# The Local Ledoit-Péché Law

Van Latimer  and  Benjamin D. Robinson

###### Abstract.

Ledoit and Péché, in [12], proved convergence of certain functions of a random
covariance matrix’s resolvent; we refer to this as the _Ledoit-Péché law_. One
important application of their result is shrinkage covariance estimation with
respect to so-called Minimum Variance (MV) loss, discussed in the work of
Ledoit and Wolf [13]. We provide an essentially optimal rate of convergence
and hypothesize it to be the smallest possible rate of excess MV loss within
the shrinkage class.

##  1\. Introduction

Let $X$ be an $M\times N$ matrix; we assume that $\frac{M}{N}$ converges to a
limit $\phi$ as both $M$ and $N$ tend to $\infty$ (although this may be
relaxed). Let

(1.1) $\Sigma=VDV^{*},\quad V=\begin{pmatrix}\mid&&\mid\\\
\mathbf{v}_{1}&\dots&\mathbf{v}_{M}\\\ \mid&&\mid\end{pmatrix},\quad
D=\operatorname{\mathrm{diag}}(\tau_{1},\dots,\tau_{M})$

with $\tau_{1}\geq\cdots\geq\tau_{M}$, be an $M\times M$ real symmetric or
complex Hermitian positive definite matrix together with its
eigendecomposition.

We will make the following assumption about the “training-data” matrix $X$.

######  Assumption 1.

$\Sigma$ is diagonal and the $M\times N$ matrix $X$ has i.i.d. columns
$\sim\mathcal{N}(0,\Sigma)$.

As is common in Random Matrix Theory, the dimensions $N$ and $M$ of $X$ and
$\Sigma$, and of most every other matrix we will study, are assumed to go to
infinity. Thus practically every major quantity of interest is a sequence of
quantities, and all properties that we desire to study are those which emerge
in the large dimensional limit. We therefore always, except perhaps when
special emphasis is needed, suppress the dependence of matrices and functions
thereof on the dimensions $N$, $M$, etc. We consider the sample covariance
matrix

(1.2) $S=\Sigma^{1/2}XX^{*}\Sigma^{1/2}$

We let

(1.3) $S=ULU^{*},\quad U=\begin{pmatrix}\mid&&\mid\\\
\mathbf{u}_{1}&\dots&\mathbf{u}_{M}\\\ \mid&&\mid\end{pmatrix},\quad
L=\operatorname{\mathrm{diag}}(\lambda_{1},\dots,\lambda_{M})$

with $\lambda_{1}\geq\cdots\geq\lambda_{M}$ be its spectral decomposition.

It is a problem of great theoretical and practical interest to understand how
the properties of the sample covariance matrix relate to properties of the
population covariance matrix $\Sigma$.

Random Matrix Theory has had great success in the last decade in getting very
fine control of random matrices $H$ by way of their resolvent

(1.4) $(H-zI)^{-1}$

This was first done by Marčenko and Pastur ([19]). Their approach was to show
that the trace of the resolvent of a random matrix approximately satisfies
some self-consistent equation and then to reason that trace of the resolvent,
which is also the Stieltjes transform of the emperical eigenvalue measure,
must be close to the true solution to the self-consistent equation. This has
remained a popular and powerful technique.

One finds that the resolvent $R_{M}:=R_{M}(z):=(S-zI)^{-1}$ of $S$ can be
written as

(1.5) $\sum_{i=1}^{M}\frac{1}{\lambda_{i}-z}\mathbf{u}_{i}\mathbf{u}_{i}^{*}$

Ledoit and Péché, in their paper [12], consider functions of the form

(1.6)
$\Theta(z):=\sum_{i=1}^{M}\sum_{j=1}^{M}\frac{1}{\lambda_{i}-z}\mathbf{u}_{i}\mathbf{u}_{i}^{*}g(\tau_{j})\mathbf{v}_{j}\mathbf{v}_{j}^{*}$

for some function $g:\mathbb{R}\to\mathbb{R}$ with finitely many
discontinuities, which amounts to a weighting of the spectral decomposition of
the resolvent; components of the resolvent in different eigendirections of the
population covariance matrix are weighted according to the value of $g$
applied to the associated eigenvalue of the population covariance matrix.
$\Theta$ may be simplified as follows:

