We introduce here some notation very useful in probability and statistics. Definition 1 For a given sequence of random variables $latex {R_{n}}&fg=000000$, $latex {(i)}&fg=000000$ $latex {X_{n}=o_{P}(R_{n})}&fg=000000$ means $latex {X_{n}=Y_{n}R_{n}}&fg=000000$ with $latex {Y_{n}}&fg=000000$ converging to $latex 0&fg=000000$ in probability. $latex {(ii)}&fg=000000$ $latex {X_{n}=O_{P}(R_{n})}&fg=000000$ means $latex {X_{n}=Y_{n}R_{n}}&fg=000000$ with the family $latex {(Y_{n})_{n}}&fg=000000$ uniformly thigh.

http://youtu.be/sLuDOEuwwso

“Sometimes You Just Have to Jump out the Window and Grow Wings on the Way Down”

Working on the “Elaboration of physics models for road traffic” (http://www.math.univ-toulouse.fr/SEME/) Abstract (in French) La modélisation du trafic routier est un sujet abordé dans de nombreux domaines. L’objectif est de mettre en parallèle deux d’entre eux: la physique et la statistique. La modélisation physique du trafic routier repose sur la loi de conservation des véhicules …

Few days ago I signed-up into AuthorAID. This program helps to researchers from developing countries to get experience preparing scientific articles for publication in peer-reviewed journals. You can check my profile at http://www.authoraid.info/author/maikolsolis Moreover, the news section gives useful tips and resources about things related to this preparation. http://www.authoraid.info/news Let me know in the comments if you …

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 …

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.

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 …

Photo of (left to right) Henry Berthold Mann and Abraham Wald. Sources: Mathematics Dept. Ohio State and MacTutor. Let $latex {d(x,y)}&fg=000000$ be the Euclidean distance in $latex {{\mathbb R}^{k}}&fg=000000$ $latex \displaystyle d(x,y)=\Vert x-y\Vert=\left(\sum_{i=1}^{k}(x_{i}-y_{i})^{2}\right)^{1/2}. &fg=000000$ A random variable sequence $latex {X_{n}}&fg=000000$ is said to converge in probability to $latex {X}&fg=000000$ if for all $latex {\varepsilon>0}&fg=000000$ $latex \displaystyle \mathbb P(d(X_{n},X)>\varepsilon)\rightarrow0. …

1. Preliminaries Given a random variable $latex {X}&fg=000000$, we define the cumulative distribution function(or distribution function) as follows,