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 $L \geq 1$, 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.