(1.7) $\Theta(z)=\operatorname{\mathrm{Tr}}((S-zI)^{-1}g(\Sigma))$

Here we recall that for a function $g:\mathbb{R}\to\mathbb{R}$ and a diagonal
matrix $D$, we define a matrix $g(D)$ by

(1.8) $[g(D)]_{ij}:=g(D_{ij})\delta_{ij},$

and for a real symmetric or complex Hermitian matrix $A$ with spectral
decomposition $A=U^{*}DU$, we define

(1.9) $g(A):=U^{*}g(D)U$

The case $g\equiv 1$ is of course of interest. In this case we have

(1.10)
$\Theta(z)=\operatorname{\mathrm{Tr}}((S-zI)^{-1})=\mathcal{S}\left(\mu\right),\quad\mu:=\sum_{i=1}^{M}\delta_{\lambda_{i}}$

where for a measure $\sigma$ we define the Stieltjes transform
$\mathcal{S}(\sigma):\mathbb{H}\to\mathbb{H}$ by

(1.11) $\mathcal{S}(\sigma)(z):=\int\frac{d\sigma(x)}{x-z}$

and where by $\mathbb{H}$ we denote the complex upper half-plane

(1.12) $\mathbb{H}:=\set{z=E+i\eta\in\mathbb{C}\mid\eta>0}$

We note that $E$ and $\eta$ are usually the way we will denote the real and
imaginary parts of a complex argument to a Stieltjes transform.

The quantity (1.10) has been deeply understood for a fairly general class of
matrices, which we will detail soon.

We will also particularly interested in the case that $g$ is the identity, ie,
$g(x)\equiv x$. In this case, $\Theta$ has another simplification:

(1.13) $\Theta=\mathcal{S}\left(\nu\right)$

where

(1.14)
$\nu:=\sum_{i=1}^{M}\mathbf{u}_{i}^{*}\Sigma\mathbf{u}_{i}\delta_{\lambda_{i}}$

The reason this case of $g$ is of particular interest to us is that the
quantities

(1.15) $\mathbf{u}_{i}^{*}\Sigma\mathbf{u}_{i}$

are precisely the quantities which describe the Frobenius-norm optimal
rotation equivariant shrinkage estimator for the population covariance matrix,
as shown in [12]. Another result of the same paper was bounding $\nu$ close to
a deterministic measure was one result of, although they did not provide rates
of convergence. In this paper we improve their result by providing essentially
optimal rate of convergence.

######  Definition 1.1 (Shrinkage Estimators and Loss Function).

Given a realization of the sample covariance matrix $S$, we define a (rotation
equivariant) shrinkage estimator $\hat{\Sigma}$ for the population covariance
matrix $\Sigma$ via

(1.16) $\hat{\Sigma}=U\hat{D}U^{*}$

for some diagonal matrix $\hat{D}$. That is, we estimate the $\Sigma$ from $S$
by keeping the eigenvectors and changing the eigenvalues, presumably
“shrinking” them since $S$ has the tendency to “spread out” the eigenvalues of
$\Sigma$.

To measure the success of $\hat{\Sigma}$ we define the loss function

(1.17)
$\mathcal{L}^{MV}(\hat{\Sigma},\Sigma):=\frac{\operatorname{\mathrm{Tr}}(\hat{\Sigma}^{-1}\Sigma\hat{\Sigma}^{-1})/N}{\left[\operatorname{\mathrm{Tr}}\left(\hat{\Sigma}^{-1}\right)/N\right]^{2}}-\frac{1}{\operatorname{\mathrm{Tr}}\left(\Sigma^{-1}\right)/N}$

Here $MV$ stands for minimum variance, and “$\mathcal{L}^{MV}$ represents the
_true_ variance of the linear combination of the original variables that has
the minimum _estimated_ variance.” (See [13] for the quote and for more
discussion of the suitability of this loss function.)

######  Lemma 1.2.

With respect to $\mathcal{L}^{MV}$, the optimal shrinkage estimator is given
when $\hat{D}_{ii}=\beta\mathbf{u}_{i}^{*}\Sigma\mathbf{u}$, where $\beta\neq
0$ is a scaling constant that we will take to be 1.

######  Definition 1.3.

We define

(1.18)
$D^{\mathrm{or}}:=\mathrm{diag}(\mathbf{u}_{i}^{*}\Sigma\mathbf{u}_{i})_{i},\quad\Sigma^{\mathrm{or}}:=UD^{\mathrm{or}}U^{*}$

and we call $\Sigma^{\mathrm{or}}$ the _shrinkage oracle_.

