The Annals of Probability

A Gaussian upper bound for martingale small-ball probabilities

James R. Lee, Yuval Peres, and Charles K. Smart

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 60G50: Sums of independent random variables; random walks

Martingales random walks on groups small-ball probabilities


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.

Export citation


  • [1] Armstrong, S. N. and Zeitouni, O. (2014). Local asymptotics for controlled martingales. Preprint. Available at arXiv:1402.2402.
  • [2] Barenblatt, G. I. and Sivashinskii, G. I. (1969). Self-similar solutions of the second kind in nonlinear filtration. J. Appl. Math. Mech. 33 836–845 (1970).
  • [3] Gurel-Gurevich, O., Peres, Y. and Zeitouni, O. (2014). Localization for controlled random walks and martingales. Electron. Commun. Probab. 19 no. 24, 8.
  • [4] Kallenberg, O. and Sztencel, R. (1991). Some dimension-free features of vector-valued martingales. Probab. Theory Related Fields 88 215–247.
  • [5] Kwapień, S. and Woyczyński, W. A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston, MA.
  • [6] Lee, J. R. and Peres, Y. (2013). Harmonic maps on amenable groups and a diffusive lower bound for random walks. Ann. Probab. 41 3392–3419.
  • [7] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.