A Gaussian upper bound for martingale small-ball probabilities

Consider a discrete-time martingale {Xt} taking values in a Hilbert space H. We show that if for some L.1, the bounds E[.Xt+1.Xt.2H|Xt]=1 and .Xt+1.Xt.H.L are satisfied for all times t.0, then there is a constant c=c(L) such that for 1.R.t., 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 G with bounded degree, there is a constant CG>0 such that if {Zt} is the simple random walk on G, then for every .>0 and t.1/.2, P(distG(Zt,Z0)..t.).CG., where distG denotes the graph distance in G.