在<i>R<sup>d</sup></i>的VC类上Wiener测度的上界

在R^d的VC类上Wiener测度的上界

摘要: 本文考察指标集为d维欧氏空间R^d的子集类c的Wiener过程W的样本轨道性质．在某些情形中，对0＜a_n≤n^d，a_n是不减的且a_n／n^d是不增的假设下，求得函数Ψn使 \limsup _n \rightarrow \infty \sup \left\\left|W\left(C^(n)\right)\right| / \psi_n: C^(n) \in \mathcalC,\left|C^(n)\right| \leq a_n\right\=1, \quad \text a.s.. 给出了Wiener测度在R^d的VC类的界，它也推广了Csörgö-Révész关于Wiener过程增量的结果于这一一般情形。

Abstract: Sample path behavior is studied for the Wiener processes W indexed by classes C of subsetsof the Borel δ-field of R^d. For 0＜a_n ≤n^d, a_n is nondecreasing and a_n/n^d is nonmcreasing, afunction Ψn is found in some cases such that limsup_n→∝sup｛|W（C^（n））|/Ψn:C^（n）∈C|C^（n）|≤a_n｝=1 a.s. This generalies the Csörgö-Révész’s increments results for a Wiener process.