I am revising lecture slides and came accross this definition but it lacks any further explanation:

Def:A stochastic process ${y_t}_t>=1$ is strictly stationary if for all $t_1, ..., t_n$ and all $h>=0$; $P(y_{t1},...,y_{tn}) = P(y_{t1+h}, \ldots, y_{tn+h})$

What is h here?

  • 1
    $\begingroup$ a spacing between data points, so $y_{t},y_{t+3}$ is an example where $h=3$ $\endgroup$ Commented Oct 20, 2022 at 15:21
  • 2
    $\begingroup$ Either you are not correctly transcribing what the lecture slides say, or there are typos in the slides. Your equation should read $$P(y_{t_1}, y_{t_2}, \ldots, y_{t_n}) = Py_{t_1+h}, y_{t_2+h}, \ldots, y_{t_n+h})$$ $\endgroup$ Commented Oct 20, 2022 at 15:49
  • $\begingroup$ This is explained and used in my post at stats.stackexchange.com/a/566582/919. $\endgroup$
    – whuber
    Commented Oct 20, 2022 at 19:13

1 Answer 1


$h$ is a number greater than zero.

But conceptually, it's the amount of time that you shift your time series by. The idea this definition is trying to convey is that if the process is stationary, the joint distribution doesn't change if you shift the set of times that you have all by the same amount. So if you have times $\{1, 2, 3\}$, you'd get the same joint distribution as if you had $\{2, 3, 4\}$ or $\{3, 4, 5\}$ or $\{100001, 100002, 100003\}$, etc.


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