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An application of the generalised Markov inequality gives 

<img src='content_image/318560.jpg'>

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 

<img src='content_image/318561.jpg'>

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 

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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 

