## 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?