Martingales and Tail Probabilities

At the twenty-eighth Actuarial Research Conference of the Society of Actuarial, WILLMOT and LIN (1993) presented a paper whose central result IS a bound on the tail probability of a random sum. In the subsequent discussion, Professor Buhlmann raised the question, if this bound could be derived by martingale methods. The purpose of this note is to show how it can be done.