
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 σ2 should be corrected to
σ2:=min1≤i,j≤nVar(Xij∣|Xi,j|≤a)>0.
It was erroneously written
σ2:=min1≤i,j≤nVar(Xij1|Xi,j|≤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 (Ω,F,P) and A∈F be an event. First the real number E(X∣A)=E(X∣1A=1) is not the random variable E(X∣1A). We have
E(X∣1A)=E(X1A)P(A)⏟E(X∣A)1A+E(X1Ac)P(Ac)⏟E(X∣Ac)1Ac.
Note that this formula still makes sense when P(A)=0 or P(A)=1.
The quantity E(X∣A) makes sense only if P(A)>0, and in this case, the conditional variance of X given the event A is the real number given by
Var(X∣A)=E((X−E(X∣A))2∣A)=E(X2∣A)−E(X∣A)2=E(X21A)P(A)−E(X1A)2P(A)2=E(X21A)P(A)−E(X1A)2P(A)2=EA(X2)−EA(X)2=:VarA(X)
where EA is the expectation with respect to the probability measure with density 1A/P(A) with respect to P. In particular, by the Cauchy--Schwarz inequality,
Var(X∣A)≥0
with equality if and only if X and 1A are colinear.
Of course Var(X∣A)=0 if X is constant. However Var(X∣A) may vanish for a non-constant X. Indeed if A={|X|≤a} and if X∼12δa/2+12δ2a then X∣A is constant and equal to a/2. In this example, since X1A is not a constant, this shows also that one cannot lower bound Var(X∣A) with the variance of the truncation
Var(X1A)=E(X21A)−E(X1A)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
supz∈C|nφn,1(√nz)−π−11[0,1](|z|)|=0
the compact set C must be taken in {z∈C:|z|≠1} and not on the whole complex plane C. Indeed, when |z|=1, nφn,1(√nz) tends as n→∞ to 1/2, and not to π−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.
Yet another correction. Page 39 line 9 replace Mμ(a) by Mμ(q).
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