A Gaussian upper bound for martingale small-ball probabilities

2014/05/23 by James R. Lee, Yuval Peres, Lee, James R. +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1405.5980

arxiv created 2015/09/09 · arxiv updated 2015/09/10

Abstract

Consider a discrete-time martingale \X_t\ taking values in a Hilbert space \mathcal H. We show that if for some L ≥ 1, the bounds 𝔼 [‖X_t+1-X_t‖_{\mathcal H}² | X_t]=1 and ‖X_t+1-X_t‖_{\mathcal H} ≤ L are satisfied for all times t ≥ 0, then there is a constant c = c(L) such that for 1 ≤ R ≤ √(t), ℙ(‖X_t‖_{\mathcal H} ≤ R | X₀ = x₀) ≤ c (R)/(√(t)) e^-‖x₀‖_{\mathcal H}²/(6 L² t) . Following [Lee-Peres, Ann. Probab. 2013], this has applications to diffusive estimates for random walks on vertex-transitive graphs.

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