## 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 […]

## Some relationships between different modes of convergence

We are going to show some relations between the different modes of convergence . These results are very important in practical examples. In the next post we will explain some of them. To proof this theorem, we shall use several times the Portmanteau’s lemma.

## 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 […]