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Besov-Orlicz moduli of Brownian motion and polygonal partial sum processes

Fabian Mies

Abstract

The sample paths of Brownian motion are known to admit the exact Besov-type smoothness exponent 1/2 when measured in the sub-Gaussian Orlicz norm. We extend these regularity results by deriving the exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces, and we provide a rate of convergence towards this limiting value. The central technique is a new chaining bound for the Orlicz modulus of a stochastic process. The latter also applies to polyogonal partial sum processes of functional random variables and allows us to strengthen Donsker's invariance principle to all function spaces on the Besov-Orlicz scale up to the exact modulus with exponent 1/2. For the critical case, we establish the thresholded weak convergence of the Besov-Orlicz seminorm of the partial sum process. The analytical results find application in a nonparametric statistical testing problem, where Besov-Orlicz statistics are shown to detect a broader range of alternatives compared to Hölderian multiscale statistics.

Besov-Orlicz moduli of Brownian motion and polygonal partial sum processes

Abstract

The sample paths of Brownian motion are known to admit the exact Besov-type smoothness exponent 1/2 when measured in the sub-Gaussian Orlicz norm. We extend these regularity results by deriving the exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces, and we provide a rate of convergence towards this limiting value. The central technique is a new chaining bound for the Orlicz modulus of a stochastic process. The latter also applies to polyogonal partial sum processes of functional random variables and allows us to strengthen Donsker's invariance principle to all function spaces on the Besov-Orlicz scale up to the exact modulus with exponent 1/2. For the critical case, we establish the thresholded weak convergence of the Besov-Orlicz seminorm of the partial sum process. The analytical results find application in a nonparametric statistical testing problem, where Besov-Orlicz statistics are shown to detect a broader range of alternatives compared to Hölderian multiscale statistics.

Paper Structure

This paper contains 10 sections, 16 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on Besov-Orlicz spaces
  4. Besov-Orlicz regularity of stochastic processes with sub-Gaussian increments
  5. Modulus of continuity of stochastic processes with sub-Gaussian increments
  6. Regularity of vector-valued Brownian motion
  7. Functional central limit theorems
  8. Relation to the signal discovery problem
  9. Discussion
  10. Proofs

Key Result

Proposition 2.4

Let $\mathcal{X}$ be a separable Banach space, $\rho$ a modulus of continuity such that $ch\leq \rho(h) \leq C h^{r}$ for some $r>0$ and $0<c<C<\infty$, and $\rho^*(h)=\rho(h)\sqrt{|\log h|}$. Then there exists a continuous embedding Moreover, let $K\subset B^{\rho}_{\Psi_2,\infty}$ be bounded such that $K_t=\{f(t)\,:\, f\in K\}\subset \mathcal{X}$ is compact for every $t\in [0,1]$. Then $K$ is r

Figures (2)

  • Figure 1: Example test signal (a) and ROC curves (b-d) for the Besov-Orlicz statistic $\|Y_n\|_{B_{\Psi_2, \infty}^{1/2}}$ and the Hölder statistic $\|Y_n\|_{C^{\rho_{1/2}}}$ in the white noise model, with alternatives $f_{\textsc{flip}}$. Reported quantities are based on $10^4$ Monte Carlo simulations.
  • Figure 2: Example test signal (a) and ROC curves (b-d) for the Besov-Orlicz statistic $\|Y_n\|_{B_{\Psi_2, \infty}^{1/2}}$ and the Hölder statistic $\|Y_n\|_{C^{\rho_{1/2}}}$ in the white noise model, with alternatives $f_{\textsc{doppler}}$. Reported quantities are based on $10^4$ Monte Carlo simulations.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: Besov-Orlicz embeddings
  • Theorem 3.1
  • Remark 1
  • Theorem 3.2
  • Lemma 3.3
  • Remark 2
  • Theorem 3.4
  • ...and 22 more