Large deviations of the extreme eigenvalues of random deformations of matrices

Introduction

In the last twenty years, many features of the asymptotics of the spectrum of large random matrices have been understood. For a wide variety of classical models of random matrices (the canonical examples hereafter will be Wigner matrices , or Wishart matrices ), it has been shown that the spectral measure converges almost surely. The extreme eigenvalues converge for most of these models to the boundaries of the limiting spectral measure (see e.g. or ). Fluctuations of the spectral measure and the extreme eigenvalues of these models could also be studied under a fair generality over the entries of the matrices; we refer to and , or and for reviews. Recently, even the fluctuations of the eigenvalues inside the bulk could be studied for rather general entries and were shown to be universal (see e.g. or ). Concentration of measure phenomenon and moderate deviations could also be established in .

Yet, the understanding of the large deviations of the spectrum of large random matrices is still very scarce and exists only in very specific cases. Indeed, the spectrum of a matrix is a very complicated function of the entries, so that usual large deviation theorems, mainly based on independence, do not apply. Moreover, large deviations rate functions have to depend on the distribution of the entries and only guessing their definition is still a widely open question. In the case of Gaussian Wigner matrices, where the joint law of the eigenvalues is simply given by a Coulomb gas Gibbs measure, things are much easier and a full large deviation principle for the law of the spectral measure of such matrices was proved in . This extends to other ensembles distributed according to similar Gibbs measure, for instance Gaussian Wishart matrices . Similar large deviation results hold in discrete situations with a Coulomb gas distribution . A large deviation principle was also established in for the law of the spectral measure of a random matrix given as the sum of a self-adjoint Gaussian Wigner random matrix and a deterministic self-adjoint matrix (or as a Gaussian Wishart matrix with non trivial covariance matrix). In this case, the proof uses stochastic analysis and Dyson’s Brownian motion, as there is no explicit joint law for the eigenvalues, but again relies heavily on the fact that the random matrix has Gaussian entries. The large deviations for the law of the extreme eigenvalues were studied in a slightly more general setting. Again relying on the explicit joint law of the eigenvalues, a large deviation principle was derived in for the same Gaussian type models. The large deviations of extreme eigenvalues of Gaussian Wishart matrices were studied in . In the case where the Wishart matrix is of the form $XX^{*}$ with $X$ a $n\times r$ rectangular matrix so that the ratio $r/n$ of its dimensions goes to zero, large deviations bounds for the extreme eigenvalues could be derived under more general assumptions on the entries in . Our approaches allow also to obtain a full large deviation for the spectrum of such Wishart matrices when $r$ is kept fixed while $n$ goes to infinity (see Section 7).

In this article, we shall be concerned with the effect of finite rank deformations on the deviations of the extreme eigenvalues of random matrices. In fact, using Weyl’s interlacing property, it is easy to check that such finite rank perturbations do not change the deviations of the spectral measure. But it strongly affects the behavior of a few extreme eigenvalues, not only at the level of deviations but also as far as convergence and fluctuations are concerned. In the case of Gaussian Wishart matrices, the asymptotics of these extreme eigenvalues were established in and a sharp phase transition, known as the BBP transition, was exhibited. According to the strength of the perturbation, the extreme eigenvalues converge to the edge of the bulk or away from the bulk. The fluctuations of these eigenvalues were also shown in to be given either by the Tracy-Widom distribution in the first case, or by the Gaussian distribution in the second case. Universality (and non-universality) of the fluctuations in BBP transition was studied for various models, see e.g. .

Our approach is based, as in , on the characterization of the eigenvalues via the determinant of a matrix with fixed size : it is an $r\times r$ matrix whose entries are the Stieltjes transforms of the non-deformed matrix evaluated along the random vectors of the perturbation. We obtain a large deviation principle for the law of this characteristic polynomial (seen as a continuous function outside of the spectrum of the deterministic matrix) by classical large deviation techniques. Even though the application which associate to a function its zeroes is not continuous for the weak topology, we deduce from the latter a large deviation principle for the law of the zeroes of this characteristic polynomial, that is the extreme eigenvalues of the deformed matrix model.

Statement of the results

Let $X_{n}$ be a real diagonal matrix of size $n\times n$ with eigenvalues $\lambda_{1}^{n}\geq\lambda_{2}^{n}\geq\ldots\geq\lambda_{n}^{n}.$

We perturb $X_{n}$ by a random matrix whose rank does not depend on $n$. More precisely, let $m,r$ be fixed positive integers and $\theta_{1}\geq\theta_{2}\geq\ldots\geq\theta_{m}>0>\theta_{m+1}\geq\ldots\geq\theta_{r}$ be fixed, let $G=(g_{1},\ldots,g_{r})$ be a random vector and $(G(k)=(g_{1}(k),\ldots,g_{r}(k)))_{k\geq 1}$ be independent copies of $G.$ We then define the $r$ vectors with dimension $n$

and study the eigenvalues $\widetilde{\lambda}_{1}^{n}\geq\cdots\geq\widetilde{\lambda}_{n}^{n}$ of the deformed matrices

In the sequel, we will refer to the model (1) as the i.i.d. perturbation model.

and refer in the sequel to the model (2) as the orthonormalized perturbation model.