######  Remark 1.4.

As noted above, the same choice of $\hat{D}$ is optimal for a Frobenius norm
loss function [12].

The optimal shrunken eigenvalues $\mathbf{u}_{i}^{*}\Sigma\mathbf{u}_{i}$ are
experimentally unavailable to us, so for statistical purposes Lemma 1.2 is of
limited use to us. However, just as was done for $\mu$, in [2], namely,
bounding it optimally close to a determinstic limit, we will do for $\nu$.

Given our random matrix with independent entries $X$ and our population
covariance matrix $\Sigma$, we define the resolvent, or Green function
introduced in [2]:

(1.19) $G(z):=\begin{pmatrix}-\Sigma^{-1}&X\\\ X^{*}&-zI\end{pmatrix}$

The reader accustomed to Random Matrix Theory will note that this is not the
usual defintion of the resolvent. It is however an important realization made
in [2] that the more familiar resolvent

(1.20) $R_{M}:=(S-zI)^{-1}$

can be neatly gotten from $G$, as well as the related resolvent

(1.21) $R_{N}:=(X^{*}\Sigma X-zI)^{-1}$

and that this single matrix containing both resolvents unlocks powerful tools
for studying resolvent estimates developed in the context of Wigner matrices.
In fact, $G$ may be decomposed in block form as

(1.22) $\begin{pmatrix}z\Sigma^{1/2}R_{M}\Sigma^{1/2}&\cdot\\\
\cdot&R_{N}\end{pmatrix}$

where the blocks labeled $\cdot$ are not of interest to us currently. The
conjugation of $R_{M}$ by $\Sigma^{1/2}$ is not of great importance to the
authors of [2], but it is very fortunate for us, the reason being that
$\Theta$, our object of greatest interest, is precisely

(1.23)
$\Theta=\operatorname{\mathrm{Tr}}(R_{M}\Sigma)=\operatorname{\mathrm{Tr}}(\Sigma^{1/2}R_{M}\Sigma^{1/2})$

by the invariance of $\operatorname{\mathrm{Tr}}$ under cyclical permutation.
Let us make a few more definitions and then quote a result of [2].

We define the the population spectral measure, or _PSM_ , of $\Sigma$ by

(1.24) $\pi=\frac{1}{M}\sum_{j=1}^{M}\delta_{\tau_{i}}$

This is of course just the probability measure which places equal weight at
each of $\Sigma$’s eigenvalues, counted with multiplicity.

We also define the following notation of size for random variables, introduced
in [20], which has proven very helpful for formulating results in RMT.

######  Definition 1.5 (Stochastic Domination).

Given two sequences of random variables $X:=\\{X_{N}\\}_{N\in\mathbb{N}}$ and
$Y:=\\{Y_{N}\\}_{N\in\mathbb{N}}$ (note that we again suppress that quantities
of interest are sequence in $N$), we say that $Y$ _stochastically dominates_
$X$, or that $X\prec Y$, if for any (small) $\epsilon>0$, (large) $D>0$, and
sufficiently large $N$, we have

(1.25) $P(X>N^{\epsilon}Y)<N^{-D}$

A little more notation: we define the important function
$m:\mathbb{H}\to\mathbb{H}$ as the unique such value solving

(1.26) $\frac{1}{m}=-z+\phi\int\frac{x}{1+mx}\mathrm{d}\pi(x)$

for $z\in\mathbb{H}$.

We will list some things that we know about $m$.

  * •

The equation (1.26), which we have taken as the definition of $m$, is the one
which allows us to make the connection between the different definitions of
$\delta$ in the papers [12] and [2].

  * •

$m$ is also the unique solution to $f(m)=z$, where

(1.27) $f(x)=-\frac{1}{x}+\phi\sum_{\tau_{i}\neq 0}\frac{1}{x+\tau_{i}^{-1}}.$

  * •

$\lim_{\mathbb{H}\ni z\to E}m(z):=\check{m}(E)$ exists and is given by

(1.28) $\check{m}(E)=\pi\left(\mathcal{H}w(E)+iw(E)\right)$

where $w:=\frac{d\varrho}{d\lambda}$ is the Radon-Nikodym derivative of
$\varrho$ with respect to $\lambda$, and where $\mathcal{H}$ is the Hilbert
transform (see [13]). We mention Hilbert transforms because [13] presents it
this way and explains how the presence of the Hilbert transform provides a
theoretical explanation for the phenomenon of eigenvalue shrinkage, but we
will not heavily use the Hilbert transform in our treatment; we will use a
equation from [2] which $m$ satisfies to get control of $m$’s real and
imaginary parts directly.

