Testing the finiteness of the support of a distribution: a statistical look at Tsirelson's equation

Abstract

We consider the following statistical problem: based on an i.i.d. sample of size $n$ of integer valued random variables with common law $\mu$, is it possible to test whether or not the support of $\mu$ is finite as $n$ goes to infinity? This question is in particular connected to a simple case of Tsirelson's equation, for which it is natural to distinguish between two main configurations, the first one leading only to laws with finite support, and the second one including laws with infinite support. We show that it is in fact not possible to discriminate between the two situations, even using a very weak notion of statistical test.