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

Let $(S_0,S_1,\mathrm{.}\mathrm{.}\mathrm{.})$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $H_{\le 0},H_{\le 1},\mathrm{.}\mathrm{.}\mathrm{.}$, with $S_0\le 0$ 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\mathrm{.}s\mathrm{.}forevery$i=1,2,…$,where$d>0$and$\sigma_i>0$arenon-randomconstants\mathrm{.}Let$T_n:=Z_1+…+Z_n$,where$Z_1,…,Z_n$arei\mathrm{.}i\mathrm{.}d\mathrm{.}r\mathrm{.}v{\mathrm{.}}^{\mathrm{\prime }}seachtakingononlytwovalues,oneofwhichis$d$,andsatisfyingtheconditions$\mathsf {E}Z_i=0$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 $$\mathsf{P}(S_n\ge y)\le c{\mathsf{P}}^{\mathsf{L}in,\mathsf{L}C}(T_n\ge y+\text{tfrach}2)\mathrm{\forall }y\in \mathbb{R}$$ is obtained, where $c:=e^2\mathrm{/}2=\mathrm{3.694...}$, $h:=d+{\sigma }^2\mathrm{/}d$, 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 $y\mapsto \mathsf {P}(T_n\ge y)$overthelatticeofallpointsoftheform$nd+kh$($k\in \mathbb {Z}$). 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.