Normal domination of (super)martingales
| dc.creator | Pinelis, Iosif | |
| dc.date | 2005-12-15 | |
| dc.date.accessioned | 2026-07-07T08:36:57Z | |
| dc.date.available | 2026-07-07T08:36:57Z | |
| dc.description | Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of σ-algebras (H_{\le0},H_{\le1},...), with S_0\le0 almost surely (a.s.) and differences X_i:=S_i-S_{i-1}. Suppose that for every i=1,2,... there exist H_{\le(i-1)}-measurable r.v.'s C_{i-1} and D_{i-1} and a positive real number s_i such that C_{i-1}\le X_i\le D_{i-1} and D_{i-1}-C_{i-1}\le 2 s_i a.s. Then for all real t and natural n one has \E f_t(S_n)\le\E f_t(sZ), where f_t(x):=\max(0,x-t)^5, s:=\sqrt{s_1^2+...+s_n^2}, and Z is N(0,1). In particular, this implies P(S_n\ge x)\le c_{5,0}P(Z\ge x/s) for all x in \R, where c_{5,0}=5!(e/5)^5=5.699.... Results for \max_{0\le k\le n}S_k in place of S_n and for concentration of measure also follow. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/math/0512382 | |
| dc.identifier | http://arxiv.org/abs/math/0512382 | |
| dc.identifier | Electronic Journal of Probability, Vol. 11 (2006), Paper 39, 1049-1070 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140238 | |
| dc.subject | Probability | |
| dc.subject | 60E15; 60J65; 60E05; 60G15; 60G50; 60J30 | |
| dc.title | Normal domination of (super)martingales | |
| dc.type | text |