A few months ago, I did publish a post (in French) following my reading of Leonard Mlodinow’s the Drunkard’s Walk. More precisely, I mentioned a paragraph that I found extremely informative

But it looks like those gangsters were not only stealing money. They were also stealing ideas, here from a British statistician, manely Leonard Henry Caleb Tippett. Leonard Tippett is famous in Extreme Value Theory for his theorem (the so-called Fisher-Tippett theorem, which gives the possible limiting distributions for a normalized version of the maximum from an i.i.d. sequence, see old posts). According to Martin Gardner, Leonard Tippett suggested to use middle numbers (not the last ones) of larger ones to generate (pseudo) random sequences, or more precisely, in 1927, “published a table of 41,600 random numbers, obtained by taking the middle digits of the area of parishes in England”

I could not get a copy of the book Random Sampling Numbers by Leonard Tippett (I could only find reviews, e.g. Nair (1938)) but I do believe that this technique should work to generate sequences that do look like sequences of random numbers. Note that several techniques were mentioned in previous posts (in French) published a few years ago.
Now, I should also take some time to apologize because, sometimes, I am the one playing the gangster: I do steal a lot of illustrations on the internet. And I would like to apologize to the authors. On my previous blog, I did try – once – to add a short line at the end of a post, explaining where the illustration was coming from (trying to give credit to the illustrator). Less than 10 days after adding this short line, I received an email from a ‘publisher’, telling me that there were rights attached to the picture, and that I had 24 hours to remove it (if not, their lawyers will see what to do). Of course, I did remove the picture, and the mention. Now, I use pictures, and no mention. And I feel guilty. So I wanted to apologize for stealing others’ work. I am still discussing to hire an illustrator, to illustrate my blog. Work in progress….
such that
where
‘s are i.i.d. with cumulative distribution function
. They had supporting arguments, but no (rigorous) proof. Nevertheless, the obtained that the only possible types for G were



as possible limit. Richard von Mises gave in 1936 sufficient, but not necessary conditions for their (max) domain of attraction, i.e. characterization of function
such that the maxima converges to some specific function 
is regularly varying at infinity, with index
(even if the term “regular variation” was not mentioned in the paper). Similar results were derived to characterize functions in the (max) domain of attraction of Weibull. For the (max) domain of attraction of
, Boris Gnedenko obtained that a necessary and sufficient condition was that there exists a function
such 




(with a much shorter proof).
. For bounded distribution, consider e.g. the uniform distribution on the unit interval, i.e.
on the unit interval. Let
and
. Then, for all
and
,

. Let
and
, then

. Let
and
, then

sample. We can use the following approximation of the cumulative distribution function (based on l’Hopital’s rule)
. Let
and
. Then we can get
. I.e. the limiting distribution of the maximum of a Gaussian sample is Gumbel’s. But what we do not see here is that for a Gaussian sample, the convergence is extremely slow, i.e., with 100 observations, we are still far away from Gumbel distribution,

, if


