Institute of Mathematical Statistics Lecture Notes - Monograph Series

Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above

Iosif Pinelis

Full-text: Open access


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 $X_i\le d$ and $\Var(X_i|H_{\le i-1})\le \si_i^2$ a.s.\ for every $i=1,2,\dots$, where $d>0$ and $\si_i>0$ are non-random constants. Let $T_n:=Z_1+\dots+Z_n$, where $Z_1,\dots,Z_n$ are i.i.d.\ r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\E Z_i=0$ and $\Var Z_i=\si^2:=\frac1n(\si_1^2+\dots+\si_n^2)$. Then, based on a comparison inequality between generalized moments of $S_n$ and $T_n$ for a rich class of generalized moment functions, the tail comparison inequality

center$$ \PP(S_n\ge y) \le c\, \PP^{\lin,\lc}(T_n\ge y+\tfrac h2)\quad\forall y\in\R $$

is obtained, where $c:=e^2/2=3.694\dots$, $h:=d+\si^2/d$, and the function $y\mapsto\PP^{\lin,\lc}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto\PP(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in\Z$). An explicit formula for $\PP^{\lin,\lc}(T_n\ge y+\tfrac h2)$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.

Chapter information

Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn, eds., High Dimensional Probability: Proceedings of the Fourth International Conference (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 33-52

First available in Project Euclid: 28 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60G42: Martingales with discrete parameter 60G48: Generalizations of martingales 60G50: Sums of independent random variables; random walks
Secondary: 60E05: Distributions: general theory 60G15: Gaussian processes

supermartingales martingales upper bounds probability inequalities generalized moments

Copyright © 2006, Institute of Mathematical Statistics


Pinelis, Iosif. Binomial upper bounds on generalized moments and tail probabilities of (super)martingales with differences bounded from above. High Dimensional Probability, 33--52, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000743.

Export citation