## The Annals of Probability

### 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 $L\geq1$, the bounds $\mathbb{E}[\|X_{t+1}-X_{t}\|_{\mathcal{H}}^{2}\vert X_{t}]=1$ and $\|X_{t+1}-X_{t}\|_{\mathcal{H}}\leq L$ are satisfied for all times $t\geq0$, then there is a constant $c=c(L)$ such that for $1\leq R\leq\sqrt{t}$,

$\mathbb{P}(\|X_{t}-X_{0}\|_{\mathcal{H}}\leq R)\leq c\frac{R}{\sqrt{t}}.$ Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph $G$ with bounded degree, there is a constant $C_{G}>0$ such that if $\{Z_{t}\}$ is the simple random walk on $G$, then for every $\varepsilon>0$ and $t\geq1/\varepsilon^{2}$,

$\mathbb{P}(\mathsf{dist}_{G}(Z_{t},Z_{0})\leq \varepsilon \sqrt{t})\leq C_{G}\varepsilon,$ where $\mathsf{dist}_{G}$ denotes the graph distance in $G$.

#### Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 4184-4197.

Dates
Revised: September 2015
First available in Project Euclid: 14 November 2016

https://projecteuclid.org/euclid.aop/1479114273

Digital Object Identifier
doi:10.1214/15-AOP1073

Mathematical Reviews number (MathSciNet)
MR3572334

Zentralblatt MATH identifier
1377.60057

#### Citation

Lee, James R.; Peres, Yuval; Smart, Charles K. A Gaussian upper bound for martingale small-ball probabilities. Ann. Probab. 44 (2016), no. 6, 4184--4197. doi:10.1214/15-AOP1073. https://projecteuclid.org/euclid.aop/1479114273

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