Independent sampling of a stochastic process (Q1805749)

Sampling a stochastic process \(X=\{Z(t) : t\geq 0\}\) at the times of an independent point process \(\psi \) leads to the same empirical distribution as the time-average limiting distribution of \(X\) under some conditions. Two cases are considered. The first is when \(X\) is asymptotically stationary and ergodic, and \(\psi \) satisfies a mixing condition. The other case is when \(X\) is only assumed to have a constant finite time average and \(\psi \) is assumed a positive recurrent renewal process with a spread-out cycle length distribution.