Harris chain

Let {X_n} be a Markov chain on a general state space Ω with stochastic kernel K. The kernel represents a generalized one-step transition probability law, so that P[X_n+1 ∈ C | X_n = x] = K(x, C) for all states x in Ω and all measurable sets C ⊆ Ω. The chain {X_n} is a Harris chain[2] if there exists A ⊆ Ω, ϵ > 0, and probability measure ρ with ρ(Ω) = 1 such that

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

The first part of the definition ensures that the chain returns to some state within A with probability 1, regardless of where it starts. It follows that it visits state A infinitely often (with probability 1). The second part implies that once the Markov chain is in state A, its next-state can be generated with the help of an independent Bernoulli coin flip. To see this, first note that the parameter ε must be between 0 and 1 (this can be shown by applying the second part of the definition to the set C = Ω). Now let x be a point in A and suppose X_n = x. To choose the next-state X_n+1, independently flip a biased coin with success probability ϵ. If the coin flip is successful, choose a next-state X_n+1 ∈ Ω according to the probability measure ρ. Else (and if ϵ < 1), choose a next-state X_n+1 according to the measure P[X_n+1 ∈ C | X_n = x] = (K(x, C) − ερ(C))/(1 − ε) (defined for all measurable subsets C ⊆ Ω).

Two random processes {X_n} and {Y_n} that have the same probability law and are Harris chains according to the above definition can be coupled as follows: Suppose that X_n=x and Y_n = y, where x and y are points in A. Using the same coin flip to decide the next-state of both processes, it follows that the next states are the same with probability at least ε.

In the following ${\displaystyle R:=\inf\{n\geq 1:X_{n}\in A\}}$; i.e. ${\displaystyle R}$ is the first time after time 0 that the process enters region ${\displaystyle A}$. Let ${\displaystyle \mu }$ denote the initial distribution of the Markov chain, i.e. ${\displaystyle X_{0}\sim \mu }$.

Definition: If for all ${\displaystyle \mu }$, ${\displaystyle P[R<\infty |X_{0}\in A]=1}$, then the Harris chain is called recurrent.

Definition: A recurrent Harris chain ${\displaystyle X_{n}}$ is aperiodic if ${\displaystyle \exists N}$, such that ${\displaystyle \forall n\geq N}$, ${\displaystyle \forall \mu ,P[X_{n}\in A|X_{0}\in A]>0}$

Theorem: Let ${\displaystyle X_{n}}$ be an aperiodic recurrent Harris chain with stationary distribution ${\displaystyle \pi }$. If ${\displaystyle P[R<\infty |X_{0}\in A]=1}$ then as ${\displaystyle n\rightarrow \infty }$, ${\displaystyle ||{\mathcal {L}}(X_{n}|X_{0})-\pi ||_{TV}\rightarrow 0}$ where ${\displaystyle ||\cdot ||_{TV}}$ denotes the total variation for signed measures defined on the same measurable space.