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November 2016 A Gaussian upper bound for martingale small-ball probabilities
James R. Lee, Yuval Peres, Charles K. Smart
Ann. Probab. 44(6): 4184-4197 (November 2016). DOI: 10.1214/15-AOP1073


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 $G$.


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James R. Lee. Yuval Peres. Charles K. Smart. "A Gaussian upper bound for martingale small-ball probabilities." Ann. Probab. 44 (6) 4184 - 4197, November 2016.


Received: 1 February 2015; Revised: 1 September 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1377.60057
MathSciNet: MR3572334
Digital Object Identifier: 10.1214/15-AOP1073

Primary: 60G42 , 60G50

Keywords: Martingales , random walks on groups , small-ball probabilities

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 6 • November 2016
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