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Source. Liu--Sawhney, On further questions regarding unit fractions, arXiv:2404.07113v1, Lemma 2.6, p. 8; see the source digest.

Statement. Let $(X_k)_{k=0}^n$ be a real-valued martingale with respect to a filtration $(\mathcal F_k)_{k=0}^n$. Suppose the deterministic constants $c_k\geq0$ satisfy

$$|X_k-X_{k-1}|\leq c_k\quad\text{almost surely}, \qquad 1\leq k\leq n.$$

For $t>0$, when $\sum_{k=1}^n c_k^2>0$,

$$\mathbb P(|X_n-X_0|\geq t) \leq2\exp\!\left(-\frac{t^2}{2\sum_{k=1}^n c_k^2}\right).$$

If every $c_k=0$, then $X_n=X_0$ almost surely and the tail probability is zero. The source calls $\sum c_k^2$ the variance proxy.

External input. This is the standard Azuma--Hoeffding inequality. The paper cites Janson, Łuczak, and Ruciński, Random Graphs, Wiley-Interscience (2000), Theorem 2.25. It gives the statement without proof; the theorem is treated here as an external input.

Bears on. #298 and #299, through the concentration bounds in the quantitative reciprocal-sum argument.