# Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound.

Chernoff bound:

Let $X_1,\ldots,X_n$ be independent random variables and $X_i \in \{0,1\}$. Let $X=\sum_{i=1}^n X_i$. Then for any $\delta>0$,

$\Pr\left(X \ge (1+\delta)EX \right) \le e^{-c\cdot(EX)\delta ^2},$

where $c$ is some absolute constant.

Now we consider dependent random variables. A slight variant of Azuma's inequality states the following.

Azuma's Inequality:

Let $X_1,\ldots,X_n$ be (dependent) random variables and $X_i \in \{0,1\}$. Assume that there exists $m$, such that $\Pr\left( \sum_{i=1}^n \mathbb{E}[X_i|X_{<i}] \le m\right) = 1.$ Let $X=\sum_{i=1}^n X_i$. Then for any $\lambda > 0$,

$\Pr\left(X \ge m+\lambda \right) \le e^{-2 \lambda^2/n}.$

Clearly Azuma's inequality is additive. My question is that does a multiplicative version of Azuma's inequality such as the following hold?

My question:

Let $X_1,\ldots,X_n$ be (dependent) random variables and $X_i \in \{0,1\}$. Assume that there exists $m$, such that $\Pr\left( \sum_{i=1}^n \mathbb{E}[X_i|X_{<i}] \le m\right) = 1.$ Let $X=\sum_{i=1}^n X_i$. Then for any $\delta >0$

$\Pr\left(X \ge (1+\delta)m \right) \le e^{-c\cdot m \delta^2},$

where $c$ is some absolute constant.

Note that the standard Azuma's inequality does not imply the multiplicative version when $m \ll \sqrt{n}$.

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