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?