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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 $i_0<i_1 < \cdots i_{k-1}$ 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.)

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This only answers the first part of your question: The strongly homogenous processes are , in particular, stationary. Extreme points of stationary processes are exactly the stationary ergodic processes. The i.i.d. measures are ergodic, hence they are extreme among stationary processes, which implies they are extreme among strongly homogenous processes.

Strongly homogenous processes that are also strong mixing are necessarily i.i.d., but this still requires dealing with ergodic strongly homogenous processes that are not strong mixing.

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  • $\begingroup$ Thank you very much. Do you have any good references for some of these statements (such as ergodic being extreme among stationary processes)? $\endgroup$ Sep 21, 2020 at 20:46
  • $\begingroup$ 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. $\endgroup$ Sep 22, 2020 at 16:30

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