# A 'reverse' Hoeffding Inequality

Hoeffding's Inequality gives, for independent Bernoulli random variables, that $$\mathbb{P}\{|\sum_{i=1}^n X_i| > t\} \leq 2\text{exp}(-\frac{t^2}{2})$$. I'm wondering if this is also a lower bound for $$\mathbb{P}\{|\sum_{i=1}^n X_i| \leq t\} \geq 2\text{exp}(-\frac{t^2}{2})$$. I think that this is true intuitively but cannot quite show it.

## 1 Answer

What you propose is not true. The event in your second equation is the opposite event, therefore what you can say is that its probability is $$\ge 1-2e^{-t^2/2}$$