An ergodic theorem for Fréchet-valued random variables

Abstract:

We generalize the classical ergodic theorem from real-valued random variables to Fréchet space-valued random variables and obtain this generalization as a direct corollary of the classical theorem. As an application we obtain several strong laws of large numbers for Fréchet-valued random variables. In a similar way we obtain a martingale theorem for Fréchet-valued random variables.