Sommes de variables aléatoires

Dans un exercice à rendre vendredi, on parle de lois de densité

qui ressemble à ça,

Pour rappel (?), si on considère la somme de deux variables indépendantes suivant une telle loi, sa densité est donnée par

qui est une convolée. Les amateurs peuvent faire les calculs, mais on peut aussi faire des calculs numériques aussi,

integrate(function(x) f(t-x)*f(x),-3,3)$value
ce qui permet de tracer la densité suivante

On continue ? Après avoir sommé deux variables (indépendantes), ont peut en sommer trois, en notant que la densité s’écrit en convolant

Cette fois, numériquement, on peut écrire

integrate(function(x) f(t-x)*sapply(x,f2),-3,3)$value
Et on peut continuer à itérer (ce qui revient à faire des gros calculs intégrales) et ça finira par converger vers une loi normale… qu’il faudra décrire pour vendredi.

“standardized” version of the maximum

For the first homework, there was a tricky question in problem 29, chapter 5. Here is the maximum of n random variables i.i.d. uniformly distributed on the unit interval I gave a hint last week about the cumulative distribution function for the maximum, i.e.

is equal to the probability that all are smaller than,

Then, we use independent to obtain that this probability is a product, of equal quantities since all random variables are identically distributed, i.e.

Then, the exercise ask the following

i.e. find a standardized version of that the maximum so that the cumulated distribution of that standardized version has a (non degenerated) limiting value. A hint is given in the answers, at the end of the book,

Actually, the question is not that simple (see here for the history of that question).
What I said during the course is that if is a random variable with finite variance, then

is a standardized (or normalized) version of, in the sense that it is centered, i.e.

and with a unit variance, i.e.

This is the kind of standardization (or normalization) that is used in the central limit theorem i.e. it is interesting when we study the core of our distribution (i.e. the mean).
Here we focus on the maxima (not on the expected value). Note that here


(up to some typing mistakes). Thus, our previous standardization would be

that can be simplified as

Hence, that random variable can be approximated by

since as Here, it is then possible to get

since if, then (see the prof of the central limit theorem we got a few days ago).
But this is usually not the way we work with maxima. Actually, Fréchet, Fisher, Tippett, Gnedenko proved that the appropriate standardization to work with maxima is to consider

where is the cumulative distribution of the‘s (the random variables used to build up the maximum). This work since the‘s have a finite support, i.e. the bounded, with an upper limit (here 1).
Note that

assuming that the density associated with exists. Hence, here the standardization becomes

which is exactly the one that John Rice is suggesting… And the proper motivation comes from extreme value theory, but it is a bit far away from what we shall see in that course…