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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 39 open journal systems 


On normal domination of (super)martingales

Iosif Pinelis, Michigan Technological University


Abstract
Let $(S_0,S_1,dots)$ be a supermartingale relative to a nondecreasing sequence of $sigma$-algebras $(H_{le0},H_{le1},dots)$, with $S_0le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that for every $i=1,2,dots$ 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 $C_{i-1}le X_ile D_{i-1}$ and $D_{i-1}-C_{i-1}le 2 s_i$ a.s. Then for all %real $t$ and natural $n$ and all functions $f$ satisfying certain convexity conditions begin{equation*}%label{eq:bernoulli} E f(S_n)leE f(sZ), end{equation*} where %$f_t(x):=max(0,x-t)^5$, $s:=sqrt{s_1^2+dots+s_n^2}$ %, and $Zsim N(0,1)$. In particular, this implies begin{equation*}%label{eq:bernoulli} P(S_nge x)le c_{5,0}P(sZge x)quadforall xinR, end{equation*} where $c_{5,0}=5!(e/5)^5=5.699dots.$ Results for $max_{0le kle n}S_k$ in place of $S_n$ and for concentration of measure also follow.


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Pages: 1049-1070

Published on: November 21, 2006
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Electronic Journal of Probability. ISSN: 1083-6489