User:Prakash.balachandran

Definition[edit]

A Markov chain {X_n} on state space Ω with kernel K is a Harris chain^[1] if there exists 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 ).

Reducibility and Periodicity[edit]

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 call a Harris chain is 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) then as n → ∞, dist_TV (L(X_n |X₀ = x), π) → 0.

References[edit]