一类N参数Gauss过程的异常震动点集合的Hausdorff维数

Hausdorff Dimension of the Set Generated by Exceptional Oscillations of a Class of N-Parameter Gaussian Processes

引进了一类N参数Gauss过程,它具有比N参数Wiener过程更为一般的性质.给出了此类N参数Gauss过程的异常震动点集的定义,并且定义了此异常震动点集的Hausdorff维数.研究了此类过程的异常震动点集Hausdorff维数,给出了它的一个确切的表达式,从而获得了与Zacharie(2001)的有关两参数Wiener过程的类似的结果.考虑的参数点集是一般的超长方体.而不是Zacharie(2001)考虑的超正方体.在此更为一般的情况下,首先建立了文中引进的过程的Fernique不等式.利用此不等式和Slepian引理,证明了过程的Lévy连续模定理.Zacharie(2001)关于Hausdorff维数公式的证明依赖于两参数Wiener过程的独立增量性,而这里引进的过程不具有这种性质,因此,必须采用新的证明途径.

A class of N-parameter Gaussian processes were introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of class of these processes was given. And then the Hausdorff dimension of this set was defined. The Hausdorff dimensions of these processes were studied and an exact represeatative for them was given, which is similar to that for the two-parameter Wiener process by Zacharie （2001）. Moreover, the time set considered is a hyperrectangle which is more general than a hyper-square used by Zacharie （2001 ）. For this more general case, a Femique-type inequality was established and then using this inequality and the Slepian lemma, a Lévy＇s continuity modulus theorem was shown. Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie （2001）. This property is absent for the processes introduced here, so a different way is to be found.