# Concentration vs. Anti-Concentration

Suppose $$X_1, \dots, X_n$$ are $$n$$ independent copies of a real random variable $$X$$. Then, what is the largest value of:

$$\alpha_X = \min_{a_1,\dots, a_n \in [1,R]} \Pr\left[\sum_i a_i X_i <1\right]\cdot \Pr[X\geq 1]?$$

When $$R$$ is small, then we can take $$X$$ to be a random Bernoulli or a normal variable. In that case, the above product is of the order of $$\frac1{R\sqrt{n}} \cdot 1$$.

When $$R$$ is large, we can take $$X$$ to be $$0$$ with probability $$1-1/n$$ and $$1$$ with probability $$1/n$$. Then the above product is of the order of $$1\cdot \frac1n$$.

It seems to me that $$\alpha_X$$ can never be much larger than $$\max(\frac{1}{R\sqrt{n}}, \frac{1}{n})$$. For the small $$R$$ case, I think one can use the local limit theorem to argue. But any ideas about how to analyze the full regime?

• Is $X$ assumed to be non-negative? Otherwise the random variable that is $1$ with probability $1/2$ and a huge negative number with probability $1/2$ gives you $\alpha_X=\frac 12(1-2^{-n})$ regardless of $R$. – fedja May 16 at 11:06