If X is a nonnegative sigma-subGaussian random variable with P(X=0) >= p, what is a good upper bound for P(X >= h)?

Tamara Anthony

Tamara Anthony

Open question

2022-08-20

If X is a nonnegative σ-subGaussian random variable with P ( X = 0 ) p, what is a good upper bound for P ( X h )?

Answer & Explanation

lywyk0

lywyk0

Beginner2022-08-21Added 12 answers

Step 1
I have tried using some simple inequalities to get (hopefully) something useful:
I have focused on the case when p is small.
Since P ( X = 0 ) = p, the subgaussian concentration implies that p 2 exp ( ( E X ) 2 2 σ 2 ) . This gives us the following:
E X σ 2 log 2 p =: h 0 .
Step 2
Given this bound on E X, we can now calculate upper tail inequalities: for any h = α h 0 for α 1, we have that
P ( X h ) = P ( X E X ( α 1 ) h 0 ) 2 exp ( ( α 1 ) 2 h 0 2 2 σ 2 ) = 2 exp ( ( α 1 ) 2 log 2 p ) = 2 ( p 2 ) ( α 1 ) 2 .
For comparison, the bound in paper has h 0 := σ p ( 1 p ) and gives the result for any h = α h 0 , P ( X α h 0 ) p ( 1 p ) α 2 1 + p .
Thus subgaussianity allows us to get much tighter bounds, especially when p 0. Please let me know if I made an error somewhere.

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