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- W2010681305 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 α P 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." @default.
- W2010681305 created "2016-06-24" @default.
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- W2010681305 date "2003-10-01" @default.
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- W2010681305 title "A Gaussian correlation inequality and its applications to the existence of small ball constant" @default.
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- W2010681305 doi "https://doi.org/10.1016/s0304-4149(03)00084-x" @default.
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