Normal domination of (super)martingales

Details Normal domination of (super)martingales Normal domination of (super)martingales

2005-12-15

arxiv.org/abs/math/0512382v1

Electronic Journal of Probability, Vol. 11 (2006), Paper 39, 1049-1070

Mathematics

Probability

20 pages

Scientific paper

Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of \sigma-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.

Affiliated with

Also associated with

No associations

Rating

Normal domination of (super)martingales does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Normal domination of (super)martingales, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Normal domination of (super)martingales will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-112320