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Around the circular law : erratum

O'Rourke Arms
O’Rourke Arms

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 the circular law (2012) that I wrote years ago in collaboration with Charles Bordenave.

Indeed the definition of \( {\sigma^2} \) should be corrected to

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

It was erroneously written

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

Let us take this occasion for a back to basics about conditional variance and variance of truncation. 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

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

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

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

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

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,

\[ \mathrm{Var}(X\mid A) \geq 0 \]

with equality if and only if \( {X} \) and \( {\mathbf{1}_A} \) are colinear.

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

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

Another notable correction. Mylène Maïda pointed out to me on February 27 2018 that at the bottom of page 14, just before the statement

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

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 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.

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