Terminology for stochastic processes: "marginalizing" over time? Suppose I have a stochastic process $X(t)$ which outputs symbols in some state space $S$.
Given each possible symbol $s$ in $S$, I would like to determine some kind of "marginal" probability representing how frequently each symbol occurs, in a sort of average of the entire behavior of the stochastic process on the entire time coordinate.
However, the time coordinate is not actually a random variable, so we are not technically marginalizing anything, so the question I am asking is really what to call this thing. Basically, we are asking how likely the stochastic process is to output each symbol "in general."
To keep it simple, let's assume that our time index $t \in \Bbb N$. Then in particular, the quantity I would like to look at is this:
$$
P(X = s) := \lim_{N \to \infty} \frac{\text{expected # of occurrences of s from t=0..N}}{N}
$$
where the $P(X = s)$ is the notation I am using for this quantity.
In other words, we compute the expected number of occurrences of our stochastic process emitting $s$ within the first $N$ time steps. Then we let $N$ go to infinity. This is the probability of the stochastic process outputting the symbol $s$ "in general."
(For now let's suppose that $S$ is a finite or at most countable set, although this question could also be asked for a real-valued stochastic process, where we'd look at something like a probability density instead.)
Basic questions:

*

*Most importantly: does this quantity have a name? It's almost like we are "marginalizing" on the time coordinate, except the time isn't a random variable.

*For what stochastic processes is this quantity guaranteed to exist?

*Is there some formally rigorous way to do this same thing with e.g. a time coordinate in $\Bbb R$?

 A: There is some confusion. A stochastic process $(X_t)_{t\in\mathbb{T}}$ (regardless if $\mathbb T$ is $\mathbb N$ or $\mathbb R_+$) can be seen as a map
$$
X:\mathbb T\times\Omega\to S\,.
$$
The probability measure $P$ is defined on $(\Omega,{\cal A})$ and not on $\mathbb T\,.$

*

*The expression $P(X=s)$ is quite ambiguous. For which $t\in\mathbb T$ are you looking at $X_t$ being $s$ ?


*It sounds like you are rather interested in (the limit of)
$$\tag{1}
\sum_{n=1}^N\frac{1_{\{X_n=s\}}}{N}
$$
which is a random variable on $(\Omega,{\cal A})$, or in (the limit of)
$$\tag{2}
E\Bigg[\sum_{n=1}^N\frac{1_{\{X_n=s\}}}{N}\Bigg]
$$
which is the expectation of (1).


*When the $X_n$ are i.i.d. then by he srong law of large numbers
$$\tag{3}
\lim_{N\to\infty}\sum_{n=1}^N\frac{1_{\{X_n=s\}}}{N}=E\Big[1_{\{X_1=s\}}\Big]=P(X_1=s)\,.
$$
The same holds for the limit of (2).


*When $\mathbb T=\mathbb R_+$ then we could consider
$$\tag{4}
\frac{1}{T}\int_0^T1_{\{X_t=s\}}\,dt\,
$$
instead of (2). This is however not very useful when $X$ is a Brownian motion.
Therefore, see Local time.


*(1) and (4) can be called the pathwise average time that $X$ equals $s\,.$
