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edited Apr 13, 2017 at 12:58

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:

The answer to Q2 is yes; the structure of the proof is exactly as I outlined in the update. Details can be found in section 5 (in particular, Corollary 100) of my notes http://wwwf.imperial.ac.uk/~jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf.

The answer to Q1 is also yes: Since $f$ is bounded, it is sufficient just to consider the limit as $T$ tends to $\infty$ in the integers. By the positive answer to the question Is it true that all stationary measurable stochastic processes are "measurably stationary"?Is it true that all stationary measurable stochastic processes are "measurably stationary"?, the discrete-time stochastic process $\left(\int_n^{n+1} f(X_t(\cdot)) \, dt \right)_{n \geq 0}$ is stationary, and therefore Birkhoff's ergodic theorem (applied to the shift map on $\mathbb{R}^{\mathbb{N}_0}$) gives the desired convergence.

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:

The answer to Q1 is also yes: Since $f$ is bounded, it is sufficient just to consider the limit as $T$ tends to $\infty$ in the integers. By the positive answer to the question Is it true that all stationary measurable stochastic processes are "measurably stationary"?, the discrete-time stochastic process $\left(\int_n^{n+1} f(X_t(\cdot)) \, dt \right)_{n \geq 0}$ is stationary, and therefore Birkhoff's ergodic theorem (applied to the shift map on $\mathbb{R}^{\mathbb{N}_0}$) gives the desired convergence.

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:

The answer to Q1 is also yes: Since $f$ is bounded, it is sufficient just to consider the limit as $T$ tends to $\infty$ in the integers. By the positive answer to the question Is it true that all stationary measurable stochastic processes are "measurably stationary"?, the discrete-time stochastic process $\left(\int_n^{n+1} f(X_t(\cdot)) \, dt \right)_{n \geq 0}$ is stationary, and therefore Birkhoff's ergodic theorem (applied to the shift map on $\mathbb{R}^{\mathbb{N}_0}$) gives the desired convergence.

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created Mar 10, 2015 at 3:06

Julian Newman

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:

The answer to Q1 is also yes: Since $f$ is bounded, it is sufficient just to consider the limit as $T$ tends to $\infty$ in the integers. By the positive answer to the question Is it true that all stationary measurable stochastic processes are "measurably stationary"?, the discrete-time stochastic process $\left(\int_n^{n+1} f(X_t(\cdot)) \, dt \right)_{n \geq 0}$ is stationary, and therefore Birkhoff's ergodic theorem (applied to the shift map on $\mathbb{R}^{\mathbb{N}_0}$) gives the desired convergence.