{"id":10047,"date":"2018-01-01T13:10:44","date_gmt":"2018-01-01T12:10:44","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=10047"},"modified":"2018-07-10T15:08:21","modified_gmt":"2018-07-10T13:08:21","slug":"around-the-circular-law-erratum","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2018\/01\/01\/around-the-circular-law-erratum\/","title":{"rendered":"Around the circular law : erratum"},"content":{"rendered":"
\"O'Rourke<\/a>
O'Rourke Arms<\/figcaption><\/figure>\n

Sean O'Rourke<\/a> pointed out on December 30, 2017 that a notation should be corrected in the statement of Lemma A.1 in the probability survey Around the circular law<\/a> (2012) that I wrote years ago in collaboration with Charles Bordenave<\/a>.<\/p>\n

Indeed the definition of \\( {\\sigma^2} \\) should be corrected to<\/p>\n

\\[ \\sigma^2 :=\\min_{1\\leq i,j\\leq n}\\mathrm{Var}(X_{ij}\\mid|X_{i,j}|\\leq a)>0. \\]<\/p>\n

It was erroneously written<\/p>\n

\\[ \\sigma^2 :=\\min_{1\\leq i,j\\leq n}\\mathrm{Var}(X_{ij}\\mathbf{1}_{|X_{i,j}|\\leq a})>0. \\]<\/p>\n

Let us take this occasion for a back to basics about conditional variance<\/b> and variance of truncation<\/b>. Let \\( {X} \\) be a real random variable on \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) and \\( {A\\in\\mathcal{F}} \\) be an event. First the real number \\( {\\mathbb{E}(X\\mid A)=\\mathbb{E}(X\\mid\\mathbf{1}_A=1)} \\) is not the random variable \\( {\\mathbb{E}(X\\mid\\mathbf{1}_A)} \\). We have<\/p>\n

\\[ \\mathbb{E}(X\\mid\\mathbf{1}_A) =\\underbrace{\\frac{\\mathbb{E}(X\\mathbf{1}_A)}{\\mathbb{P}(A)}}_{\\mathbb{E}(X\\mid A)}\\mathbf{1}_A +\\underbrace{\\frac{\\mathbb{E}(X\\mathbf{1}_{A^c})}{\\mathbb{P}(A^c)}}_{\\mathbb{E}(X\\mid A^c)}\\mathbf{1}_{A^c}. \\]<\/p>\n

Note that this formula still makes sense when \\( {\\mathbb{P}(A)=0} \\) or \\( {\\mathbb{P}(A)=1} \\).<\/p>\n

The quantity \\( {\\mathbb{E}(X\\mid A)} \\) makes sense only if \\( {\\mathbb{P}(A)>0} \\), and in this case, the conditional variance of \\( {X} \\) given the event \\( {A} \\) is the real number given by<\/p>\n

\\[ \\begin{array}{rcl} \\mathrm{Var}(X\\mid A) &=&\\mathbb{E}((X-\\mathbb{E}(X\\mid A))^2\\mid A)\\\\ &=&\\mathbb{E}(X^2\\mid A)-\\mathbb{E}(X\\mid A)^2\\\\ &=&\\frac{\\mathbb{E}(X^2\\mathbf{1}_A)}{\\mathbb{P}(A)} -\\frac{\\mathbb{E}(X\\mathbf{1}_A)^2}{\\mathbb{P}(A)^2}\\\\ &=& \\frac{\\mathbb{E}(X^2\\mathbf{1}_A)\\mathbb{P}(A)-\\mathbb{E}(X\\mathbf{1}_A)^2}{\\mathbb{P}(A)^2}\\\\ &=&\\mathbb{E}_A(X^2)-\\mathbb{E}_A(X)^2=:\\mathrm{Var}_A(X) \\end{array} \\]<\/p>\n

where \\( {\\mathbb{E}_A} \\) is the expectation with respect to the probability measure with density \\( {\\mathbf{1}_A\/\\mathbb{P}(A)} \\) with respect to \\( {\\mathbb{P}} \\). In particular, by the Cauchy--Schwarz inequality,<\/p>\n

\\[ \\mathrm{Var}(X\\mid A) \\geq 0 \\]<\/p>\n

with equality if and only if \\( {X} \\) and \\( {\\mathbf{1}_A} \\) are colinear.<\/p>\n

Of course \\( {\\mathrm{Var}(X\\mid A)=0} \\) if \\( {X} \\) is constant. However \\( {\\mathrm{Var}(X\\mid A)} \\) may vanish for a non-constant \\( {X} \\). Indeed if \\( {A=\\{|X|\\leq a\\}} \\) and if \\( {X\\sim\\frac{1}{2}\\delta_{a\/2}+\\frac{1}{2}\\delta_{2a}} \\) then \\( {X\\mid A} \\) is constant and equal to \\( {a\/2} \\). In this example, since \\( {X\\mathbf{1}_A} \\) is not a constant, this shows also that one cannot lower bound \\( {\\mathrm{Var}(X\\mid A)} \\) with the variance of the truncation<\/p>\n

\\[ \\mathrm{Var}(X\\mathbf{1}_A)=\\mathbb{E}(X^2\\mathbf{1}_A)-\\mathbb{E}(X\\mathbf{1}_A)^2. \\]<\/p>\n

Another notable correction.<\/b> Myl\u00e8ne Ma\u00efda<\/a> pointed out to me on February 27 2018 that at the bottom of page 14, just before the statement<\/p>\n

\\[ \\sup_{z\\in C}|n\\varphi_{n,1}(\\sqrt{n}z)-\\pi^{-1}\\mathbf{1}_{[0,1]}(|z|)|=0 \\]<\/p>\n

the compact set \\( {C} \\) must be taken in \\( {\\{z\\in\\mathbb{C}:|z|\\neq1\\}} \\) and not on the whole complex plane \\( {\\mathbb{C}} \\). Indeed, when \\( {|z|=1} \\), \\( {n\\varphi_{n,1}(\\sqrt{n}z)} \\) tends as \\( {n\\rightarrow\\infty} \\) to \\( {1\/2} \\), and not to \\( {\\pi^{-1}} \\), see for instance this former post<\/a> for a one formula proof based on the central limit theorem for Poisson random variables. Anyway this is really not surprising since a sequence of continuous functions cannot converge uniformly to a discontinuous function.<\/p>\n

Yet another correction.<\/strong> Page 39 line 9 replace $M_\\mu(a)$ by $M_\\mu(q)$.<\/p>\n","protected":false},"excerpt":{"rendered":"

Sean O'Rourke pointed out on December 30, 2017 that a notation should be corrected in the statement of Lemma A.1 in the probability survey Around…<\/p>\n

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