Supercentenarian paradox

Abstract

Consider the following statement: $B(t, \Delta t)$: a $t$ years old person NN will survive another $\Delta t$ years, where $t, \Delta t\in \mathbb{R}$ are nonnegative real numbers. We know only that NN is $t$ years old and nothing about the health conditions, gender, race, nationality, etc. We bet that $B(t, \Delta t)$ holds. It seems that our odds are very good, for any $t$ provided $\Delta t$ is small enough, say, $1 / 365$ (that is, one day). However, this is not that obvious and depends on the life-time probabilistic distribution. Let $F(t)$ denote the probability to live at most $t$ years and set $\Phi(t) = 1 - F(t)$. Clearly, $\Phi(t) \rightarrow 0$ as $t \rightarrow \infty$. It is not difficult to verify that $Pr(B(t, \Delta t)) \rightarrow 0$ as $t \rightarrow \infty$, for any fixed $\Delta t$, whenever the convergence of $\Phi$ is fast enough (say, super-exponential). Statistics provide arguments (based on an extrapolation yet) that this is the case. Hence, for an arbitrarily small positive $\Delta t $ and $\epsilon$ there exists a sufficiently large $t$ such that $Pr(B(t, \Delta t)) < \epsilon$, which means that we should not bet... in theory. However, in practice we can bet safely, because for the inequality $Pr(B(t, \Delta t)) < 1/2$ a very large $t$ is required. For example, $\Delta t = 1/365$ may require $t > 125$ years for some typical distributions $F$ considered in the literature. Yet, on Earth there is no person of such age. Thus, our odds are good, either because the chosen testee NN is not old enough, or for technical (or, more precisely, statistical) reasons -- absence of a testee. This situation is similar to the famous St.Petersburg Paradox.