Editing Markov chain Monte Carlo (section)

=== Harris recurrent ===

;Definition (Harris recurrence)
A set <math>A</math> is '''Harris recurrent''' if <math>P_x(\eta_A = \infty) = 1</math> for all <math>x \in A</math>, where <math>\eta_A = \sum_{n=1}^\infty \mathbb{I}_A(X_n)</math> is the number of visits of the chain <math>(X_n)</math> to the set <math>A</math>.

The chain <math>(X_n)</math> is said to be '''Harris recurrent''' if there exists a measure <math>\psi</math> such that the chain is <math>\psi</math>-irreducible and every measurable set <math>A</math> with <math>\psi(A) > 0</math> is Harris recurrent.

A useful criterion for verifying Harris recurrence is the following:

;Proposition
If for every <math>A \in \mathcal{B}(\mathcal{X})</math>, we have <math>P_x(\tau_A < \infty) = 1</math> for every <math>x \in A</math>, then <math>P_x(\eta_A = \infty) = 1</math> for all <math>x \in \mathcal{X}</math>, and the chain <math>(X_n)</math> is Harris recurrent. 

This definition is only needed when the state space <math>\mathcal{X}</math> is uncountable. In the countable case, recurrence corresponds to <math>\mathbb{E}_x[\eta_x] = \infty</math>, which is equivalent to <math>P_x(\tau_x < \infty) = 1</math> for all <math>x \in \mathcal{X}</math>.

;Definition (Invariant measure)
A <math>\sigma</math>-finite measure <math>\pi</math> is said to be '''invariant''' for the transition kernel <math>K(\cdot, \cdot)</math> (and the associated chain) if:

:<math>\pi(B) = \int_{\mathcal{X}} K(x, B) \, \pi(dx), \qquad \forall B \in \mathcal{B}(\mathcal{X}).</math>
When there exists an ''invariant probability measure'' for a '''&psi;-irreducible''' (hence recurrent) chain, the chain is said to be '''positive recurrent'''.  
Recurrent chains that do not allow for a finite invariant measure are called '''null recurrent'''.

In applications of Markov Chain Monte Carlo (MCMC), a very useful criterion for Harris recurrence involves the use of bounded harmonic functions.

;Definition (Harmonic function)
A measurable function <math>h</math> is said to be '''harmonic''' for the chain <math>(X_n)</math> if:

:<math>\mathbb{E}[h(X_{n+1}) \mid x_n] = h(x_n)</math>

These functions are ''invariant'' under the transition kernel in the functional sense, and they help characterize Harris recurrence.

;Proposition
''For a positive Markov chain, if the only bounded harmonic functions are the constant functions, then the chain is Harris recurrent.''