Electronic Journal of Probability

On normal domination of (super)martingales

Iosif Pinelis

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Abstract

Let $(S_0,S_1,\dots)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $(H_{\le0},H_{\le1},\dots)$, with $S_0\le0$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730574

Digital Object Identifier
doi:10.1214/EJP.v11-371

Mathematical Reviews number (MathSciNet)
MR2268536

Zentralblatt MATH identifier
1130.60019

Subjects
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

Keywords
supermartingales martingales upper bounds probability inequalities generalized moments

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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