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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.

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  • $\begingroup$ Hi Richard: I like the first one, $\{x_t\}$ , and I think that is what is commonly used in books. $\endgroup$
    – mlofton
    Commented Nov 4, 2021 at 15:32
  • $\begingroup$ @mlofton, thanks. I have forgotten what is used in the textbooks, especially since the distinction is not always made and the readers are then just supposed to figure this out from the context. Good to know I would not be inventing something that has never been used before. $\endgroup$
    – Richard Hardy
    Commented Nov 4, 2021 at 15:35
  • $\begingroup$ no problem. glad to help a little. $\endgroup$
    – mlofton
    Commented Nov 5, 2021 at 22:49
  • $\begingroup$ @RichardHardy: It seems to me this question is ready to be accepted and closed; if not, feel free to ask about the outstanding part. $\endgroup$
    – ColorStatistics
    Commented Nov 12, 2021 at 20:01
  • $\begingroup$ @ColorStatistics, 43 views is not that high, so I would usually wait a bit more. But I like your answer, so I am going to accept it right away. Thanks for some good input! $\endgroup$
    – Richard Hardy
    Commented Nov 12, 2021 at 20:39

1 Answer 1

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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.

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  • $\begingroup$ Thanks! Looks good. $\endgroup$
    – Richard Hardy
    Commented Nov 4, 2021 at 16:36
  • $\begingroup$ This is an important question. Thank you for asking it. I added a further distinction between a realization of the process and the observed time series/the data at hand. $\endgroup$
    – ColorStatistics
    Commented Nov 5, 2021 at 15:13
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    $\begingroup$ I would be tempted to use $(x_1,x_2,\dots,x_n)$ instead of $\{x_1,x_2,\dots,x_n\}$ because we have an ordered sequence rather than an unordered set. Is $(\cdot)$ less common than $\{\cdot\}$ in such instances? $\endgroup$
    – Richard Hardy
    Commented Nov 5, 2021 at 15:24
  • $\begingroup$ I think that make intuitive sense; I gave you Brockwell & Davis' notation style $\endgroup$
    – ColorStatistics
    Commented Nov 5, 2021 at 15:31
  • $\begingroup$ The distinction you make is a very good one. Hamilton does exactly as you suggested (see page 25 for example). I will edit the answer. $\endgroup$
    – ColorStatistics
    Commented Nov 5, 2021 at 15:37

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