关于高斯过程的最大值的重对数律的一点注记

A NOTE ON TNE ITERATED LOGARITHM LAW FOR THE MAXIMUM IN A GAUSSIAN PROCESS

令{X(t),-∞<t<∞}是在概率空间(Ω, ,P)上定义的平稳可分实高斯过程。设EX(t)≡0,EX^2(t)≡1,相关函数r(t)=E(X(s)X(s+t))是连续的且满足r(t)=1-|t|~aH(|t|)+o(|t|~aH(|t|)),(1)其中o<a≤2,H是在o点的缓变函数,不失一般性,可以假设缓变函数H是“标准化”的。我们研究M(t)≡ X(s)当t→∞时的渐近性质,得到如下的重对数律:

Let{X(t),-∞<t<∞} denote a separable stationary Gaussian process with mean zero and variane one.Suppos the covariance function r(t)=1-|t|~αH(|t|)+0(|t|~αH(|t|)), where 0<α≤2 and H is a normalized slowly varying function at o.An iterated logarithm law for M(t)≡sup X(s) is obtained as follows: If for some λ>0, then We can easily generalize the result to the non-stationary case.