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

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

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.