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

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