Let $\epsilon_i$ be i.i.d. Rademacher random variables (i.e., $\epsilon_i$ takes value $\pm 1$ with equal probability). The upper bound $\mathbb{E} |\sum_{i=1}^n \epsilon_i| \le \sqrt{n}$ follows from Jensen's inequality. [Can this be sharpened easily?]
My main question: is there a simple lower bound for this expectation?
Best Answer
My answer here gives an explicit formula for $\mathbb{E}\left|\sum_{i=1}^{2n} \epsilon_i\right|$. You can get the lower bound $$\mathbb{E}\left|\sum_{i=1}^{2n} \epsilon_i\right|\geq \sqrt{n}, \tag1$$ if you combine (1) with the bound at the link there.