Geometric ergodicity

A Markov kernel is geometrically ergodic when its iterates converge to stationarity in total variation at a geometric rate, with a finite state-dependent prefactor.

For an aperiodic positive Harris recurrent Markov chain with geometric ergodicity, an observable with stationary moment $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$ satisfies a central limit theorem. Its asymptotic variance is ${\gamma }_h(0)+2{\sum }_{k\ge 1}{\gamma }_h(k)$, with stationary covariances. Ergodicity alone does not guarantee this theorem.

Uniform geometric ergodicity strengthens geometric ergodicity by bounding the prefactor uniformly over the starting state. If $P(x,\cdot )\ge \epsilon \mu (\cdot )$ globally and $\mu$ is invariant, thenWrite $P=\epsilon \Pi +(1-\epsilon )R$, where $\Pi$ draws independently from $\mu$, to obtain the bound.

Applying a Markov kernel to two initial laws cannot increase their total variation distance, even for the distribution of an entire subsequent finite trajectory. If a bounded summand has absolute value at most $B$, the variance of a length-$N$ path sum differs between these laws by at most $6B^2{N}^2$ times their total variation distance. Thus a geometrically long enough burn-in makes this difference negligible relative to $N$.

For a reversible Markov chain with invariant law $\mu$, its Markov kernel acts as a self-adjoint operator $P$ on $L^2(\mu )$. An absolute spectral gap means that $\Vert P\Vert$ on the mean-zero functions is at most $r<1$. Geometric ergodicity implies this gap for a reversible chain. Indeed, for bounded mean-zero $g$ supported where the geometric prefactor is bounded, the total-variation estimate gives $⟨g,P^{2n}g⟩\le C_g{r}^{2n}$. The spectral theorem for normal operators puts its spectral measure in $[-r,r]$. Such test functions are dense in the mean-zero subspace.

If $\Vert P{\Vert }_{L_0^2(\mu )}\le r<1$, a stationary reversible Markov chain satisfiesConsequently the variance of a sum of $N$ consecutive values is at most $N(1+r)/(1-r)$ times the stationary variance of one value. This bounds the Markov chain Monte Carlo asymptotic variance uniformly over square-integrable functions.

For a stationary ergodic Markov chain with invariant distribution $\pi$ and an integrable function $h$, the sample mean $n^{-1}{\sum }_{t=1}^{n}h(X_t)$ converges almost surely to $\int hd\pi$. Geometric ergodicity supplies standard sufficient conditions for this conclusion and stronger limit theorems.

A minorization condition gives a common component of transition laws: $K^m(x,\cdot )\ge \alpha \nu (\cdot )$ for every $x$ in a specified set, some $m\ge 1$, $\alpha >0$, and a probability measure $\nu$.

A Doeblin condition is a minorization of the whole state space. It implies uniform geometric convergence in total variation distance.

A set $A$ is $\alpha$-small when some iterate satisfies $K^m(x,\cdot )\ge \alpha \nu (\cdot )$ for every $x\in A$ and some probability measure $\nu$.

A geometric drift $KV\le \lambda V+b{1}_A$ with $\lambda <1$, together with a small-set minorisation on $A$ and irreducibility and aperiodicity, implies geometric ergodicity.

A geometric drift condition requires a measurable $V\ge 1$, a small set $C$, constants $\lambda <1$ and $b<\infty$, andTogether with irreducibility and aperiodicity, it implies geometric ergodicity. A global Doeblin condition allows $C$ to be the entire state space and $V=1$.

Suppose a Markov kernel satisfies $PV(x)\le M<\infty$ for a measurable $V\ge 1$. For $0<\rho <1$ and $R>M/\rho$, the sublevel set $C=\{V\le R\}$ gives $PV\le \rho V+M{1}_C$. If $C$ is a small set, this is a geometric drift condition. The hypotheses of an irreducible Markov chain and an aperiodic Markov chain are also required for the usual geometric ergodicity theorem.