# If a strictly stationary process is also independent, does this imply i.i.d.?

Suppose I have a time series process $$\{X_t\}$$ that is strictly stationary in the sense that the joint distribution of $$[X_{t_1},...,X_{t_k}]$$ and $$[X_{t_1+a},...,X_{t_k+a}]$$ are the same for any set of integers $$t_1,...,t_k$$ and any integer $$a$$. If in addition, I know that this process have mutually independent observations. Does this imply that $$\{X_t\}$$ is an i.i.d. process? My guess is that it is. Because strict stationarity implies that every term in this time series have the same distribution as $$X_1$$, and thus when independence also hold, it is an i.i.d. process. Does this look correct?

• "Because strict stationarity implies that every term in this time series have the same distribution as $X_1$" This is not what strict stationarity means. This contradicts the definition that you gave, which is about the joint distribution, and not the marginals. Apr 23 at 23:05
• @mhdadk Yeah, it's about joint distribution, but I meant it IMPLIES marginals are have identical distribution. Just take $t_1,...,t_k$ as 1, the definition says $X_1$ has the same distribution as $X_{1+a}$ for any $a$. What's wrong with this? Apr 23 at 23:16

Yes, that is correct. Strict stationarity implies a common marginal distribution for the variables in the series, which is the ID part in IID. If you combine this with an assumption of independence you then get IID.

• Thanks a lot! This is very helpful. Apr 24 at 4:02
• But isn't the assumption of "mutual independence" in the question weaker than independence? That distinction appears to be the crux of the matter.
– whuber
May 6 at 15:38
• @whuber: Isn't mutual independence the stronger one (i.e., independent conditional on any of the observations in the series)? I was under the impression that pairwise, triplewise, etc., were the weaker and mutual independence is full independence.
– Ben
Oct 3 at 5:02
• You could be right: I had understood "mutual" as meaning "pairwise," for otherwise why modify the word "independence" at all?
– whuber
Oct 3 at 13:10
• @whuber: For now I'll leave the answer unchanged, since I think that mutual independence is the strong form. Like you, I always take the unqualified term "independence" as meaning the strong (mutual) form, so don't need the qualifier, but perhaps OP wanted to specify to be sure.
– Ben
Oct 3 at 15:02