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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