Monthly Archives: May 2012

JdS 2012: Efficient estimation of conditional covariance matrices for dimension reduction

In the framework of the Journées de Statisque 2012 in Bruxelles, I presented the paper “Efficient estimation of conditional covariance matrices” made under Jean-Michel Loubes and Clement Marteau direction. You could check the program and the slides of the presentation. Today I will present you some ideas about the problem studied and the solution found …

The Slutsky’s lemma as an application of the continuous mapping theorem and uniform weak convergence

Photo of Evgeny Evgenievich Slutsky. Sources: MacTutor and Bomkj. Applying the continuous mapping theorem and $latex {(v)}&fg=000000$ from the last post, we get the following theorem Lemma (Slutsky). Let be $latex {X_{n}}&fg=000000$, $latex {X}&fg=000000$ and $latex {Y_{n}}&fg=000000$ random vectors and $latex {c}&fg=000000$ a constant vector. If $latex {X_{n}\rightsquigarrow X}&fg=000000$ and $latex {Y_{n}\rightsquigarrow c}&fg=000000$, then $latex …

Equivalence between weak convergence and uniform tightness.

From left to right: Eduard Helly, Yurii Vasilevich Prokhorov and Andrei Andreyevich Markov. Source: MacTutor (1, 2, 3) and TellOfVisions. Let me start with a technical lemma that it will be very useful to show the equivalence between weak convergence and uniform tightness (Prohorov’s theorem). 1. The Helly‘s lemma Lemma (Helly’s Lemma) Let $latex {(F_{n})_{n}}&fg=000000$ a …