Notation around processes: Is Yt the process or a single observation?

Question

I have troubles interpreting this paragraph in an introduction to Stationarity:

A time series {Y_t} is said to be stationary if for every integer m, the set of variables Y_t1, Y_t2, ..., Y_tm depends only on the distance between the times t₁, t₂, ..., t_m, rather than on their actual values. So a stationary process {Y_t} tends to behave in a homogeneous manner as it moves through time. The means and variances of the Y_t are the same for all t, that is, E(Y_t) = μ and Var(Y_t) = σ² are constant, for all t.

In E(Y_t) and Var(Y_t), does Y_t refer to the process or, or to a single observation of it, or to something else ?

Then the paragraph then goes with

So we may express the autocovariance function as
Cov(Y_t,Y_t-s) = E[(Y_t - μ)(Y_t-1 - μ)]

Here, are Y_t and Y_t-s referring to single observations of the process, or to something else ?

Answer 1

The time series is $Y$, not $Y_t$ or any $Y_{t_1},Y_{t_2}$.
However, one usually defines a time series by the following notations :$(Y_t, t \in \mathbb{N})$, $(Y_t, t \in \mathbb{Z})$.
The reason why they do that is that they want to highlight the fact that a time series is a collection of random variables indexed by some set. In the previous examples, those random variables are indexed by $\mathbb{N}$ and $\mathbb{Z}$.
Also, by that manner, they want to be clear on how they will call out the value at specific instants. Because, there usually are many ways to call out point observations. For example, for a time series $Y$, its observation at the instant $t$ may also be denoted by $Y^t, Y(t),$ etc.
Thirdly, and perhaps, the most common reason is that it is just a symbol convention. Probabilists like to save the majuscule letters for real random variables, but not any other kind of random elements. Only writing $Y$ for a time series maybe lead to confusion at some points.
Finally, in some lazy writings (or just because authors want to get rid of some unimportant details), one may just write $(Y_t)$ for their time series. Nevertheless, what they mean by $(Y_t)$ is always $(Y_t, t \in T)$ for some set $T$.