Prediction over countable alphabets

We consider the problem of predicting finite upper bounds on unseen samples of an unknown distribution p over the set of natural numbers, using only observations generated i.i.d. from it. While p is unknown, it belongs to a known collection P of possible models. This problem is motivated from an insurance setup. The distribution p is a probabilistic model for loss, and each sample from p stands for the total loss incurred by the insured at a particular time step. The upper bound plays the role of the total built up reserves of an insurer, including past premiums after paying out past losses, as well as the current premium charged in order to cover future losses. Thus, if an insurer can accurately upper bound future unseen losses, premiums can be set so that the insurer will not be bankrupted. However, is it possible for the insurer to set premiums so that the probability of bankruptcy can be made arbitrarily small-even when the possible loss is unbounded, the underlying loss model unknown, and the game proceeds for an infinitely long time? Equivalently, when is P insurable? We derive a condition that is both necessary and sufficient for any class P of distributions to be insurable.