For any \\( {u\\in[0,1]} \\), let us consider a binary expansion<\/p>\n
\\[ u=0.b_1b_2\\ldots=\\sum_{n=1}^\\infty b_n2^{-n} \\]<\/p>\n
where \\( {b_1,b_2,\\ldots} \\) belong to \\( {\\{0,1\\}} \\) (bits). This expansion is not unique when \\( {u} \\) is rational, e.g.<\/p>\n
\\[ 0.011111\\cdots=0.10000\\cdots. \\]<\/p>\n
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\\}} \\):<\/p>\n
\\[ \\mathbb{P}(b_1=\\varepsilon_1,\\ldots,b_n=\\varepsilon_n)=2^{-n} \\]<\/p>\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<\/p>\n
\\[ U:=\\sum_{n=1}^\\infty b_n2^{-n} \\]<\/p>\n
follows the uniform law on \\( {[0,1]} \\). Actually the odd\/even separation map<\/p>\n
\\[ 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). \\]<\/p>\n
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<\/p>\n
\\[ \\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} \\]<\/p>\n
This reduces the simulation of any law to the simulation of the Bernoulli law.<\/p>\n","protected":false},"excerpt":{"rendered":"
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).…<\/p>\n