Ergodic Theorems with Random Weights for Stationary Random Processes and Fields

Let X(t) be an ergodic stationary random process or an ergodic homogeneous random field on Rm,m >=;2, and let M (B) be a mixing homogeneous locally finite randomBorel measure with mean density gamma onRm,m >= 1. We assume thatXandMare inde-pendent and possess finite expectations. If{Tn}is an increasing sequence of boundedconvex sets, containing balls of radiirn ->infinity, then(sic)Special cases are ergodic theorems with averages over finite random sets. Example:IfSis an independent-of-XPoisson random set inRmwith mean density gamma, then (sic)