A Gaussian correlation inequality and its applications to the existence of small ball constant

Abstract

Let X 1,…, X n be jointly Gaussian random variables with mean zero. It is shown that ∀ x>0 and ∀1⩽ k< n P max 1⩽i⩽n |X i|⩽x ⩽(1/ρ)P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x , P max 1⩽i⩽n |X i|⩽x ⩾ρP max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i|⩽x and P max 1⩽i⩽n |X i|⩽x ⩾2 − min(k,n−k)/2 P max 1⩽i⩽k |X i|⩽x P max k<i⩽n |X i |⩽x , where ρ=(| Σ|/(| Σ 11| | Σ 22|)) 1/2, Σ, Σ 11 and Σ 22 are the covariance matrices of (X 1,…,X n), (X 1,…,X k) and ( X k+1 ,…, X n ), respectively. In particular, for fractional Brownian motion { X( t), t⩾0} of order α (0< α<1), there exists d α >0 such that P sup 0⩽s⩽a |X(t)|⩽x, sup a⩽t⩽b |X(t)−X(a)|⩽y ⩾d αP sup 0⩽s⩽a |X(t)|⩽x P sup a⩽t⩽b |X(t)−X(a)|⩽y for any 0<a<b, x>0 and y>0. As an application, it is proved that the small ball constant for the fractional Brownian motion of order α exists.