At this time, let us also define the shrinkage function

(1.29) $\delta(x):=\frac{x}{[\pi cxw(x)]^{2}+[1-c-\pi cx\mathcal{H}w(x)]^{2}}$

This function appeared first in a slightly different form in [12] and then in
its stated form in [13]; in both cases it is useful to us as an approximator
to the values $\mathbf{u}_{i}^{*}\Sigma\mathbf{u}_{i}$ which describe the
optimal shrinkage estimator.

######  Remark 1.6.

We note that there is a small discrepancy between our definition of $m$ and
the definition of $m$ in the context of [12] for $M/N\gtrsim 1+\epsilon$.
However, this discrepancy only amounts to how the limiting empirical spectral
measure weights 0, and thus can be easily accounted for.

######  Theorem 1.7 (Informal Statement of [2]’s main result).

Define the matrix

(1.30) $\Pi:=\Pi(z):=\begin{pmatrix}-\Sigma(I+m(z)\Sigma)^{-1}&0\\\
0&m(z)I\end{pmatrix}$

Then element by element, $G$ is very close to $\Pi$.

######  Theorem 1.8 (Slightly more Formal Statement of part of Theorem 1.7).

If $\Sigma$’s population spectral measure $\pi$ satisfies some mild regularity
constraints, then

(1.31)
$\frac{1}{N}\operatorname{\mathrm{Tr}}(R_{N}(z))-m(z)=O_{\prec}\left((N\eta)^{-1}\right)$

This result is essentially optimal (up to the definitions in $\prec$) and
cannot be gotten naively. The paper also provides essentially optimal bounds
on individual resolvent elements; individual diagonal entries of $R_{N}$ are
themselves close to $m$, but the difference is of an order $(N\eta)^{-1/2}$ in
the bulk spectrum; this means that in averaging the diagonal elements to get
the normalized trace, there is a fair bit of cancellation between different
diagonal elements, as there is between independent random variables. The task
of finding the “parts” of the random variables $(R_{N})_{ii}$ which are
independent to one another and thus provide this cancellation is the content
of a “Fluctuating Averaging Lemma” in random matrix theory.

A corollary of this result is the “Marchenko-Pastur law on small scales”

######  Corollary 1.8.1.

For any interval $I\subseteq\mathbb{R}$, we have

(1.32) $\mu(I)=\varrho(I)+O_{\prec}\left(N^{-1}\right)$

This corollary is important in that it captures the fact that statements about
Stieltjes transforms of measures, which we have in great strength thanks to
the techniques of [2], can be translated into statements about the measures
themselves. This statement captures the fact that empirical eigenvalue
distribution is given very accurately by a certain deterministic distribution,
even on very fine scales: it says “even in intervals which are predicted to
contain only $N^{\epsilon}$ eigenvalues of $S$ according the to deterministic
measure $\varrho$, we do have that the prediction is correct to leading order
with very high probability.”

What we will first do in this note is adapt the proof of 1.8 to deal also with
the quantity

(1.33)
$\frac{1}{M}\operatorname{\mathrm{Tr}}\left(zR_{M}\Sigma+\Sigma(I+m(z)\Sigma)^{-1})\right)$

which leads us to our first main result:

######  Theorem 1.9.

If $\Sigma$ satisfies the same regularity conditions as required for Theorem
1.8, then we have

(1.34)
$\frac{1}{M}\operatorname{\mathrm{Tr}}\left(zR_{M}\Sigma+\Sigma(I+m(z)\Sigma)^{-1}\right)=O_{\prec}\left((N\eta)^{-1}\right)$

Using equation (1.26), we may rewrite the limit
$M^{-1}\operatorname{\mathrm{Tr}}\left(-\Sigma(I+m(z)\Sigma)^{-1}\right)$ as

