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

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

2005-12-14

arxiv.org/abs/math/0512301v2

IMS Lecture Notes Monograph Series 2006, Vol. 51, 33-52

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000743 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000743

Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf {Var}(X_i|H_{\le i-1})\le \sigma_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $\sigma_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where $Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf {Var}Z_i=\sigma ^2:=\frac{1}{n}(\sigma_1^2+...+\sigma_n^2)$. 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 $$ \mathsf P(S_n\ge y) \le c \mathsf P^{\mathsf Lin,\mathsf L C}(T_n\ge y+\tfrach2)\quad\forall y\in \mathbb R$$ is obtained, where $c:=e^2/2=3.694...$, $h:=d+\sigma ^2/d$, and the function $y\mapsto \mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto \mathsf {P}(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in \mathbb {Z}$). An explicit formula for $\mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y+\tfrac{h}{2})$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.

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