleyFernique theorem, is proved for such processes. Sub-nth Gaussian chaos concentration inequalities for the supremum are obtained, using Malliavin derivative conditions; for n = 1, this generalizes the BorellSudakov inequality to a class of sub-Gaussian processes, with a particularly simple and e¢ cient proof; for n = 2 a natural extension to sub-2nd Gaussian chaos processes is established; for n 3 a slightly less e¢ cient Malliavin derivative condition is needed. Key words and phrases:.Stochastic analysis, Malliavin derivative, Wiener chaos, sub-Gaussian process, concentration, suprema of processes, Dudley-Fernique theorem, Borell-Sudakov inequality. AMS 2000 MSC codes: primary 60H07; secondary 60G15, 60G17. 1 Introduction Gaussian analysis, and in particular the Malliavin calculus, are powerful and versatile tools in contemporary probability theory and stochastic analysis. The latter has applications ranging from other areas of probability theory, to physics, to …nance, to name a few; a very short selection of references might include [2], [5], [6], [7], [12], [13], [14], [15], [16], [17], [22]. We will not attempt to give an overview of such a wide array of areas. Instead, this article presents a new way of using Malliavin derivatives to uncover sub-Gaussian and other non-Gaussian properties of functionals of random …elds, with particular attention to the estimation of suprema. After introducing some standard material on Wiener chaoses and the Malliavin derivative in what we hope is a streamlined and didactic way (Section 2), we introduce the fundamental lemma that serves as a basis and a springboard for non-Gaussian results: it is the observation that if a random variable X has a Malliavin derivative whose norm in L [0; 1] is almost surely bounded, then X is sub-Gaussian (Lemma 3.3). In Section 3, this lemma is exploited to analyze sub-Gaussian processes. Even though the proofs of the results therein are quite elementary, we believe they may have far-reaching consequences in probability and its applications. For example, even though it is not stated so explicitly, Lemma 3.3 is the key ingredient in the new proofs of existence of Lyapunov exponents for the continuous space stochastic Anderson model and the Brownian directed polymer in a Gaussian environment, obtained respectively in [8] and [18]; these existence results had been open problems for many years (see e.g. [4]). Lemma 3.3, and its application to sub-Gaussian deviations of the supremum of a sub-Gaussian random …eld (Theorem 3.6, which is a generalization of the so-called Borell-Sudakov inequality, see [1]), are techniques applied in [21] for statistical estimation problems for non-linear fractional Brownian functionals. Inspired by the power of such applications, we postulate that in order to generalize the concept of subGaussian random variables, one would be well-advised to investigate the properties of random …elds whose nth Malliavin derivatives are bounded. Our study chooses to de…ne the concept of sub-nth Gaussian chaos (or sub-nth chaos, for short) random …elds slightly di¤erently, in order to facilitate the study of such processes’ concentration properties as well as those of their suprema. This is done in Section 4, which also includes an analysis of the relation between the sub-nth chaos property and boundedness of nth Malliavin derivatives. Our proofs in Section 4 are inspired by some of the techniques that worked well in the sub-Gaussian case of Section 3; yet when n 3, many technical di¢ culties arise, and our work opens up as many new problems as it solves in that case. While we prefer to provide full statements of our results in the main body of this paper, we include here some typical consequences of our work under a simplifying assumption which is nonetheless relevant for some applications, leaving it to the reader to check that the results now given do follow from our theorems. Assumption Let n be a positive integer. Let X be a centered separable random …eld on an index set I. Assume that there exists a non-random metric on I I such that almost surely, for all x; y 2 I, for all 0 sn s2 s1 1, jDsn Ds2Ds1 (X (x) X (y))j (x; y) : (1) Conclusions Let N (") be the smallest number of balls of radius " in the metric needed to cover I. There is a constant Cn depending only on n such that, if the assumption above holds, the following conclusions hold: Sub-nth Gaussian chaos property: (see Theorem 4.7) E " exp 1 Cn X (x) X (y) (s; y) 2=n !# 2; sub-nth Gaussian chaos extension of the Dudley-Fernique upper bound : (see Theorem 4.5) := E sup x2I X (x) Cn Z 1 0 (logN (")) n=2 d"; sub-nth Gaussian chaos extension of the Borell-Sudakov concentration inequality : (see Corollary 4.15). With = ess sup !2 fsup jDsn Ds2Ds1X (x)j : x 2 I; 0 sn s2 s1 1g ; for all " > 0, for u large enough, P sup x2I X (x) > u 2 (1 + ") exp 1 (1 + ") u 2=n :