(1.35) $-\phi^{-1}\left(\frac{1}{zm}+1\right)$

Just as Marchenko-Pastur Law at small scales followed from Theorem 1.8, so
does our second main result. First we observe that [12, Theorem 4] is
equivalent to saying that the function $\delta$ from (1.29) is the Radon-
Nikodym derivative of the limiting measure for $\nu$ against the deformed
Marchenko Pastur law. Our second main result adds a rate of convergence to
this limiting behavior, as follows:

######  Corollary 1.9.1.

We have for any interval $I\subseteq\mathbb{R}$ that

(1.36)
$\mathrm{d}\nu(I)-\delta\mathrm{d}\varrho(I)=O_{\prec}\left(N^{-1}\right).$

If $\tilde{\Sigma}$ is the shrinkage estimator
$\sum_{i=1}^{M}\delta(\lambda_{i})\mathbf{u}_{i}\mathbf{u}_{i}^{*}$, then the
above implies an order of error between $\tilde{\Sigma}$ and
$\Sigma^{\text{or}}$:

(1.37)
$\left|\mathrm{tr}\left(\Sigma^{\text{or}}-\tilde{\Sigma}\right)\right|\prec
N^{-1}.$

Boundedness and continuity of $1/\delta$, together with the Portmanteau
theorem, can then be used to show:

(1.38)
$\mathcal{L}^{\text{MV}}(\Sigma^{\text{or}},\Sigma)-\mathcal{L}^{\text{MV}}(\tilde{\Sigma},\Sigma)=O_{\prec}(N^{-1}).$

Further, we hypothesize that no _bona fide_ shrinkage estimator can make this
error asymptotically smaller, in which case this would be the smallest
possible excess MV loss within the shrinkage class, as claimed in the
abstract.

##  2\. Relation to Previous Works

This paper is, firstly, a direct successor to the papers [12] and [13] which
advances a program established therein using recent advances in RMT. Another
important connection is to the papers [18] and [17], which discuss a measure
which is related to ours: for a fixed unit vector $\mathbf{x}$, they study

(2.1)
$F_{Q_{1},\mathbf{x}}:=\sum_{i=1}^{M}\left\lvert\mathbf{u}_{i}^{*}\mathbf{x}\right\rvert^{2}\delta_{\lambda_{i}}$

As in our context, they prove that this measure is close to a deterministic
limit, which [17] calls $F_{1c,\mathbf{x}}$ (up to adjusting from their
context to ours). In particular, they prove

(2.2) $F_{Q_{1},\mathbf{x}}(I)-F_{1c,\mathbf{x}}(I)\prec M^{-1/2}$

The earlier paper [18] establishes the optimality of the factor $M^{-1/2}$ by
remarkably establishing the joint asymptotic Gaussian distribution of any $k$
different analytic functions integrated against $F_{Q_{1},\mathbf{x}}$ (one
way that [17] differs from or improves on [18] is in the very high probability
with which the error bounds hold).

We can recover our measure $\mu$ from $F_{Q_{1},\mathbf{x}}$: indeed, if
$\\{\mathbf{v}_{1},\dots,\mathbf{v}_{M}\\}$ are the eigenvectors of $\Sigma$,
then

(2.3) $\mu=M^{-1}\sum_{j=1}^{M}\pi_{j}F_{Q_{1},\mathbf{v}_{j}}$

Similiarly, the limiting deterministic measures satisfy

(2.4)
$\delta\mathrm{d}\varrho=M^{-1}\sum_{j=1}^{M}\pi_{j}F_{1c,\mathbf{v}_{j}}$

So our main result, with the error weakened from
$O_{\prec}\left(N^{-1}\right)$ to $O_{\prec}\left(N^{-1/2}\right)$, is a
consequence of the results of [17].

The improvement by a factor of $M^{-1/2}$ that occurs after averaging is
exactly reminiscent of the central limit theorem, which hints at a sort of
independence between the meaures $F_{Q_{1},\mathbf{v}_{j}}$ and
$F_{Q_{1},\mathbf{v}_{j^{\prime}}}$. Also note that this improvement of
$M^{-1/2}$ does not reflect an improvement of our work over theirs; the error
bound of $O_{\prec}\left(N^{-1/2}\right)$ gotten in [17] is optimal, and it is
only after the averaging over many different deterministic vectors
$\mathbf{x}$ that one sees the the improvement. Thus, our work is to their
work as an averaged local law is to an entry-wise local law.

