Random sampling of continuous-parameter stationary processes: Statistical properties of joint density estimators

Abstract

Let { X( t), −∞ < t < ∞} be a real-valued stationary process with a bivariate probability density function f( x 1, x 2; t), t > 0, and let { t j } be a renewal point processes on [0, ∞). Estimates f ̂ n(x 1, x 2; t) of f( x 1, x 2; t), based on the discretetime observations { X( t j ), t j } j = 1 n , are considered and their statistical properties are investigated. The quadratic-mean consistency of f ̂ n(x 1, x 2; t) and central limit theorems for f ̂ n(x 1, x 2; t) are established for mixing processes { X( t), −∞ < t < ∞}. Similar results are obtained for estimators of multivariate densities of the process { X( t), −∞ < t < ∞}.