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The Cantelli inequality. If S is square-integrable, i.e. σ 2 := Var( S ) is finite, then 

<img src='content_image/306499.jpg'>

This is the Cantelli inequality, and it is derived from the Markov inequality. As S is a compound process, 

<img src='content_image/306500.jpg'>

the second equality following because N is Poisson. The second expectation is simply m 

<img src='content_image/306501.jpg'>

We expect the Cantelli bound will perform much better than the Markov bound both because it exploits the fact that S is square integrable, and because its derivation involves an optimisation step. It is almost as cheap to compute, and so it is really a free upgrade. 

The moment inequality. This inequality and the Chernoff inequality below use the generalised Markov inequality: if g is increasing, then S ≥ s ⇐⇒ g ( S ) ≥ g ( s ), and so 

for any g that is increasing and non-negative. 

<img src='content_image/306502.jpg'>

