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

Let {X(t), -[infinity] < t < [infinity]} be a real-valued stationary process with a bivariate probability density function f(x1, x2; t), t > 0, and let {tj} be a renewal point processes on [0, [infinity]). Estimates of f(x1, x2; t), based on the discretetime observations {X(tj), tj}j = 1n, are considered and their statistical properties are investigated. The quadratic-mean consistency of and central limit theorems for are established for mixing processes {X(t), -[infinity] < t < [infinity]}. Similar results are obtained for estimators of multivariate densities of the process {X(t), -[infinity] < t < [infinity]}.