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On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence

Luisa F. Estrada, Michael A. Högele, Alexander Steinicke

TL;DR

The paper develops a quantitative theory linking almost sure martingale convergence to a tradeoff between error tolerance sequences and mean deviation frequencies, via a refined Borel-Cantelli framework grounded in the overlap count $\mathcal{O}_{\varepsilon,n_0}$ and the modulus $\mathbbm{m}_{\varepsilon,n_0}$. It provides a general lemma that ensures $\limsup_{n\to\infty} |X_n-X|/\varepsilon_n\le 1$ a.s. under a finite weighted sum $K(a,\varepsilon,n_0)$ and yields MDF bounds for various tail decays (polynomial, exponential, Weibull). The framework is then applied to a broad class of martingale settings, including $L^2$ martingales, Azuma-Hoeffding, MDs in $L^p$, BK-type strong laws, and exponential-moment regimes, with concrete results for multicolor Pólya urns, GCRP, a.s. convergence of M-estimators, and Galton–Watson excursion dynamics. It also clarifies the relationship between MDF convergence and Ky Fan metric convergence, showing that MDF insights provide a quantitative bridge to probabilistic topologies and practical stopping criteria. Overall, the work offers a versatile toolkit to quantify and compare almost sure convergence rates and deviation frequencies across a wide spectrum of martingale-type models and applications.

Abstract

In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in $L^2$, and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in $L^p$, $p\geq 2$, using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric.

On the tradeoff between almost sure error tolerance and mean deviation frequency in martingale convergence

TL;DR

The paper develops a quantitative theory linking almost sure martingale convergence to a tradeoff between error tolerance sequences and mean deviation frequencies, via a refined Borel-Cantelli framework grounded in the overlap count and the modulus . It provides a general lemma that ensures a.s. under a finite weighted sum and yields MDF bounds for various tail decays (polynomial, exponential, Weibull). The framework is then applied to a broad class of martingale settings, including martingales, Azuma-Hoeffding, MDs in , BK-type strong laws, and exponential-moment regimes, with concrete results for multicolor Pólya urns, GCRP, a.s. convergence of M-estimators, and Galton–Watson excursion dynamics. It also clarifies the relationship between MDF convergence and Ky Fan metric convergence, showing that MDF insights provide a quantitative bridge to probabilistic topologies and practical stopping criteria. Overall, the work offers a versatile toolkit to quantify and compare almost sure convergence rates and deviation frequencies across a wide spectrum of martingale-type models and applications.

Abstract

In this article we quantify almost sure martingale convergence theorems in terms of the tradeoff between asymptotic almost sure rates of convergence (error tolerance) and the respective modulus of convergence. For this purpose we generalize {an} elementary quantitative version of the first Borel-Cantelli lemma on the statistics of the deviation frequencies (error incidence), which was recently established by the authors. First we study martingale convergence in , and in the setting of the Azuma-Hoeffding inequality. In a second step we study the strong law of large numbers for martingale differences in two settings: uniformly bounded increments in , , using the respective Baum-Katz-Stoica theorems, and uniformly bounded exponential moments with the help of the martingale estimates by Lesigne and Volný. We also present applications for the tradeoff for the multicolor generalized Pólya urn process, the Generalized Chinese restaurant process, statistical M-estimators, as well as the a.s.~excursion frequencies of the Galton-Watson branching process. Finally, we relate the tradeoff concept to the convergence in the Ky Fan metric.
Paper Structure (22 sections, 31 theorems, 277 equations, 1 figure)

This paper contains 22 sections, 31 theorems, 277 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A quantitative version of the first Borel-Cantelli lemma
  3. The tradeoff in almost sure martingale convergence theorems
  4. The tradeoff for martingales bounded in $\mathbf{L^p}$, $\mathbf{p\geqslant2}$
  5. The tradeoff for martingale convergence by the Azuma-Hoeffding inequality
  6. The tradeoff in the strong law for martingale differences (MDs)
  7. The tradeoff in the strong law for MDs in $L^p$
  8. The tradeoff for Baum-Katz-Nagaev type strong laws for MDs uniformly bounded in $L^p$
  9. The tradeoff in a strong law for MDs with uniformly bounded exponential moments
  10. Applications
  11. The tradeoff in multicolor Pólya urn models
  12. Applications of Pólya's urn models in machine learning
  13. Applications of Pólya's urn models in genetics, psychology and biology
  14. The tradeoff for the Generalized Chinese Restaurant Process (GCRP)
  15. A tradeoff quantification of a.s. convergent M-estimators
  16. ...and 7 more sections

Key Result

Lemma 1

Given a probability space $(\Omega, \mathcal{A}, \mathbb{P})$, $n_0\in \mathbb{N}_0$, and a sequence of events $(A_n)_{n\geqslant n_0}$, such that Then for any positive, nondecresing sequence $(a_n)_{n\geqslant n_0}$, the following statements are true:

Figures (1)

  • Figure 2: Overlap statistic $\mathcal{O}_{n_0, N}$ of the nested events $A_{n_0}\supseteq A_{n_0+1}\supseteq \dots \supseteq A_{n_0+N}$

Theorems & Definitions (91)

  • Definition 1
  • Lemma 1: Quantitative version of the first Borel-Cantelli lemma
  • Remark 1
  • proof : Proof of Lemma \ref{['lem:BC1']}:
  • Example 1: Polynomial probability decay
  • Example 2: Exponential probability decay
  • Example 3: Weibull type probability decay
  • Definition 2
  • Remark 2
  • Lemma 2: Almost sure error tolerance and mean deviation frequency
  • ...and 81 more