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

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

lywyk0 · Accepted Answer

Step 1I 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 2Given 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.

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

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