KERNEL DENSITY-ESTIMATION FOR LINEAR-PROCESSES

Let X1,...,X(n) be n consecutive observations of a linear process X(t) = mu + SIGMA(r=0)infinity a(r)Z(t-r), where mu is a constant and {Z(t)} is an innovation process consisting of independent and identically distributed random variables with mean zero and finite variance. Assume that X1 has a probability density f. Uniform strong consistency of kernel density estimators of f is established, and their rates of convergence are obtained. The estimators can achieve the rate of convergence (n-1 log n)1/3 in L(infinity) norm restricted to compacts under weak conditions.