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Wild regenerative block bootstrap for Harris recurrent Markov chains

Kyuseong Choi, Gabriella Ciolek

TL;DR

The paper develops non-asymptotic Gaussian approximation and a Gaussian multiplier wild regenerative block bootstrap for the supremum of empirical processes over possibly growing VC-type function classes indexed by Harris recurrent Markov chains. Central to the theory is Nummelin splitting, which decomposes the chain into independent or one-dependent blocks, enabling sharp finite-sample error bounds and practical bootstrap procedures. The authors establish a polynomial-rate uniform inference method that does not rely on the Smirnov–Bickel–Rosenblatt condition, and apply it to construct a uniform confidence band for the stationary density of a diffusion process from discrete observations. These results advance nonparametric inference for dependent data and diffusion-type models by providing tractable, finite-sample guarantees for high-dimensional, non-Donsker settings.

Abstract

We consider Gaussian and bootstrap approximations for the supremum of additive functionals of aperiodic Harris recurrent Markov chains. The supremum is taken over a function class that may depend on the sample size, which allows for non-Donsker settings; that is, the empirical process need not have a weak limit in the space of bounded functions. We first establish a non-asymptotic Gaussian approximation error, which holds at rates comparable to those for sums of high-dimensional independent or one-dependent vectors. Key to our derivation is the Nummelin splitting technique, which enables us to decompose the chain into either independent or one-dependent random blocks. Additionally, building upon the Nummelin splitting, we propose a Gaussian multiplier bootstrap for practical inference and establish its finite-sample guarantees in the strongly aperiodic case. Finally, we apply our bootstrap to construct a uniform confidence band for an invariant density within a certain class of diffusion processes.

Wild regenerative block bootstrap for Harris recurrent Markov chains

TL;DR

The paper develops non-asymptotic Gaussian approximation and a Gaussian multiplier wild regenerative block bootstrap for the supremum of empirical processes over possibly growing VC-type function classes indexed by Harris recurrent Markov chains. Central to the theory is Nummelin splitting, which decomposes the chain into independent or one-dependent blocks, enabling sharp finite-sample error bounds and practical bootstrap procedures. The authors establish a polynomial-rate uniform inference method that does not rely on the Smirnov–Bickel–Rosenblatt condition, and apply it to construct a uniform confidence band for the stationary density of a diffusion process from discrete observations. These results advance nonparametric inference for dependent data and diffusion-type models by providing tractable, finite-sample guarantees for high-dimensional, non-Donsker settings.

Abstract

We consider Gaussian and bootstrap approximations for the supremum of additive functionals of aperiodic Harris recurrent Markov chains. The supremum is taken over a function class that may depend on the sample size, which allows for non-Donsker settings; that is, the empirical process need not have a weak limit in the space of bounded functions. We first establish a non-asymptotic Gaussian approximation error, which holds at rates comparable to those for sums of high-dimensional independent or one-dependent vectors. Key to our derivation is the Nummelin splitting technique, which enables us to decompose the chain into either independent or one-dependent random blocks. Additionally, building upon the Nummelin splitting, we propose a Gaussian multiplier bootstrap for practical inference and establish its finite-sample guarantees in the strongly aperiodic case. Finally, we apply our bootstrap to construct a uniform confidence band for an invariant density within a certain class of diffusion processes.
Paper Structure (31 sections, 23 theorems, 237 equations)

This paper contains 31 sections, 23 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Organization
  4. Notation
  5. Gaussian Approximation
  6. Nummelin splitting technique
  7. Gaussian approximation
  8. Wild regenerative block bootstrap
  9. Approximate Nummelin splitting technique
  10. Gaussian multiplier bootstrap consistency
  11. Technical tool: Covariance function approximation
  12. Drift condition for controlling moments of stopping times
  13. Application : Uniform confidence band for diffusion processes
  14. Samples from a regular diffusion process
  15. Uniform confidence band
  16. ...and 16 more sections

Key Result

Theorem 2.3

Suppose the chain $\{X_t\}_{t \in \mathbb N_0}$ and the function class $\mathcal{F}$ satisfy Assumption assump : Gauss approx. Set Given positive constants $\underline c, \underline \gamma > 0$, let $\delta_n$ be a nonincreasing sequence and $M_n$ a nondecreasing sequence satisfying $M_n \geq n^{\underline \gamma/2}$ and Then, for each value of $m$, there exists a zero-mean Gaussian process $G$

Theorems & Definitions (51)

  • Definition 2.1: VC type class
  • Theorem 2.3: Gaussian approximation
  • Corollary 2.4
  • Remark 2.5: Comparison of rates for strongly mixing chains
  • Remark 3.1: Multiplier bootstrap for Markov chain data
  • Remark 3.3: Assumption \ref{['assump : bootstrap']}
  • Theorem 3.4: Bootstrap consistency
  • Remark 3.5: Theorem \ref{['thm: bootstrap consistency']}
  • Lemma 3.6
  • Remark 3.7: Lemma \ref{['lem:cov-est']}
  • ...and 41 more