## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Number 51, 2006, 33-52

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

#### 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 $X_i\le d$ 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

**Source***High Dimensional Probability: Proceedings of the Fourth International Conference* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196284102

**Digital Object Identifier**

doi:10.1214/074921706000000743

**Mathematical Reviews number (MathSciNet)**

MR2387759

**Zentralblatt MATH identifier**

1125.60017

**Subjects**

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

**Keywords**

supermartingales martingales upper bounds probability inequalities generalized moments

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

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. https://projecteuclid.org/euclid.lnms/1196284102