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Month: January 2018

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 σ2 should be corrected to

σ2:=min1i,jnVar(Xij|Xi,j|a)>0.

It was erroneously written

σ2:=min1i,jnVar(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 AF be an event. First the real number E(XA)=E(X1A=1) is not the random variable E(X1A). We have

E(X1A)=E(X1A)P(A)E(XA)1A+E(X1Ac)P(Ac)E(XAc)1Ac.

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

The quantity E(XA) 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(XA)=E((XE(XA))2A)=E(X2A)E(XA)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(XA)0

with equality if and only if X and 1A are colinear.

Of course Var(XA)=0 if X is constant. However Var(XA) may vanish for a non-constant X. Indeed if A={|X|a} and if X12δa/2+12δ2a then XA 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(XA) 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

supzC|nφn,1(nz)π11[0,1](|z|)|=0

the compact set C must be taken in {zC:|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 π1, 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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