We first establish a general version of the Birkhoff Ergodic Theorem for quasi-integrable extended real-valued random variables without assuming ergodicity. The key argument involves the Poincaré Recurrence Theorem. Our extension of the Birkhoff Ergodic Theorem is also shown to hold for asymptotic mean stationary sequences. This is formulated in terms of necessary and sufficient conditions. In particular, we examine the case where the probability space is endowed with a metric and we discuss the validity of the Birkhoff Ergodic Theorem for continuous random variables. The interest of our results is illustrated by an application to the convergence of statistical transforms, such as the moment generating function or the characteristic function, to their theoretical counterparts.

### Ergodic Theorems for extended-real valued random variables

#### Abstract

We first establish a general version of the Birkhoff Ergodic Theorem for quasi-integrable extended real-valued random variables without assuming ergodicity. The key argument involves the Poincaré Recurrence Theorem. Our extension of the Birkhoff Ergodic Theorem is also shown to hold for asymptotic mean stationary sequences. This is formulated in terms of necessary and sufficient conditions. In particular, we examine the case where the probability space is endowed with a metric and we discuss the validity of the Birkhoff Ergodic Theorem for continuous random variables. The interest of our results is illustrated by an application to the convergence of statistical transforms, such as the moment generating function or the characteristic function, to their theoretical counterparts.
##### Scheda breve Scheda completa Scheda completa (DC) STOCHASTIC PROCESSES AND THEIR APPLICATIONS
http://dx.doi.org/10.1016/j.spa.2010.05.008
Birkhoff's Ergodic Theorem; Asymptotic mean stationarity; Extended real-valued random variables; Non-integrable random variables; Cesaro convergence; Conditional expectation
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11383/1717445`
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