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

TL;DR

This paper investigates whether it is statistically possible to determine if a distribution's support is finite based on i.i.d. samples, concluding that such discrimination is fundamentally impossible even with weak tests.

Contribution

It demonstrates the fundamental impossibility of testing the finiteness of a distribution's support in the context of Tsirelson's equation, highlighting limitations in statistical inference.

Findings

Impossible to distinguish finite from infinite support distributions asymptotically

Supports the conjecture that support finiteness cannot be tested with any statistical test

Connects the problem to Tsirelson's equation and its implications for statistical testing

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 m, is it possible to test whether or not the support of m 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.