Lastly, one should note that an ultimate goal of the program investigated in
[17] is to establish the convergence of the CDF of $F_{Q_{1},\mathbf{x}}$ to a
Brownian bridge, which would amount to finding some “internal” independence
inside $F_{Q_{1},\mathbf{x}}$ in the form of independence between the
quantities $\left\langle\mathbf{u}_{i},\mathbf{x}\right\rangle$. The
“external” indepdence that we have hinted at between the measures
$F_{Q_{1},\mathbf{v}_{j}}$ and $F_{Q_{1},\mathbf{v}_{j^{\prime}}}$ is not
unrelated to this.

The connections between the papers [18], [17] and ours are perhaps deeper than
we have realized; future work will hopefully bring further connections to
light.

##  3\. Tools

First, we extend the definition of matrix multiplication to matrices indexed
by arbitrary sets.

######  Definition 3.1.

Let $\mathcal{I}_{i}$ be a finite set for $i\in\\{1,2,3,4\\}$. Let $A$ be a
$\mathcal{I}_{1}\times\mathcal{I}_{2}$ matrix, and let $B$ be a
$\mathcal{I}_{3}\times\mathcal{I}_{4}$ matrix. We define the matrix product
$AB$ to be a $\mathcal{I}_{1}\times\mathcal{I}_{4}$ matrix satisfying

(3.1) $(AB)_{ij}=\sum_{k\in\mathcal{I}_{2}\cap\mathcal{I}_{3}}A_{ik}B_{kj}$

######  Definition 3.2.

Let $A$ be an invertible matrix $\mathcal{J}\times\mathcal{J}$ matrix and let
$S$ be its inverse. We define the minor $S^{(i)}$ via

(3.2) $S^{(i)}=\left((A_{jk}:j,k\in\mathcal{J}\setminus\\{i\\})\right)^{-1}$

######  Lemma 3.3 (Resolvent Identities).

Let $A$ be an invertible $\mathcal{J}\times\mathcal{J}$ matrix and $S=A^{-1}$.

  1. (1)

If $j,k\in\mathcal{J}\setminus\\{i\\}$,

(3.3) $S^{(i)}_{jk}=S_{jk}-\frac{S_{ji}S_{ik}}{S_{ii}}$

  2. (2)

If $i\neq j$,

(3.4) $S_{ij}=-S_{ii}\left(AS^{(i)}\right)_{ij}$

  3. (3)

We have

(3.5) $\frac{1}{S_{ii}}=S_{ii}-(A^{*}S^{(i)}A)_{ii}$

###### Proof.

  1. (1)

By the definition of an the inverse, it suffices to show that

(3.6)
$\sum_{l\in\mathcal{J}\setminus\\{i\\}}\left(S_{jl}-\frac{S_{ji}S_{il}}{S_{ii}}\right)A_{lk}=\delta_{jk}$

But the left-hand side indeed yields

(3.7)
$\sum_{l\in\mathcal{J}\setminus\\{i\\}}S_{jl}A_{lk}-\frac{S_{ji}}{S_{ii}}\sum_{l\in\mathcal{J}\setminus\\{i\\}}S_{il}A_{lk}=(\delta_{jk}-S_{ji}A_{ik})-\frac{S_{ji}}{S_{ii}}(\delta_{ik}-S_{ii}A_{ik})=\delta_{jk}$

since we have assumed $k\neq i$.

  2. (2)

This proof is taken from [16]. Using part $(1)$, we get

(3.8)
$\begin{split}\left(AS^{(i)}\right)_{ij}=\sum_{k\in\mathcal{J}\setminus\\{i\\}}A_{ik}S^{(i)}_{kj}=\sum_{k\in\mathcal{J}\setminus\\{i\\}}A_{ik}\left(S_{kj}-\frac{S_{ki}S_{ij}}{S_{ii}}\right)\\\
=-A_{ii}S_{ij}-\frac{S_{ij}}{S_{ii}}(1-A_{ii}S_{ii})=-\frac{S_{ij}}{S_{ii}}\end{split}$

as desired.

  3. (3)

This is an immediate consequence of Schur’s complement formula, wherein if

(3.9) $M=\begin{pmatrix}B&C\\\ D&E\end{pmatrix}$

then

(3.10) $M^{-1}=\begin{pmatrix}(B-CE^{-1}D)^{-1}&*\\\ *&*\end{pmatrix}$

provided all the inverses exist. To see how Schur’s complement formula
applies, it is helpful to write

(3.11) $(A^{*}S^{(i)}A)_{ii}=(e_{i}^{*}A^{*})S^{(i)}(Ae_{i})$

It is of course crucial that we use the correct definition of matrix
multiplication here.

