Azuma-Hoeffding inequality

Let $D_k=M_k-M_{k-1}$ and $V={\sum }_{k=1}^n{c}_k^2$. For $\lambda >0$, F1, F2, and F5 give $\begin{array}{rl}{\displaystyle \mathbb{E}{e}^{\lambda \sum _{k=1}^n{D}_k}}& {\displaystyle =\mathbb{E}\text{\hspace{0.17em}⁣}[e^{\lambda \sum _{k<n}{D}_k}\mathbb{E}(e^{\lambda D_n}\mid {\mathcal{F}}_{n-1})]}\\ {\displaystyle }& {\displaystyle \le e^{{\lambda }^2{c}_n^2\mathrm{/}8}\mathbb{E}{e}^{\lambda \sum _{k<n}{D}_k}\le e^{{\lambda }^{2}V\mathrm{/}8}\mathrm{.}}\end{array}$ The final inequality follows by finite induction, with the empty sum at time $0$.

Markov applied to $e^{\lambda (M_n-M_0)}$ yields $\mathbb{P}(M_n-M_0\ge t)\le \exp (-\lambda t+{\lambda }^{2}V\mathrm{/}8)\mathrm{.}$ If $V>0$, the quadratic is minimized at $\lambda =4t\mathrm{/}V$, giving $\exp (-2t^2\mathrm{/}V)$.

If $V=0$, every $c_k=0$. F1's hypotheses then force every $D_k=0$ almost surely, so the event is empty for $t>0$, agreeing with the stated convention. Apply step 1.1, step 2.1 to $-M$, whose endpoints are $-B_k,-A_k$, to obtain the lower-tail bound. AC has exactly the inherited role in F4.