Fix a positive integer $n$. Let $X = {X_i}_{i in mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a ${0,dots,n-1}$-valued random variable. Suppose that the joint probability distributions of any finite sequence of $X_i$‘s only depends on the order of their indices, or to be more precise suppose that $X$ satisfies the following:

- For any $kin mathbb{N}$, any two increasing sequences of indices $i_0<i_1 < cdots i_{k-1}$ and $j_0 < j_1<cdots<j_{k-1}$, and any function $f : {0,dots,k-1} to {0,dots,n-1}$, $$mathbb{P}(X_{i_0} = f(0) wedge X_{i_1} = f(1) wedge cdots wedge X_{i_{k-1}} = f(k-1)) = mathbb{P}(X_{j_0} = f(0) wedge X_{j_1} = f(1) wedge cdots wedge X_{j_{k-1}} = f(k-1)).$$

Call such a stochastic process ‘strongly homogeneous.’ I’m trying to understand what the set of strongly homogeneous stochastic processes looks like. This is my approach so far:

The set of ${0,dots,n-1}$-valued discrete time stochastic processes can be understood as the set of Borel probability measures on the (compact) space $A = {0,dots,n-1}^{mathbb{N}}$. This is a subset of Banach space of (regular Borel) signed measures on $A$. Let $S$ be the set of such measures corresponding to a strongly homogeneous stochastic process. It’s not hard to check that $S$ is a convex, weak* closed set, and therefore that the Krein-Milman theorem applies to it. This gives us that every element of $S$ is in the weak* closure of the convex hull of the set of extreme points of $S$ (where a point is extreme if it is not the convex combination of any other elements of $S$). This leads to my precise question.

**Question:** What are the extreme points of $S$?

Note that the extreme points of the set of all probability measures on $A$ is precisely the set of Dirac measures on $A$, but a similar statement here is not sufficient. For instance, if $n=2$, then the only strongly homogeneous Dirac measures are those concentrated on the constant $0$ sequence or the constant $1$ sequence, but convex combinations of these do not have the measure corresponding to a sequence of i.i.d. fair coin flips in their weak* closure.

My suspicion is that the measures corresponding to i.i.d. sequences are the extreme points, but I haven’t proved either that they are all extreme points or that all extreme points are of that form. (Proving that they are all extreme points should be easy, however.)