## Concentration vs Anti-concentration

Leave $$S_n$$ be the sum of $$n$$ i.i.d. copies of a real random variable $$X$$. I'm interested in how big the amount is:
$$alpha_X = Pr[|S_n|<1/R] cdot Pr[S_1 geq 1]$$
it can be, for parameters $$R$$ Y $$n$$.

Yes $$X$$ It is a Rademacher or normal variable, then $$alpha_X = Theta (1 / R sqrt {n})$$. Yes $$X$$ is the distribution that is $$0$$ with probability $$1-1 / n$$ Y $$1$$ with probability $$1 / n$$, so $$alpha_X = Theta (1 / n)$$.

I would guess that $$alpha_X$$ It can not be bigger than $$max (1 / R sqrt {n}, 1 / n)$$. Is this correct?

by $$R = 1$$, the result is derived essentially from the local limit theorems. But I do not see how to argue in favor of big $$R$$.

## Probability: anti-concentration: upper limit for \$ P ( sup_ {a in mathbb S_ {n-1}} sum_ {i = 1} ^ na_i ^ 2Z_i ^ 2 ge epsilon) \$

Leave $$mathbb S_ {n-1}$$ be the unitary sphere in $$mathbb R ^ n$$ Y $$z_1, ldots, z_n$$ be a sample i.i.d of $$mathcal N (0, 1)$$.

Dice $$epsilon> 0$$ (It can be assumed that it is very small), which is reasonable upper limit for the tail probability $$P ( sup_ {a in mathbb S_ {n-1}} sum_ {i = 1} ^ na_i ^ 2z_i ^ 2 ge epsilon)$$ ?

• Using ideas from this other answer (MO link), you can set the not uniform Limit of anti-concentration: $$P ( sum_ {i = 1} ^ na_i ^ 2z_i ^ 2 le epsilon) le sqrt {e epsilon}$$ for all $$a in mathbb S_ {n-1}$$.

• The uniform analog is another story. Can coverage numbers be used?