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?