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

,

and similarly

.

(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).

**References.**

Grenander, Ulf (1965). Some direct estimates of the mode. Ann. Math. Statist. 36:131-138