Agnostic insurability of model classes

Motivated by problems in insurance, our task is to predict finite upper bounds on a future draw from an unknown distribution $p$ over the set of natural numbers. We can only use past observations generated independently and identically distributed according to $p$. While $p$ is unknown, it is known to belong to a given collection ${\cal P}$ of probability distributions on the natural numbers. The support of the distributions $p \in {\cal P}$ may be unbounded, and the prediction game goes on for \emph{infinitely} many draws. We are allowed to make observations without predicting upper bounds for some time. But we must, with probability 1, start and then continue to predict upper bounds after a finite time irrespective of which $p \in {\cal P}$ governs the data. If it is possible, without knowledge of $p$ and for any prescribed confidence however close to 1, to come up with a sequence of upper bounds that is never violated over an infinite time window with confidence at least as big as prescribed, we say the model class ${\cal P}$ is \emph{insurable}. We completely characterize the insurability of any class ${\cal P}$ of distributions over natural numbers by means of a condition on how the neighborhoods of distributions in ${\cal P}$ should be, one that is both necessary and sufficient.

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