A Central Limit Theorem for the Renormalized Self-Intersection Local Time of a Stationary Vector Gaussian Process

Abstract

Let $\mathbf{X}(t)$ be a stationary vector Gaussian process in $R^m$ whose components are independent copies of a real stationary Gaussian process with covariance function $r(t)$. Let $\phi(z)$ be the standard normal density and, for $t > 0, \varepsilon > 0$, consider the double integral $\int^t_0\int^t_0\varepsilon^{-m} \prod^m_{j=1} \phi(\varepsilon^{-1}(X_j(s) - X_j(s')))ds ds',$ which represents an approximate self-intersection local time of $\mathbf{X}(s), 0 \leq s \leq t$. Under the sole condition $r \in L_2$, the double integral has, upon suitable normalization, a limiting normal distribution under a class of limit operations in which $t \rightarrow \infty$ and $\varepsilon = \varepsilon(t)$ tends to 0 sufficiently slowly. The expected value and standard deviation of the double integral, which are the normalizing functions, are asymptotically equal to constant multiples of $t^2$ and $t^{3/2}$, respectively. These results are valid without any restrictions on the behavior of $r(t)$ for $t \rightarrow 0$ other than continuity.