Conditional Hoeffding bound for bounded martingale differences

The inequalities and the deterministic width give $D-c\le A\le D\le B\le D+c$, so $A,B\in L^1$. Conditional order and F2 yield $A=\mathbb{E}[A\mid \mathcal{G}]\le 0\le \mathbb{E}[B\mid \mathcal{G}]=B\mathrm{.}$ Thus $\mathrm{\mid }D\mathrm{\mid }\le B-A\le c$, making $e^{\lambda D}$ bounded and its conditional expectation well-defined.

On $\{A=B\}$, step 1.1 forces $D=A=B=0$, so the result is equality. On $\{A<B\}$ write $w=B-A$ and $\theta =(D-A)\mathrm{/}w\in [0,1]$. F3 gives $e^{\lambda D}\le \frac{B-D}{w}{e}^{\lambda A}+\frac{D-A}{w}{e}^{\lambda B}\mathrm{.}$ The ratios $B\mathrm{/}w$ and $-A\mathrm{/}w$ lie in $[0,1]$, while the $\mathcal{G}$-measurable divided difference $K=(e^{\lambda B}-e^{\lambda A})\mathrm{/}w$ is bounded by $\mathrm{\mid }\lambda \mathrm{\mid }{e}^{\mathrm{\mid }\lambda \mathrm{\mid }c}$. Thus the right side is the sum of the two displayed bounded endpoint terms and $KD$; conditional linearity, the known-variable rule, F5, and $\mathbb{E}[D\mid \mathcal{G}]=0$ turn its conditional expectation into $\frac{B}{w}{e}^{\lambda A}+\frac{-A}{w}{e}^{\lambda B}\mathrm{.}$ This justifies pulling out the potentially small-width coefficient rather than formally dividing inside a conditional expectation.

Put $u=-A\mathrm{/}w\in [0,1]$ and $z=\lambda w$. The last expression is $h_u(z)=(1-u)e^{-uz}+u{e}^{(1-u)z}\mathrm{.}$ For $g_u=\log h_u$, direct differentiation gives $g_u(0)=g_u^{\mathrm{\prime }}(0)=0$ and $0\le g_u^{\mathrm{\prime }\mathrm{\prime }}(z)=p_z(1-p_z)\le \frac{1}{4},$ where $p_z=u{e}^{(1-u)z}\mathrm{/}{h}_u(z)\in [0,1]$. Integrating the second-derivative bound from $0$ to $z$ (with reversed limits when $z<0$) gives $g_u(z)\le z^2\mathrm{/}8$. Hence $h_u(z)\le e^{z^2\mathrm{/}8}\le e^{{\lambda }^2{c}^2\mathrm{/}8}$.

Combining the two measurable cases proves the conditional inequality. Random endpoints cause no hidden selection; only their deterministic width enters the final bound. AC is used exactly as recorded in F4.