Electronic Journal of Probability

On normal domination of (super)martingales

Iosif Pinelis

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Let $(S_0,S_1,\dots)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $(H_{\le0},H_{\le1},\dots)$, with $S_0\le0$ 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 $C_{i-1}\le X_i\le D_{i-1}$ and $D_{i-1}-C_{i-1}\le 2 s_i$ a.s. Then for all real $t$ and natural $n$ and all functions $f$ satisfying certain convexity conditions $ E f(S_n)\le E f(sZ)$, where $f_t(x):=\max(0,x-t)^5$, $s:=\sqrt{s_1^2+\dots+s_n^2}$, and $Z\sim N(0,1)$. In particular, this implies $ P(S_n\ge x)\le c_{5,0}P(sZ\ge x)\quad\forall x\in R$, where $c_{5,0}=5!(e/5)^5=5.699\dots.$ Results for $\max_{0\le k\le n}S_k$ in place of $S_n$ and for concentration of measure also follow.

Article information

Electron. J. Probab., Volume 11 (2006), paper no. 39, 1049-1070.

Accepted: 21 November 2006
First available in Project Euclid: 31 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60J65: Brownian motion [See also 58J65]
Secondary: 60E05: Distributions: general theory 60G15: Gaussian processes 60G50: Sums of independent random variables; random walks 60J30

supermartingales martingales upper bounds probability inequalities generalized moments

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Pinelis, Iosif. On normal domination of (super)martingales. Electron. J. Probab. 11 (2006), paper no. 39, 1049--1070. doi:10.1214/EJP.v11-371. https://projecteuclid.org/euclid.ejp/1464730574

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