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For example, when we have a univariate time series, we are assuming that we monitor only one random variable $X_{t}$. But when we get down to the actual deduction and calculation we will replace $X_{t}$ with a sequence of random variables, i.e. random variables, where $t = 1,2,3......n$, and $X_{1},X_{2},X_{3}...,X_{n}$ each represents a random variable at time $t$. Therefore when we approach the $PMF/PDF$, we would have to calculate the joint version.

Stochastic process is a sequence of random variables. I could imagine that we have a sequence of random variables, each of which represents a difference factor for the analysis. e.g $x_{1}$ is room size, $x_{2}$ is property location, $x_{3}$ is which floor the apartment is on. But why do we choose to model time series as stochastic process, where the sequence of random variables all represent just one random variable?

my question is:

  1. Can we assume that $X_{t}$ is indeed a random variable and $x_{1},x_{2},x_{3}...,x_{n}$ only represents the values $X_{t}$ take at different times?
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  • $\begingroup$ I'm not the best person to answer such questions, but remember that time series are considered stochastic processes, and generally speaking, stochastic processes (or subsets thereof) are not considered random vectors. Rather, recall that time series are considered a sequence of random variables, rather than a vector of random variables. See also math.stackexchange.com/questions/569951/…. $\endgroup$
    – Clarinetist
    Jan 30, 2018 at 12:51
  • $\begingroup$ thanks very much @Clarinetist I think I shouldn't be using the term random vector; I'll edit the question now. $\endgroup$
    – stucash
    Jan 30, 2018 at 13:02
  • $\begingroup$ @Clarinetist i think you are a perfect fit on this question though :) or actually there's a bit of granularity in some specific field? $\endgroup$
    – stucash
    Jan 30, 2018 at 13:21

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