高斯差分方程Wick幂的几乎必然极限定理
An almost sure limit theorem for Wick powers of Gaussian differences quotients
摘要：
设G={G（x），xPDMATHABMM R +}，G（0）=0，是具有$E(G(x)-G(y))^2=σ^2(x-y) $的均值为零的高斯过程.设$ ρ(x)= \frac12{d^{2}\over dx^2}σ^2(x)$，$x\ne 0 $。当$ρ^{k}$在零处可积并满足某些附加的正则性条件时，对所有$g\in B_{0}(R^{+})$，\[ \lim_{h\downarrow 0} \int :(\frac{G(x+h)-G(x)}{h})^{k}:g(x) dx=:(G') ^{k}:(g){.3 in}a.s. \]是$R_+$上具有紧支集的有界Lebesgue可测函数集.这里，$G'$是$G$的广义导数，而$:(\cd)^{k}:$是$k$阶威克幂。
英文摘要：
Let G={G(x), x\in R_+}, G(0)=0, be a mean zero Gaussian process with $E(G(x)-G(y))^2=σ^2(x-y) $. Let $ ρ(x)= \frac12{d^{2}\over dx^2}σ^2(x)$, $x\ne 0 $. When $ρ^{k}$ is integrable at zero and satisfies some additional regularity conditions, \[ \lim_{h\downarrow 0} \int :(\frac{G(x+h)-G(x)}{h})^{k}:g(x) dx=:(G') ^{k}:(g){.3 in}a.s. \] for all $g\in B_{0}(R^{+})$, the set of bounded Lebesgue measurable functions on $R_+$ with compact support. Here $G'$ is a generalized derivative of $G$ and $:(\cd)^{k}:$ is the $k$--th order Wick power.
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