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Month: March 2011

Philippe Flajolet

Il y a quelques jours s’éteignait Philippe Flajolet (1948 – 2011). Voilà un  scientifique dont l’âme était aussi grande que l’esprit, et dont la technicité n’occultait pas l’intellectualité. Ils sont si rares, ces personnages à la fois riches, curieux, ouverts, accessibles et attentifs à tous, bien au delà de la comédie humaine. Je mesure à présent la chance que j’ai eu de le rencontrer. C’était aussi un fabuleux conteur (compteur), qui savait faire vibrer l’émerveillement et l’imagination de l’enfance.

Philippe Flajolet
Philippe Flajolet alias Algorithmix
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No Dice!

Dice

Do you use dice as a symbol of randomness in your writings? According to the probabilist David Aldous, this is not a good idea, because “dice are greatly overused, both as a verbal metaphor and as a visual image, and because dice are simply unrepresentative of the way we really do encounter chance in the real world“. He proposes to use for instance dart throws. Personally, I never play darts, and I do believe that dice are  conceptually beautiful and  truly real for  game players. We may also use cards, coins, stocks, … any concrete cultural situation expressing randomness. But nothing replaces the pure and minimalist beauty of dice.

PS : I have learned the Aldous opinion on dice from the probabilist Marc Lelarge.

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Uniform bits

For any \( {u\in[0,1]} \), let us consider a binary expansion

\[ u=0.b_1b_2\ldots=\sum_{n=1}^\infty b_n2^{-n} \]

where \( {b_1,b_2,\ldots} \) belong to \( {\{0,1\}} \) (bits). This expansion is not unique when \( {u} \) is rational, e.g.

\[ 0.011111\cdots=0.10000\cdots. \]

If \( {U} \) is a uniform random variable on \( {[0,1]} \) then almost surely, \( {U} \) is irrational and its binary expansion is unique with \( {b_1,b_2,\ldots} \) independent uniform random variables on \( {\{0,1\}} \):

\[ \mathbb{P}(b_1=\varepsilon_1,\ldots,b_n=\varepsilon_n)=2^{-n} \]

for any \( {n\geq1} \) and every \( {\varepsilon_1,\ldots,\varepsilon_n} \) in \( {\{0,1\}} \). Conversely, if \( {b_1,b_2,\ldots} \) are independent uniform random variables on \( {\{0,1\}} \) then the random variable

\[ U:=\sum_{n=1}^\infty b_n2^{-n} \]

follows the uniform law on \( {[0,1]} \). Actually the odd/even separation map

\[ U=\sum_{n=1}^\infty b_n2^{-n}\mapsto (V_1,V_2):=\left(\sum_{n=1}^\infty b_{2n}2^{-n},\sum_{n=1}^\infty b_{2n-1}2^{-n}\right). \]

allows to extract from \( {U} \) a couple \( {(V_1,V_2)} \) of independent uniform random variables on \( {[0,1]} \). More generally, one can extract from \( {U} \) a countable family \( {{(W_n)}_{n\in\mathbb{Z}}} \) of independent uniform random variables on \( {[0,1]} \) by considering the diagonals (or the columns, or the rows) in

\[ \begin{array}{ccccc} b_1 & b_2 & b_5 & b_{10} & \cdots \\ b_4 & b_3 & b_6 & b_{11} & \cdots \\ b_9 & b_8 & b_7 & b_{12} & \cdots \\ b_{16} & b_{15} & b_{14} & b_{13} & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \]

This reduces the simulation of any law to the simulation of the Bernoulli law.

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Beta laws: arcsine, uniform, semicircle

The Beta law on \( {[0,1]} \) with parameters \( {a>0} \) and \( {b>0} \) has density

\[ x\mapsto \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} x^{a-1}(1-x)^{b-1}\mathbf{1}_{[0,1]}(x). \]

This family of laws allows to interpolate between the arcsine law \( {a=b=1/2} \) and the semicircle law \( {a=b=3/2} \), passing thru the uniform law \( {a=b=1} \). It is sometimes more convenient to work on the interval \( {[-1,1]} \) instead of \( {[0,1]} \). The Jacobi polynomials are orthogonal for this \( {(a,b)} \)-model. We recover the Chebyshev polynomials of the first and second kind in the arcsine and semicircle cases respectively.

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