A Gaussian upper bound for martingale small-ball probabilities

TL;DR

This paper establishes a Gaussian upper bound on the probability that a bounded, mean-zero martingale in a Hilbert space remains small, with implications for random walk estimates on graphs.

Contribution

It provides a novel Gaussian upper bound for small-ball probabilities of Hilbert space martingales under bounded increments and variance conditions.

Findings

Derived a Gaussian upper bound for small-ball probabilities

Bounded the probability by a term involving initial position and time

Applicable to diffusive estimates for random walks on graphs

Abstract

Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal H} \leq L$ are satisfied for all times $t \geq 0$, then there is a constant $c = c(L)$ such that for $1 \leq R \leq \sqrt{t}$, \[\mathbb{P}(\|X_t\|_{\mathcal H} \leq R \mid X_0 = x_0) \leq c \frac{R}{\sqrt{t}} e^{-\|x_0\|_{\mathcal H}^2/(6 L^2 t)}\,.\] Following [Lee-Peres, Ann. Probab. 2013], this has applications to diffusive estimates for random walks on vertex-transitive graphs.