Harris chain

In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. Harris chains are regenerative processes and are named after Theodore Harris.

Definition

A Markov chain {X_n} on state space Ω with stochastic kernel K is a Harris chain if there exist A, B ⊆ Ω, ϵ > 0, and probability measure ρ with ρ(B) = 1 such that

1. If τ_A := inf {n ≥ 0 : X_n ∈ A}, then P(τ_A < ∞|X₀ = x) > 0 for all x ∈ Ω.
2. If x ∈ A and C ⊆ B then K(x, C) ≥ ερ(C).

In essence, this technical definition can be rephrased as follows: given two points x₁ and x₂ in A, then there is at least an ϵ chance that they can be moved together to the same point at the next time step.

Another way to say it is that suppose that x and y are in A. Then at the next time step I first flip a Bernoulli with parameter ϵ. If it comes up one, I move the points to a point chosen using ρ. If it comes up zero, the points move independently, with x moving according to P(X_n+1 ∈ C|X_n = x) = K(x, C) − ερ(C) and y moving according to P(Y_n+1 ∈ C|Y_n = y) = K(y, C) − ερ(C).

Examples

Example 1: Countable state space

Given a countable set S and a pair (A′, B′ ) satisfying (1) and (2) in the above definition, we can without loss of generality take B′ to be a single point b. Upon setting A = {b}, pick c such that K(b, c) > 0 and set B = {c}. Then, (1) and (2) hold with A and B as singletons.

Example 2: Chains with continuous densities

Let {X_n}, X_n ∈ R^d be a Markov chain with a kernel that is absolutely continuous with respect to Lebesgue measure:

K(x, dy) = K(x, y) dy

such that K(x, y) is a continuous function.

Pick (x₀, y₀) such that K(x₀, y₀ ) > 0, and let A and B be open sets containing x₀ and y₀ respectively that are sufficiently small so that K(x, y) ≥ ε > 0 on A × B. Letting ρ(C) = |B ∩ C|/|B| where |B| is the Lebesgue measure of B, we have that (2) in the above definition holds. If (1) holds, then {X_n} is a Harris chain.

Reducibility and periodicity

In the following, R := inf {n ≥ 1 : X_n ∈ A}; i.e. R is the first time after time 0 that the process enters region A.

Definition: If for all L(X₀), P(R < ∞|X₀ ∈ A) = 1, then the Harris chain is called recurrent.

Definition: A recurrent Harris chain X_n is aperiodic if ∃N, such that ∀n ≥ N, ∀L(X₀), P(X_n ∈ A|X₀ ∈ A) > 0.

Theorem: Let X_n be an aperiodic recurrent Harris chain with stationary distribution π. If P(R < ∞|X₀ = x) =1 then as n → ∞, dist_TV (L(X_n|X₀ = x), π) → 0.