Notation for an entire time series vs. one time instance In time series analysis, $x_t$ may denote an entire time series (a time-ordered set of random variables or their realizations) or a single random variable or its realization that is specific to time $t$. What is some brief, non-clumsy notation that would allow to disambiguate between the two?
I encountered this when teaching, and I want to prevent confusion among students. So far I have considered $x_t$ for a specific time period and $\{x_t\}$ for the entire series. Would that make sense?
On the other hand, something like $\{x_t\}_{t=1}^T$ seems both too specific regarding the permitted time indices (why exactly $t=1$ but not $t=0$ or $t=-\infty$, and similarly for $T$ vs. $\infty$ vs. ...) and too clumsy to me.

I would go with the notation in Hamilton:

*

*{$X_t$} for the infinite sequence of random variables/the process

*{$x_t$} for a realization of the process (also infinite)/these are values/scalars

*($x_1,x_2,...,x_n$) for the finite observed time series/our data (a subset of a realization)

*$x_t$ for the particular realized value at time t

Uppercase letters used for random variables; lowercase letters used for realized values of random variables; {} used for infinite sequences; () used for finite sequences.
