An error bound in the Sudakov-Fernique inequality

We obtain an asymptotically sharp error bound in the classical Sudakov-Fernique comparison inequality for finite collections of gaussian random variables. Our proof is short and selfcontained, and gives an easy alternative argument for the classical inequality, extended to the case of non-centered processes. 1 Statement of the result Gaussian comparison inequalities are among the most important tools in the theory of gaussian processes, and the Sudakov-Fernique inequality (named after Sudakov [11, 12] and Fernique [3]) is perhaps the most widely used member of that class. We will concentrate on the Sudakov-Fernique inequality in this article; general discussions about comparison inequalities can be found in Adler [1], Fernique [4], Ledoux & Talagrand [9], and Lifshits [10]. The classical Sudakov-Fernique inequality goes as follows: Theorem 1.1. [Sudakov-Fernique inequality] Let {Xi, i ∈ I} and {Yi, i ∈ I} be two centered gaussian processes indexed by the same indexing set I. Suppose that both the processes are almost surely bounded. For each i, j ∈ I, let γ ij = E(Xi −Xj) and γ ij = E(Yi − Yj). If γ ij ≤ γ ij for all i, j, then E(supi∈I Xi) ≤ E(supi∈I Yi). As mentioned before, this inequality is attributed to Sudakov [11, 12] and Fernique [3]. Later proofs were given in Alexander [2] and an unpublished work of S. Chevet. Important variants were proved by Gordon [5, 6, 7] and Kahane [8]. More recently, Vitale [14] has shown, through a clever argument, that we only need E(Xi) = E(Yi) instead of E(Xi) = E(Yi) = 0 in the hypothesis of Theorem 1.1. We will prove the following result, which gives an sharp error bound when the indexing set is finite, and also contains Vitale’s extension of the Sudakov-Fernique inequality. Theorem 1.2. Let (X1, . . . ,Xn) and (Y1, . . . , Yn) be gaussian random vectors with E(Xi) = E(Yi) for each i. For 1 ≤ i, j ≤ n, let γ ij = E(Xi − Xj) and γ ij = E(Yi − Yj), and let γ = max1≤i,j≤n |γX ij − γ ij |. Then |E( max 1≤i≤n Xi)− E( max 1≤i≤n Yi)| ≤ √ γ log n. Moreover, if γ ij ≤ γ ij for all i, j, then E(maxi Xi) ≤ E(maxi Yi). 1 The asymptotic sharpness of the error bound is easy to see from the case where all the Xi’s are independent standard normals and all the Yi’s are zero. 2 Proof We first need to state the following well-known “integration by parts” lemma: Lemma 2.1. If F : R → R is a C function of moderate growth at infinity, and X = (X1, . . . ,Xn) is a centered Gaussian random vector, then for any 1 ≤ i ≤ n, E(XiF (X)) = n ∑ j=1 E(XiXj)E ( ∂F ∂xi (X) ) . A proof of this lemma can be found in the appendix of [13], for example. Proof of Theorem 1.2. Let X = (X1, . . . ,Xn) and Y = (Y1, . . . , Yn). Without loss of generality, we may assume that X and Y are defined on the same probability space and are independent. Fix β > 0, and define Fβ : R n → R as: Fβ(x) := β −1 log ( n ∑