Two-sided Azuma-Hoeffding inequality ($\mathbb{P}(|M_n-M_0|\ge x)\le 2e^{-x^2/(2{\sum }_{k=1}^n{c}_k^2)}$)

For deterministic bounds $|M_k-M_{k-1}|\le c_k$, the conditional mean-zero increment has exponential moment at most $e^{{\theta }^2{c}_k^2/2}$. Iteration, the Markov inequality and optimization at $\theta =x/\sum c_k^2$ give each one-sided tail, and the union bound gives the displayed inequality. The factor two in the exponent denominator cannot be omitted: a single symmetric increment of size one has absolute deviation one with probability one.