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.
[1] Harris chains are regenerative processes and are named after Theodore Harris.
The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces.
be a Markov chain on a general state space
The kernel represents a generalized one-step transition probability law, so that
is a Harris chain[2] if there exists
such that The first part of the definition ensures that the chain returns to some state within
The second part implies that once the Markov chain is in state
, its next-state can be generated with the help of an independent Bernoulli coin flip.
must be between 0 and 1 (this can be shown by applying the second part of the definition to the set
, independently flip a biased coin with success probability
If the coin flip is successful, choose the next state
(defined for all measurable subsets
that have the same probability law and are Harris chains according to the above definition can be coupled as follows: Suppose that
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
Let Ω be a countable state space.
The kernel K is defined by the one-step conditional transition probabilities P[Xn+1 = y | Xn = x] for x,y ∈ Ω.
The measure ρ is a probability mass function on the states, so that ρ(x) ≥ 0 for all x ∈ Ω, and the sum of the ρ(x) probabilities is equal to one.
Suppose the above definition is satisfied for a given set A ⊆ Ω and a given parameter ε > 0.
Then P[Xn+1 = c | Xn = x] ≥ ερ(c) for all x ∈ A and all c ∈ Ω.
Let {Xn}, Xn ∈ Rd be a Markov chain with a kernel that is absolutely continuous with respect to Lebesgue measure: such that K(x, y) is a continuous function.
Pick (x0, y0) such that K(x0, y0 ) > 0, and let A and Ω be open sets containing x0 and y0 respectively that are sufficiently small so that K(x, y) ≥ ε > 0 on A × Ω.
Letting ρ(C) = |Ω ∩ C|/|Ω| where |Ω| is the Lebesgue measure of Ω, we have that (2) in the above definition holds.
If (1) holds, then {Xn} is a Harris chain.
is the first time after time 0 that the process enters region
denote the initial distribution of the Markov chain, i.e.
, then the Harris chain is called recurrent.
Definition: A recurrent Harris chain
be an aperiodic recurrent Harris chain with stationary distribution
denotes the total variation for signed measures defined on the same measurable space.