A GAUSSIAN UPPER BOUND FOR MARTINGALE SMALL-BALL PROBABILITIES
成果类型：
Article
署名作者：
Lee, James R.; Peres, Yuval; Smart, Charles K.
署名单位：
University of Washington; University of Washington Seattle; Microsoft; University of Chicago
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/15-A0P1073
发表日期：
2016
页码：
4184-4197
关键词：
摘要：
Consider a discrete-time martingale {X-t} taking values in a Hilbert space H. We show that if for some L >= 1, the bounds E[parallel to Xt+1 - X-t parallel to(2)(H)parallel to X-t] = 1 and parallel to Xt+1 - Xt+1 parallel to(H) <= L are satisfied for all times t >= 0, then there is a constant c = c(L) such that for 1 <= R <= root t, P(parallel to X-t-X0 parallel to(H) <= R) <= cR/root 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 epsilon > 0 and t >= 1/epsilon(2), P(dist(G)(Z(t), Z(0)) <= epsilon root t) <= C-G epsilon, where distG denotes the graph distance in G.