∎

Next we apply these general matrix algebra facts to our specific resolvent
$G$.

######  Corollary 3.3.1 (Resolvent Identities for $G$).

  1. (1)

For $s,t\neq r$ we have

(3.12) $G^{(r)}_{st}=G_{st}-\frac{G_{sr}G_{rt}}{G_{rr}}$

  2. (2)

If $\mu\neq\nu\in\mathcal{I}_{N}$, then

(3.13)
$G_{\mu\nu}=-G_{\mu\mu}(X^{*}G^{(\mu)})_{\mu\nu}=-G_{\nu\nu}(G^{(\nu)}X)_{\mu\nu}$

Similarly if $i\neq j\in\mathcal{I}_{M}$, then

(3.14) $G_{ij}=-G_{ii}(XG^{(i)})_{ij}=-G_{jj}(G^{(j)}X^{*})_{ij}$

Lastly if $i\in\mathcal{I}_{M}$ and $\mu\in\mathcal{I}_{N}$, then

(3.15) $G_{i\mu}=-G_{\mu\mu}(G^{(\mu)}X)_{i\mu},\quad G_{\mu
i}-G_{\mu\mu}(X^{*}G^{(\mu)})_{\mu i}$

  3. (3)

We have

(3.16) $\frac{1}{G_{\mu\mu}}=-z-(X^{*}G^{(\mu)}X)_{\mu\mu}$

###### Proof.

These all follow from the lemma; only the correct definition of matrix
multiplication must be used, and one should note that many of the lemmas’
conclusions are insensitive to the diagonal entries of $A$, so that one may
see a

$\begin{pmatrix}-\Sigma^{-1}&X\\\ X^{*}&0\end{pmatrix}$

when one expects to see a

$\begin{pmatrix}-\Sigma^{-1}&X\\\ X^{*}&-zI\end{pmatrix}$

∎

Let us cite one of the main results of [2]

######  Theorem 3.4 (Entrywise Local Law).

For any deterministic unit vectors
$\mathbf{v},\mathbf{w}\in\mathbb{R}^{\mathcal{I}}$, one has

(3.17) $\left\langle\mathbf{v},(G-\Pi)\mathbf{w}\right\rangle=O_{\prec}(\Psi)$

where

(3.18)
$\Psi:=\Psi(z)=\sqrt{\frac{\operatorname{\mathrm{Im}}m(z)}{N\eta}}+\frac{1}{N\eta}$

##  4\. Proof of Theorem 1.9

First, we prove a partial result. We do not include many details, and only
show how section 5 of [2] may be quickly adapted to our setting.

######  Lemma 4.1.

If $\Sigma$ is diagonal, then Theorem 1.9 holds.

###### Proof.

This proof amounts to adapting section 5 of [2] to address the top-left corner
of thei matrix $G$. We use the notation of [2] without further comment.

Equation (5.15) of [2] reads

(4.1)
$1(\Xi)G_{ii}=1(\Xi)\frac{-\sigma_{i}}{1+m_{N}\sigma_{i}+\sigma_{i}Z_{i}+O_{\prec}\left(\sigma_{i}\Psi_{\Theta}^{2}\right)}$

A Taylor expansion on this quantity, justified because
$Z_{i}=O_{\prec}\left(\Psi_{\Theta}\right)$ by Lemma 5.2 of [2], yields

(4.2)
$1(\Xi)G_{ii}=1(\Xi)\left[\frac{-\sigma_{i}}{1+m_{N}\sigma_{i}}+\frac{\sigma_{i}}{(1+m_{N}\sigma_{i})^{2}}Z_{i}+O_{\prec}\left(\sigma_{i}\Psi_{\Theta}^{2}\right)\right]$

Averaging now over $i$ yields

(4.3)
$1(\Xi)\frac{1}{M}\sum_{i}G_{ii}=1(\Xi)\frac{1}{M}\sum_{i}\frac{-\sigma_{i}}{1+m_{N}\sigma_{i}}+[Z]_{M}+O_{\prec}\left(\Psi_{\Theta}^{2}\right)$

