[Paper Review] Geometric ergodicity for families of homogeneous Markov chains

Leonid Galtchouk, Serguei Pergamenchtchikov|arXiv (Cornell University)|Feb 11, 2010

Markov Chains and Monte Carlo Methods9 references3 citations

TL;DR

This paper establishes computable, non-asymptotic bounds for the geometric convergence rate of parametric families of homogeneous Markov chains using Lyapunov functions and coupling techniques. It proves simultaneous geometric ergodicity across a class of diffusion processes, enabling nonparametric drift estimation with explicit exponential error bounds.

ABSTRACT

In this paper we find nonasymptotic exponential upper bounds for the deviation in the ergodic theorem for families of homogeneous Markov processes. We find some sufficient conditions for geometric ergodicity uniformly over a parametric family. We apply this property to the nonasymptotic nonparametric estimation problem for ergodic diffusion processes.

Motivation & Objective

• To derive non-asymptotic, computable upper bounds for the geometric convergence rate of homogeneous Markov chains across a parametric family of transition kernels.
• To establish simultaneous geometric ergodicity for a class of Markov chains indexed by a parameter θ ∈ Θ, ensuring uniform convergence rates over the parameter space.
• To apply the results to nonparametric estimation of the drift coefficient in ergodic diffusion processes, particularly in minimax risk analysis.
• To provide explicit expressions for the convergence rate parameters R > 0 and κ > 0 in the geometric ergodicity condition, enabling practical implementation in MCMC and statistical inference.

Proposed method

• Utilizes Lyapunov functions to derive drift conditions that ensure geometric ergodicity for Markov chains with parametrized transition probabilities.
• Applies coupling renewal processes and renewal theory to bound the convergence rate of the Markov chain to its invariant measure.
• Employs moment estimates and Gronwall-type inequalities to derive uniform bounds on the second moments of the diffusion process over time.
• Uses the Chebyshev inequality on stochastic integrals to bound the probability of large deviations in the process, leading to uniform tail bounds over the parameter class.
• Derives explicit upper bounds on the probability that the process exceeds a given level K in one step, uniformly over initial states |x| ≤ K and parameters θ ∈ Θ.
• Establishes a recursive moment bound via induction on the m-th moment, leading to uniform integrability and moment control for the diffusion process.

Experimental results

Research questions

• RQ1 Can we derive non-asymptotic, computable upper bounds for the geometric convergence rate of a parametric family of homogeneous Markov chains?
• RQ2 Under what conditions is geometric ergodicity simultaneously satisfied across all parameter values in a given class Θ?
• RQ3 How can we bound the convergence rate parameters R > 0 and κ > 0 explicitly for a class of ergodic diffusion processes?
• RQ4 Can these bounds be used to analyze the minimax risk in nonparametric drift estimation for ergodic diffusions?
• RQ5 What uniform moment and tail bounds can be established for the diffusion process over a compact set and across parameter families?

Key findings

• The paper establishes a uniform upper bound on the probability that the diffusion process exceeds level K in one step: $ ext{sup}_{|x| eq K} ext{sup}_{ heta eq heta} ext{P}^{ heta}_x(|y_1| eq K) eq rac{4 heta^2_{ ext{max}}(K^2 + heta_2)}{eta(1 - e^{-eta})(K^2 - heta_1)^2} $, where $ heta_1 = ext{M}_1, heta_2 = ext{M}_2 $.
• A recursive moment bound is derived via induction: $ ext{E}^{ heta}_x[|y_t|^{2m}] eq x^{2m} + m(2m-1) ext{M}_* ext{int}_0^t e^{-meta(t-s)} z_s(m-1) ds $, ensuring uniform moment control.
• The second moment of the process is uniformly bounded: $ ext{sup}_{t eq 0} ext{E}^{ heta}_x[y_t^2] eq x^2 + ext{M}_2 $, with $ ext{M}_2 $ defined in (3.9).
• The convergence rate is quantified via the geometric ergodicity condition: $ ext{sup}_x ext{sup}_{0 eq g eq V} rac{1}{V(x)} | ext{E}^{ heta}_x[g( heta_n)] - heta(g)| eq R e^{- heta n} $, with explicit R and κ derived.
• The method ensures simultaneous geometric ergodicity over the entire parameter class Θ, enabling application to minimax risk analysis in nonparametric estimation.
• The results are applied to stochastic differential equations of the form $ dy_t = S(y_t)dt + heta(y_t)dW_t $, with explicit bounds on the convergence of empirical averages to the invariant measure.