Explicit bounds for spectral theory of geometrically ergodic Markov kernels and applications

Abstract

In this paper, we deal with a Markov chain on a measurable state space $(\mathbb{X},\mathcal{X})$ which has a transition kernel P admitting an aperiodic small-set S and satisfying the standard geometric-drift condition. Under these assumptions, there exists ${\mathit{\alpha }}_0\in (0,1]$ such that $P{V}^{{\mathit{\alpha }}_0}\le {\mathit{\delta }}^{{\mathit{\alpha }}_0}{V}^{{\mathit{\alpha }}_0}+\mathrm{\nu }(V^{{\mathit{\alpha }}_0})1_S$. Hence P is $V^{{\mathit{\alpha }}_0}$-geometrically ergodic and its “second eigenvalue” ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ provides the best rate of convergence. Setting $R:=P-\mathrm{\nu }(\cdot )1_S$ and $\mathrm{\Gamma }:=\{\mathit{\lambda }\in \mathbb{C},{\mathit{\delta }}^{{\mathit{\alpha }}_0}<|\mathit{\lambda }|<1\}$, ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ is shown to satisfy, either ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}=max\{|\mathit{\lambda }|:\mathit{\lambda }\in \mathrm{\Gamma },{\sum }_{k=1}^{+\mathrm{\infty }}{\mathit{\lambda }}^{-k}\mathrm{\nu }(R^{k-1}{1}_S)=1\}$ if this set is not empty, or ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}\le {\mathit{\delta }}^{{\mathit{\alpha }}_0}$. Actually the set is finite in the first case and is composed by the spectral values of P in Γ. The second case occurs when P has no spectral value in Γ. Moreover, a bound of the operator-norm of ${(zI-P)}^{-1}$ allows us to derive an explicit formula for the multiplicative constant in the rate of convergence, which can be evaluated provided that any information of the “second eigenvalue” is available. Such numerical computation is carried out for a classical family of reflected random walks. Moreover we obtain a simple and explicit bound of the operator-norm of ${(I-P+\mathrm{\pi }(\cdot )1_{\mathbb{X}})}^{-1}$ involved in the definition of the so-called fundamental solution to Poisson’s equation. This allows us to specify the location of the eigenvalues of P and, then, to obtain a general bound on ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$. The reversible case is also discussed. In particular, the bound of ${\mathit{\varrho }}_{{\mathit{\alpha }}_0}$ obtained for positive reversible Markov kernels is the expected one, and numerical illustrations are proposed for the Metropolis-Hastings algorithm and for the Gaussian autoregressive Markov chain. The bound for the operator-norm of ${(I-P+\mathrm{\pi }(\cdot )1_{\mathbb{X}})}^{-1}$ is derived from an estimate, only depending on ${\mathit{\delta }}^{{\mathit{\alpha }}_0}$, of the operator-norm of ${(I-R)}^{-1}$ which provides another way to get a solution to Poisson’s equation. This estimate is also shown to be of greatest interest to generalize the error bounds obtained for perturbed discrete and atomic Markov chains in Liu and Li (Adv. in Appl. Probab. 50 (2018) 645–669) to the case of general geometrically ergodic Markov chains. These error estimates are the simplest that can be expected in this context. All the estimates in this work are expressed in the standard $V^{{\mathit{\alpha }}_0}$-weighted operator norm.