Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 35206, 20 pages
doi:10.1155/JAMSA/2006/35206
Central limit theorem for solutions of random initialized differential equations: a simple proof
Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47197 Los Chaguaramos, Caracas 1041-A, Venezuela
Received 13 November 2004; Revised 12 May 2005; Accepted 13 May 2005
Copyright © 2006 Ileana Iribarren and José R. León. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We study the central and noncentral limit theorems for the
convolution of a certain kernel h with F(ξ(⋅)), where ξ is a stationary Gaussian process and F is a square integrable function with respect to the standard Gaussian measure.
Our method consists in showing that in the weak dependence case,
we can use the Lindeberg method, approaching the
initial Gaussian process by an m-dependent process. We could say
that only variance computations are needed to get the two types of
limits. Then we apply the obtained results to the solutions of the
certain differential equations.