A Gaussian upper bound for martingale small-ball probabilities

• James R. Lee ^[1] ; Yuval Peres ^[2] ; Charles K. Smart ^[3]
  1. [1] University of Washington

     University of Washington

     Estados Unidos

     
  2. [2] The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     The Microsoft Research - University of Trento Centre for Computational and Systems Biology

     Trento, Italia

     
  3. [3] University of Chicago

     University of Chicago

     City of Chicago, Estados Unidos

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 6, 2016, págs. 4184-4197
• Idioma: inglés
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• Resumen

  • 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 ε>0ε>0 and t≥1/ε2t≥1/ε2, P(distG(Zt,Z0)≤εt√)≤CGε, P(distG(Zt,Z0)≤εt)≤CGε, where distGdistG denotes the graph distance in GG.