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 $${\mathbb {R}}^m,m\ge 2$$ R m , m ≥ 2 , and let M(B) be a mixing homogeneous locally finite random Borel measure with mean density $$\gamma $$ γ on $${\mathbb {R}}^m,m\ge 1$$ R m , m ≥ 1 . We assume that X and M are independent and possess finite expectations. If $$\{T_n\}$$ { T n } is an increasing sequence of bounded convex sets, containing balls of radii $$r_n\rightarrow \infty $$ r n → ∞ , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\lambda (T_n)}\int _{T_n}X(t)M(\textrm{d}t,w)=\gamma E[X(0)]\text { a.s. and in } L^1. \end{aligned}$$ lim n → ∞ 1 λ ( T n ) ∫ T n X ( t ) M ( d t , w ) = γ E [ X ( 0 ) ] a.s. and in L 1 . Special cases are ergodic theorems with averages over finite random sets. Example: If S is an independent-of-X Poisson random set in $${\mathbb {R}}^m$$ R m with mean density $$\gamma $$ γ , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{ {\lambda (T_n)}}\sum _{t\in S \cap T_n}X(t)=\gamma E[X(0 )] \ \text {a.s. and in } L^1 \ \ (\text {card} (S\cap T_n)