The Cantelli inequality. If S is square-integrable, i.e. σ 2 := Var( S ) is finite, then This is the Cantelli inequality, and it is derived from the Markov inequality. As S is a compound process, the second equality following because N is Poisson. The second expectation is simply 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. Pr( S ≥ s ) ≤ σ 2 σ 2 + ( s − µ ) 2 for s ≥ µ . (Cant) σ 2 = E( N ) Var( Y ) + E( Y ) $^{2}$Var( N ) = λt E( Y $^{2}$) , (2) E( Y $^{2}$) = m ∑ i =1 λ$_{i}$ λ E( X 2 i ) . Pr( S ≥ s ) ≤ E { g ( S ) } g ( s ) 6