Steins lemma, Malliavin calculus, and tail bounds, with application to polymer uctuation exponent

We consider a random variable X satisfying almost-sure conditions involving G := DX; DL X where DX is X’s Malliavin derivative and L 1 is the pseudo-inverse of the generator of the OrnsteinUhlenbeck semigroup. A lower(resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [X > z]. Bounds of other natures are also given. A key ingredient is the use of Stein’s lemma, including the explicit form of the solution of Stein’s equation relative to the function 1x>z, and its relation to G. Another set of comparable results is established, without the use of Stein’s lemma, using instead a formula for the density of a random variable based on G, recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G, we show that the Brownian polymer in a Gaussian environment which is white-noise in time and positively correlated in space has deviations of Gaussian type and a ‡uctuation exponent = 1=2. We also show this exponent remains 1=2 after a non-linear transformation of the polymer’s Hamiltonian. Key words and phrases: Malliavin calculus, Wiener chaos, sub-Gaussian, Stein’s lemma, polymer, Anderson model, random media, ‡uctuation exponent. AMS 2000 MSC codes: primary 60H07; secondary 60G15, 60K37, 82D60 1 Introduction 1.1 Background and context Ivan Nourdin and Giovanni Peccati have recently made a long-awaited connection between Stein’s lemma and the Malliavin calculus: see [9], and also [10]. Our article uses crucial basic elements from their work, to investigate the behavior of square-integrable random variables whose Wiener chaos expansions are not …nite. Speci…cally we devise conditions under which the tail of a random variable is bounded below by Gaussian tails, by using Stein’s lemma and the Malliavin calculus. Our article also derives similar lower bounds by way of a new formula for the density of a random variable, established in [12], which uses Malliavin calculus, but not Stein’s lemma. Tail upper bounds are also derived, using both methods. Author’s reserach partially supported by NSF grant 0606615 Stein’s lemma has been used in the past for Gaussian upper bounds, e.g. in [3] in the context of exchangeable pairs. Malliavin derivatives have been invoked for similar upper bounds in [22]. In the current paper, the combination of these two tools yields a novel criterion for a Gaussian tail lower bound. We borrow a main idea from Nourdin and Peccati [9], and also from [12]: to understand a random variable Z which is measurable with respect to a Gaussian …eld W , it is fruitful to consider the random variable G := hDZ; DL 1ZiH ; where D is the Malliavin derivative relative to W , h ; iH is the inner product in the canonical Hilbert space H of W , and L is the generator of the Ornstein-Uhlenbeck semigroup. Details on D, H, L, and G, will be given below. The function g (z) = E [GjZ = z] has already been used to good e¤ect in the density formula discovered in [12]; this formula implied new lower bounds on the densities of some Gaussian processes’suprema. These results are made possible by fully using the Gaussian property, and in particular by exploiting both upper and lower bounds on the process’s covariance. The authors of [12] noted that, if Z has a density and an upper bound is assumed on G, in the absence of any other assumption on how Z is related to the underlying Gaussian process W , then Z’s tail is sub-Gaussian. On the other hand, the authors of [12] tried to discard any upper bound assumption, and assume instead that G was bounded below, to see if they could derive a Gaussian lower bound on Z’s tail; they succeeded in this task, but only partially, as they had to impose some additional conditions on Z’s function g, which are of upper-bound type, and which may not be easy to verify in practice. The techniques used in [12] are well adapted to studying densities of random variables under simultaneous lower and upper bound assumptions, but less so under single-sided assumptions. The point of the current paper is to show that, while the quantitative study of densities via the Malliavin calculus seems to require two-sided assumptions as in [12], single-sided assumptions on G are in essence su¢ cient to obtain single sided bounds on tails of random variables, and there are two strategies to this end: Nourdin and Peccati’s connection between Malliavin calculus and Stein’s lemma, and exploiting the Malliavin-calculus-based density formula in [12]. A key new component in our work, relative to the …rst strategy, may be characterized by saying that, in addition to a systematic exploitation of the Stein-lemma–Malliavin-calculus connection (via Lemma 3.5 below), we carefully analyze the behavior of solutions of the so-called Stein equation, and use them pro…tably, rather than simply use the fact that there exist bounded solutions with bounded derivatives. We were inspired to work this way by the similar innovative use of the solution in the context of Berry-Esséen theorems in [10]. The novelty in our second strategy is simply to note that the di¢ culties inherent to using the density formula of [12] with only one-sided assumptions, tend to disappear when one passes to tail formulas. Our work follows in the footsteps of Nourdin and Peccati’s. One major di¤erence in focus between our work and their’s, and indeed between ours and the main use of Stein’s method since its inception in [19] to the most recent results (see [3], [4], [17], and references therein) is that Stein’s method is typically concerned with convergence to the normal distribution while we are only interested in rough bounds of Gaussian or other types for single random variables (not sequences), without imposing conditions which would lead to normal or any other convergence. While the focus in [9] is on convergence theorems, its authors were already aware of the ability of Stein’s lemma and the Malliavin calculus to yield bounds for …xed r.v.’s, not sequences: their work implies that a bound on the deviation of a single G from the value 1 has clear implications for the distance from Z’s distribution to the normal law. In fact the main technical tool therein ([9, Theorem 3.1]) is stated for …xed random variables, yielding bounds on various distances between the distributions of such r.v.’s and the normal law, based on expectation calculations using G 1 explicitly; also see [9, Remark 3.6]. Nourdin and Peccati’s motivations only required them to make use of [9, Theorem 3.1] as applied to convergences of sequences.