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 [for all]x>0 and [for all]1[less-than-or-equals, slant]k<nandwhere [rho]=([Sigma]/([Sigma]11 [Sigma]22))1/2, [Sigma],[Sigma]11 and [Sigma]22 are the covariance matrices of (X1,...,Xn), (X1,...,Xk) and (Xk+1,...,Xn), respectively. In particular, for fractional Brownian motion {X(t),t[greater-or-equal, slanted]0} of order [alpha] (0<[alpha]<1), there exists d[alpha]>0 such that 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 [alpha] exists.