# Characterizing 'very homogeneous' finitely valued stochastic processes

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 $$k\in \mathbb{N}$$, any two increasing sequences of indices $$i_0 and $$j_0 < j_1<\cdots, 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.)

• Cornfeld and Sinai springer.com/gp/book/9781461569299 should have it. This is really immediate from the definition of ergodicity. In particular, if an ergodic measure $\mu$ is the average of two stationary measures $\mu_1$ and $\mu_2$, then they must be absolutely continuous to $\mu$. The Radon Nikodym derivatives $\frac{d\mu_i}{d\mu}$ are invariant functions, hence a.s. constant. Sep 22, 2020 at 16:30