In this post I give a proof of an exercise of real analysis that has found an application to statistical mode estimation (see the Remark below). I wish to thank gerard0, remarque, and La vieille for their contribution.
Proposition. Let be an upper-semicontinuous map from () to . Suppose that there is a unique (). Then for every continuous map we have
Proof. First remark that the uniqueness of implies that is not identically zero. We denote by the sequence at stake. We are looking for an upper bound of
So let . The map is continuous (at ) so there exists an open interval containing such that for all . Defining we obtain
where denotes the compact subset . It remains to show that the ratio tends to . But it is a well known result (see e.g. here for a proof relying on Chebyshev’s inequality) that
(The existence of is ensured by the upper-semicontinuity of and the compactness of .) This implies that
where is the ratio , which is strictly less than by uniqueness of . Thus, tends to , and so does . This shows that tends to , and concludes the proof.
Remark. Suppose that is actually an upper-semicontinuous probability density with a unique mode . Taking the previous result states that
where, for each , is a random variable with density . This result is at the core of the mode estimator proposed by Grenander (1965).
Grenander, Ulf (1965). Some direct estimates of the mode. Ann. Math. Statist. 36:131-138