Using that $(1-\mathbb{E}_{i})\frac{1}{G_{ii}}=Z_{i}$, we have by lemma 5.6 of
[2] that

(4.4) $[Z]_{M}=O_{\prec}\left(\Psi_{\Theta}^{2}\right)$

Using equation (3.10) of [2] to bound $\Theta$, we have
$\Psi_{\Theta}\prec\frac{1}{\sqrt{N\eta}}$, so that our equation (4.4) becomes

(4.5) $[Z]_{M}=O_{\prec}\left((N\eta)^{-1}\right)$

Equation (3.1) of [2] also yields that $\Xi$ holds with high probability, so
that equation (4.3) yields

(4.6)
$\frac{1}{M}\sum_{i}G_{ii}=\frac{1}{M}\sum_{i}\frac{-\sigma_{i}}{1+m_{N}\sigma_{i}}+O_{\prec}\left((N\eta)^{-1}\right)$

Furthermore, it is a result of section 5 of [2] that

(4.7) $\left\lvert m_{N}-m\right\rvert=O_{\prec}\left((N\eta)^{-1}\right)$

so that $m_{N}$ may be replaced with $m$ (it is noted that $\left\lvert
1+m(z)\sigma_{i}\right\rvert\geq\tau$ under our regularity assumptions).
Finally, $\frac{1}{M}\sum_{i}\frac{-\sigma_{i}}{1+m_{N}\sigma_{i}}$ is
precisely $\frac{1}{M}\sum_{i}\Pi_{ii}$, and

$\sum_{i}G_{ii}=\operatorname{\mathrm{Tr}}z\Sigma^{1/2}R_{M}\Sigma^{1/2}=z\operatorname{\mathrm{Tr}}R_{M}\Sigma$

so that we are done.

∎

######  Remark 4.2.

In the recent paper [17], a very similar result to our 4.1 is proven in their
equation (3.13), under much more general moment assumptions on $X$; however,
the resolvent used in that paper differs slightly from the one used in this
paper. We hope to use some of their techniques perhaps to weaken some of the
moment assumptions in our work.

## Acknowledgements

This work was supported by the US Air Force Office of Scientific Research, Lab
Task number 19RYCOR036. The views and opinions of this paper do not
necessarily reflect the official positions of the Air Force. The public
affairs approval number of this document is AFRL-2021-2586.

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##  Appendix A Committees

Local Organizing Committee

  * •

Amber Boehnlein (Jefferson Lab)

  * •

Latifa Elouadrhiri (Jefferson Lab)

  * •

Robert McKeown (Jefferson Lab)

  * •

Yves Roblin (Jefferson Lab)

Advisory Committee

  * •

Robert McKeown, Jefferson Lab

  * •

Peter Alonzi, University of Virginia

  * •

Oliver Baker, Yale University

  * •

Jason Detwiler, University of Washington

  * •

Carl Gagliardi, Texas A&M University

  * •

Sean Liddick, Michigan State University

  * •

Amy Lovell, Los Alamos National Lab

  * •

Bronson Messer, Oak Ridge National Lab

  * •

Curtis Meyer, Carnegie Mellon University

  * •

Zein-Eddine Meziani, Argonne National Lab

  * •

Jeff Nichols, Oak Ridge National Lab

  * •

Alan Poon, Lawrence Berkeley Lab

  * •

Alessandro Roggero, University of Washington

  * •

Phiala Shanahan, Massachusetts Institute of Technology

  * •

Torre Wenaus, Brookhaven National Lab

  * •

Ying Wu, Duke University

##  Appendix B Working Groups and Conveners

WG1. Lattice QCD and Other Quantum Field Theories

Paulo Bedaque (University of Maryland) and Kostas Orginos
(William&Mary/Jefferson Lab)

WG2. Nuclear Structure Theory

Witold Nazarewicz (Michigan State University) and Dean Lee (Michigan State
University)

WG3. Accelerator Science and Operations

Steven Lidia (Michigan State University) and Malachi Schram (Pacific Northwest
National Lab)

WG4. Experimental Methods

Michelle Kuchera (Davidson College) and Mario Cromaz (LBNL)

WG5. Event Generation and Simulation

Markus Diefenthaler (Jefferson Lab)

WG6. Bayesian Inference for Quantum Correlation Functions

Wally Melnitchouk (Jefferson Lab)