(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".)

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $(E,\mathcal{E})$ be a measurable space, and let $X:[0,\infty) \times \Omega \to E$ be a measurable function with the property that for any $n \in \mathbb{N}$, for all $\tau,t_1,\ldots,t_n \geq 0$ the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1,\omega),\ldots,X(t_n,\omega) \right)$

is equal to the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1+\tau,\omega),\ldots,X(t_n+\tau,\omega) \right)$.

Is it necessarily the case that for any $n \in \mathbb{N}$, for all $A \in \mathcal{E}^{\otimes n}$ and all $s,\tau,t_1,\ldots,t_n \geq 0$, the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [0,s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

is equal to the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [\tau,\tau+s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

(where $\textrm{Leb}$ denotes the Lebesgue measure)?

How about in the special case that $(E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$?

Now I've asked the above question in quite a general form, but what I ultimately want is: if $(X_t)_{t \geq 0}$ is a measurable stationary stochastic process in continuous time, then is the discrete-time stochastic process $\left( \int_n^{n+1} g(X_t) \, dt \right)_{n \geq 0}$ necessarily also stationary (where $g$ is a bounded measurable function)??

(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".)

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $(E,\mathcal{E})$ be a measurable space, and let $X:[0,\infty) \times \Omega \to E$ be a measurable function with the property that for any $n \in \mathbb{N}$, for all $\tau,t_1,\ldots,t_n \geq 0$ the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1,\omega),\ldots,X(t_n,\omega) \right)$

is equal to the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1+\tau,\omega),\ldots,X(t_n+\tau,\omega) \right)$.

Is it necessarily the case that for any $n \in \mathbb{N}$, for all $A \in \mathcal{E}^{\otimes n}$ and all $s,\tau,t_1,\ldots,t_n \geq 0$, the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [0,s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

is equal to the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [\tau,\tau+s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

(where $\textrm{Leb}$ denotes the Lebesgue measure)?

How about in the special case that $(E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$?

Now I've asked the above question in quite a general form, but what I ultimately want is: if $(X_t)_{t \geq 0}$ is a measurable stationary stochastic process in continuous time, then is the discrete-time stochastic process $\left( \int_n^{n+1} g(X_t) \, dt \right)_{n \geq 0}$ necessarily also stationary (where $g$ is a bounded measurable function)??

(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".)

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $(E,\mathcal{E})$ be a measurable space, and let $X:[0,\infty) \times \Omega \to E$ be a measurable function with the property that for any $n \in \mathbb{N}$, for all $\tau,t_1,\ldots,t_n \geq 0$ the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1,\omega),\ldots,X(t_n,\omega) \right)$

is equal to the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1+\tau,\omega),\ldots,X(t_n+\tau,\omega) \right)$.

Is it necessarily the case that for any $n \in \mathbb{N}$, for all $A \in \mathcal{E}^{\otimes n}$ and all $s,\tau,t_1,\ldots,t_n \geq 0$, the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [0,s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

is equal to the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [\tau,\tau+s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

(where $\textrm{Leb}$ denotes the Lebesgue measure)?

How about in the special case that $(E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$?

(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".)

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $(E,\mathcal{E})$ be a measurable space, and let $X:[0,\infty) \times \Omega \to E$ be a measurable function with the property that for any $n \in \mathbb{N}$, for all $\tau,t_1,\ldots,t_n \geq 0$ the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1,\omega),\ldots,X(t_n,\omega) \right)$

is equal to the image measure of $\mathbb{P}$ under the $E^n$-valued map

$\hspace{5mm} \omega \mapsto \left( X(t_1+\tau,\omega),\ldots,X(t_n+\tau,\omega) \right)$.

Is it necessarily the case that for any $n \in \mathbb{N}$, for all $A \in \mathcal{E}^{\otimes n}$ and all $s,\tau,t_1,\ldots,t_n \geq 0$, the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [0,s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

is equal to the image measure of $\mathbb{P}$ under the $\mathbb{R}$-valued map

$\hspace{2mm} \omega \mapsto \textrm{Leb}(t \in [\tau,\tau+s]: \left( X(t_1+t,\omega),\ldots,X(t_n+t,\omega) \right) \in A )$

(where $\textrm{Leb}$ denotes the Lebesgue measure)?

How about in the special case that $(E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$?

Now I've asked the above question in quite a general form, but what I ultimately want is: if $(X_t)_{t \geq 0}$ is a measurable stationary stochastic process in continuous time, then is the discrete-time stochastic process $\left( \int_n^{n+1} g(X_t) \, dt \right)_{n \geq 0}$ necessarily also stationary (where $g$ is a bounded measurable function)??