A CENTRAL-LIMIT-THEOREM FOR THE RENORMALIZED SELF-INTERSECTION LOCAL TIME OF A STATIONARY VECTOR GAUSSIAN PROCESS

• 成果类型: Article
• 署名作者: BERMAN, SM
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989918
• 发表日期: 1992
• 页码: 61-81
• 关键词: brownian-motion; functionals; FIELDS

摘要

Let 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, epsilon > 0, consider the double integral [GRAPHICS] which represents an approximate self-intersection local time of X(s), 0 less-than-or-equal-to s less-than-or-equal-to t. Under the sole condition r is-an-element-of L2, the double integral has, upon suitable normalization, a limiting normal distribution under a class of limit operations in which t --> infinity and epsilon = epsilon(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 t2 and t3/2, respectively. These results are valid without any restrictions on the behavior of r(t) for t --> 0 other than continuity.