Stationarity and ergodicity of a process conditional on a finite trajectory

Question

Let us say we are interested in a single time series, e.g. the daily closing share price of Tesla. We can model it as a realization of a stochastic process $\{Y_t(\omega)\}$. It corresponds to a particular subset $\Omega_0$ of the set of all possible outcomes $\Omega$ of the "statistical experiment" underlying the stochastic process. The subset $\Omega_0$ is defined by the history of the time series observed until now, $(y_1,\dots,y_t)$; it contains all $\omega$s that produce $(y_1,\dots,y_t)$.

Let us also say we do not believe there are any other realizations of the same stochastic process, i.e. no other share price has been generated from the same process that has generated Tesla's share price. So effectively we are only interested in $\Omega_0$ and not in $\Omega\backslash\Omega_0$. Or even if there were other realizations, let us say we only care about Tesla's share price; e.g. we want to forecast its distribution a few days ahead. We do not want any inference on any other realizations from the same data generating process. Let us define the process based on $\Omega_0$ by $\{X_t(\omega)\}$. This is the process that necessarily generates $(y_1,\dots,y_t)$ as its first $t$ observations. What can be said about its stationarity and ergodicity?

_{This is a follow up on my recent question "Stochastic modelling, distribution and ergodicity of a particular time series with a given finite history".}

Answer 1

Your answer here is mostly technically correct (though it is of course possible for some time-series that you might get observed values $y_1= \cdots = y_t = \mathbb{E}(X_k)$ in some other applications), but it really misses the point, because you are essentially looking at the properties of a process that tacks together a starting part that is fixed and a later part that is stochastic. That is not really what we are interested in when we apply probabilistic concepts for conditional inferences in time-series.

Presumably, what you really want to do here is to consider whether or not the remaining part of the series $\{ Y_{t+1},Y_{t+2},Y_{t+3},... \}$ is still stationary and ergodic once you condition on observing $(y_1,...y_t)$. For anything but trivial series, the remaining process will not be conditionally stationary, though it might be conditionally asymptotically stationary and this may be enough to get you the essence of ergodicity. One could frame various useful questions about the conditional properties of this remaining series, since this is the stochastic part we are dealing with when we condition on the observations.

Answer 2

$\{X_t(\omega)\}$ is clearly nonstationary, because $\mathbb{E}(X_1)=\dots=\mathbb{E}(X_t)$ does not hold. This is because $\mathbb{E}(X_1)=y_1,\dots,\mathbb{E}(X_t)=\dots=y_t$, and we know that $y_1=\dots=y_t$ does not hold, since the share price of Tesla has not been constant the whole time from day 1 to day $t$ (today or yesterday, depending on whether the stock exchange has already closed today or not). (Note that $\mathbb{E}(X_\tau)=y_\tau$ for every $\tau=1,\dots,t$ because for every $\omega\in\Omega_0$ we have $y_\tau(\omega)=y_\tau$ by the definition of $\Omega_0$.)

Since ergodicity is only defined for strictly stationary processes (see e.g. this answer by Michael), $\{X_t(\omega)\}$ is not ergodic.

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_{The fact that $\{X_t(\omega)\}$ is nonstationary unless $y_1=\dots=y_t$ seems to make modelling the conditional process $\{X_t(\omega)\}$ indefinitely more complicated than modelling the original process $\{Y_t(\omega)\}$. And while modelling $\{Y_t(\omega)\}$ may be much easier, this seems irrelevant for those who only care about the particular time series (Tesla's share price in OPs example) but not about the hypothetical other realizations (trajectories) from the same data generating process $\{Y_t(\omega)\}$. I am still trying to figure out what that implies...}