An application of the generalised Markov inequality gives because g ( s ) = s k is non-negative and increasing for all k > 0. Fractional moments can be tricky to compute, but integer moments are possible for compound Poisson processes. Hence we consider This cannot do worse that the Markov bound, which is the special case of k = 1. The integer moments of a compound Poisson process can be computed recursively, as shown in Ross (1996, sec. 2.5.1): The only new term here is At this point it would be helpful to know the Moment Generating Function (MGF, see below) of each X$_{i}$ . Although not as cheap as the Cantelli bound, this does not appear to be an expensive calculation, if the f$_{i}$ ’s have standard forms with simple known MGFs. It is legitimate to stop at any value of k , and it might be wise to 7 Pr( S ≥ s ) ≤ inf k> 0 E( S $^{k}$) s k , Pr( S ≥ s ) ≤ min k =1 , 2 ... E( S $^{k}$) s k . (Mom) E( S $^{k}$) = λt$^{k}$$^{−}$$^{1}$ ∑ j =0 $^{(}$k − 1 j ) E( S $^{j}$) E( Y k − $^{j}$) . (3) E( Y k − $^{j}$) = m ∑ i =1 λ$_{i}$ λ E( X k − j i ) .