Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence

TL;DR

This paper addresses mean estimation for sequences of Banach-space–valued, heavy-tailed observations with potentially infinite variance under martingale dependence. It extends a simple truncation-based estimator by centering around a naive mean and proving time-uniform, line-crossing and iterated-logarithm concentration bounds that depend on a centered $p$-th moment with $p\in(1,2]$, yielding dimension-free guarantees. The main contributions include a general template bound (Theorem) for the estimator, a Banach-space martingale-based analysis with explicit constants, and a law-of-the-iterated-logarithm refinement that achieves tight asymptotics up to a doubly-logarithmic factor. Empirically, the estimator shows competitive performance against geometric median-of-means and tournament MoM, while offering online update efficiency and robustness to martingale dependence, making it practically appealing for heavy-tailed, high-dimensional settings.

Abstract

We consider estimating the shared mean of a sequence of heavy-tailed random variables taking values in a Banach space. In particular, we revisit and extend a simple truncation-based mean estimator first proposed by Catoni and Giulini. While existing truncation-based approaches require a bound on the raw (non-central) second moment of observations, our results hold under a bound on either the central or non-central $p$th moment for some $p \in (1,2]$. Our analysis thus handles distributions with infinite variance. The main contributions of the paper follow from exploiting connections between truncation-based mean estimation and the concentration of martingales in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size $n$, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in $n$. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all constants in the inequalities are known and small.

Figures (2)

• Figure 1: For $p \in \{1.25, 1.5, 1.75\}$, we plot the tightness of optimized bounds associated with the sample mean, geometric median-of-means (Geo-MoM), truncation with initial sample mean estimate, and truncation with initial Geo-MoM estimate. We assume $n \in [10^2, 10^{10}]$, $v = 1.0$, $\delta=10^{-4}$, and $k = n/10$. In the case $p=2.0$, we assume a shared covariance matrix $\Sigma$ exists so we can plot the tournament median-of-means bounds assuming $\lambda_{\max}(\Sigma) = v/d$ and $d = 100$.
• Figure 2: We compare the empirical distributions of distance between the mean estimate and the true mean for a variety of estimators. We generate $n =10^6$ i.i.d. samples in $\mathbb{R}^{10}$ as outlined above, and use $k = \lfloor\sqrt{10^6}\rfloor$ samples to construct initial mean estimates. We compute these estimates of 250 runs. For truncation-based estimates, we consider $\lambda \in [0.0005, 0.005, 0.05, 0.5]$. We only include the sample mean in the first plot for readability.

Theorems & Definitions (26)

• Lemma 1.1: Naive Mean Concentration
• Theorem 2.1: Main result
• Corollary 2.2
• Corollary 2.3
• Remark 1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3