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

Let X1,…,Xn be jointly Gaussian random variables with mean zero. It is shown that ∀x>0 and ∀1⩽k<nPmax1⩽i⩽n |Xi|⩽x⩽(1/ρ)Pmax1⩽i⩽k |Xi|⩽xPmaxk<i⩽n |Xi|⩽x,Pmax1⩽i⩽n |Xi|⩽x⩾ρPmax1⩽i⩽k |Xi|⩽xPmaxk<i⩽n |Xi|⩽xandPmax1⩽i⩽n |Xi|⩽x⩾2−min(k,n−k)/2Pmax1⩽i⩽k |Xi|⩽xPmaxk<i⩽n |Xi |⩽x,where ρ=(|Σ|/(|Σ11| |Σ22|))1/2, Σ,Σ11 and Σ22 are the covariance matrices of (X1,…,Xn), (X1,…,Xk) and (Xk+1,…,Xn), respectively. In particular, for fractional Brownian motion {X(t),t⩾0} of order α (0<α<1), there exists dα>0 such that Psup0⩽s⩽a |X(t)|⩽x,supa⩽t⩽b |X(t)−X(a)|⩽y⩾dαPsup0⩽s⩽a |X(t)|⩽xPsupa⩽t⩽b |X(t)−X(a)|⩽yfor 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.