## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 1 (1992), 61-81.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 1 (1992), 61-81.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989918

**Digital Object Identifier**

doi:10.1214/aop/1176989918

**Mathematical Reviews number (MathSciNet)**

MR1143412

**Zentralblatt MATH identifier**

0749.60021

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G15: Gaussian processes 60G17: Sample path properties 60J55: Local time and additive functionals

**Keywords**

Central limit theorem mixing renormalized local time self-intersections stationary Gaussian process

#### Citation

Berman, Simeon M. A Central Limit Theorem for the Renormalized Self-Intersection Local Time of a Stationary Vector Gaussian Process. Ann. Probab. 20 (1992), no. 1, 61--81. doi:10.1214/aop/1176989918. https://projecteuclid.org/euclid.aop/1176989918