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I want to know if a strong stationary process is equivalent to a process with identically distributed RVs.

In another words : if A is the set of s.s.s processes and B is the set of processes with identically distributed RVs, is it true that A = B ?

Afaf
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  • What do you mean by an "identically distributed process"? That $X_1,X_2,\dots$ all have the same distribution? – kccu Jun 21 '19 at 17:45
  • Yes they have the same distribution – Afaf Jun 21 '19 at 22:48
  • If I am not mistaken i.i.d processes are a subset of s.s.s processes set. I wanted to know what is the processes other than i.i.d ones that forms the set of s.s.s processes – Afaf Jun 21 '19 at 22:49
  • I was thinking that since the s.s.s processes are characterised by random variables that have the same CDFs, then these random variables are identically distributed. Is this correct? – Afaf Jun 21 '19 at 22:59

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No, they are not the same. In a strong stationary process $(X_1,X_2,\dots)$ it is the case that all of the $X_n$'s have the same distribution. However, the converse does not hold.

To see why, take $X_2,X_3,\dots$ to be IID (whatever distribution you like, but not constant), and $X_1=X_2$. Then all of these terms are identically distributed, but this is not a strong stationary sequences because, for instance, $(X_1,X_2)$ is not equal in distribution to $(X_2,X_3)$.

kccu
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  • Thanks this is clear. Can you give me an example of a strong stationary process that is not an IID process ? – Afaf Jun 23 '19 at 10:12
  • I think it should work to take $X_1 \sim N(0,1)$, and define $X_n = X_1$ if $n$ is odd, and $X_n=-X_1$ if $n$ is even. This should be a strong stationary process, but it's not IID. – kccu Jun 24 '19 at 01:31
  • Do you mean an i.i.d. random process is not a strict stationary random process? And what do you mean by "(X1, X2) is not equal in distribution to (X2, X3)"? Why aren't they equal even if all Xs have identical probability density/mass functions? – starriet 주녕차 Oct 26 '22 at 15:10
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    @starriet No, an IID process is a strong stationary process. For your second question, $\mathbb{P}(X_1=X_2)=1$ while $\mathbb{P}(X_2=X_3)<1$ since $X_2$ and $X_3$ are independent and not constant. – kccu Nov 14 '22 at 16:16
  • So, any "strong(strict) stationary process" is an "identically distributed" process, but not vice versa, right? (The first paragraph of the answer is a little bit confusing) Thanks! :) – starriet 주녕차 Nov 16 '22 at 12:47
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    @starriet Yes, every strong stationary process is a sequence of identically distributed RVs, but not every sequence of identically distributed RVs is a stationary process. If you add the condition that the sequence is independent (so an IID process), then it is a stationary process. But not every stationary process is an IID process. – kccu Dec 06 '22 at 16:00