A Gaussian upper bound for martingale small-ball probabilities

Abstract

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

References in corpus (1)

• Localization for controlled random walks and martingales