A Gaussian upper bound for martingale small-ball probabilities

Consider a discrete-time martingale {Xt}{Xt} taking values in a Hilbert space HH. We show that if for some L≥1L≥1, the bounds E[∥Xt+1−Xt∥2H|Xt]=1E[‖Xt+1−Xt‖H2|Xt]=1 and ∥Xt+1−Xt∥H≤L‖Xt+1−Xt‖H≤L are satisfied for all times t≥0t≥0, then there is a constant c=c(L)c=c(L) such that for 1≤R≤t√1≤R≤t, P(∥Xt−X0∥H≤R)≤cRt√. P(‖Xt−X0‖H≤R)≤cRt. 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 GG with bounded degree, there is a constant CG>0CG>0 such that if {Zt}{Zt} is the simple random walk on GG, then for every e>0e>0 and t≥1/e2t≥1/e2, P(distG(Zt,Z0)≤et√)≤CGe, P(distG(Zt,Z0)≤et)≤CGe, where distGdistG denotes the graph distance in GG.