If $g_{1},\ldots,g_{r}$ are $r$ independent standard (real or complex) Gaussian variables, it is well known that the law of $(U_{i}^{n})_{1\leq i\leq r}$ is the uniform measure on the set of $r$ orthonormal vectors. The model (2) coincides then with the one introduced in .

Our goal will be to examine the large deviations for the $m$ largest eigenvalues of the deformed matrix $\widetilde{X_{n}},$ with $m$ the number of positive eigenvalues of the random deformation.

2. The assumptions

Concerning the spectral measure of the full rank deterministic matrix $X_{n},$ we assume the following

The empirical distribution $\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}^{n}}$ of $X_{n}$ converges weakly as $n$ goes to infinity to a compactly supported probability $\mu$.

Concerning the random vector $G$, we make the following assumption. It allows to claim that with probability one, the column vectors $G_{1}^{n},\ldots,G_{r}^{n}$ are linearly independent and is technically needed in the proof of Lemma 11.1. It is also the reason why we say that the column vectors $G_{1}^{n},\ldots,G_{r}^{n}$ or $U_{1}^{n},\ldots,U_{r}^{n}$ are delocalized with respect to the eigenvectors of $X_{n}$. Indeed, the eigenvectors of $X_{n}$ are the vectors of the canonical basis, whereas we know that with probability one, none of the entries of the $G_{i}^{n}$’s (or of the $U_{i}^{n}$’s) is zero. The i.i.d. feature of the $G(k)$’s allows even to assert that all entries of each $G_{i}^{n}$’s (or of the $U_{i}^{n}$’s) have the same distribution.

The law of $G$ could also depend on $n$ provided it satisfies the above hypothesis uniformly on $n$ and converges in law as $n$ goes to infinity.

We consider two distinct kind of assumptions on the extreme eigenvalues of $X_{n}.$ We will be first interested in the case when these extreme eigenvalues stick to the bulk (see Assumption 2.3), and then to the case with outliers, when we allow some eigenvalues of $X_{n}$ to take their limit outside the support of the limiting measure $\mu$ (see Assumption 2.5).

3. The results in the case without outliers

We first consider the case where the extreme eigenvalues of $X_{n}$ stick to the bulk.

The largest and smallest eigenvalues of $X_{n}$ tend respectively to the upper bound (denoted by $b$) and the lower bound (denoted by $a$) of the support of $\mu$.

Our main theorem is the following (see Theorem 6.1 and Theorem 6.4 for precise statements).

Moreover, this rate function achieves its minimum value at a unique $m$-tuple $(\lambda_{1}^{*},\ldots,\lambda_{m}^{*})$ towards which $(\widetilde{\lambda}_{1}^{n},\ldots,\widetilde{\lambda}_{m}^{n})$ converges almost surely.

Theorem 2.4 is true for both the i.i.d. perturbation model and the orthonormalized perturbation model, but the exact expression of the rate function $L$ is not the same for both models. As could be expected, the minimum $(\lambda_{1}^{*},\ldots,\lambda_{m}^{*})$ only depends on the $\theta_{i}$’s, on the limiting spectral distribution $\mu$ of $X_{n}$, and on the covariance matrix of the vector $G,$ this latter dependence coming from the fact that the rate function involves a Laplace transform of the law of $G$ and its behavior near the extremum will generically be governed by the second derivatives, that is the covariance.

The rate function $L$ is not explicit in general. However, in the particular case where $X_{n}=0$, $L$ can be evaluated. It amounts to consider the large deviations of the eigenvalues of matrices $W_{n}=\frac{1}{n}G_{n}^{*}\Theta G_{n}$ for $G_{n}$ an $n\times r$ matrix, with $r$ fixed and $n$ growing to infinity. $L$ is very explicit when $G$ is Gaussian but even when the entries are not Gaussian, we can recover a large deviation principle and refine a bound of about the deviations of the largest eigenvalue (see Section 7).

4. The results in the case with outliers

We now consider the case where some eigenvalues of $X_{n}$ escape from the bulk, so that Assumption 2.3 is not fulfilled. We assume that these eigenvalues, that we call outliers, converge:

In this framework, we will need to make on $G$ the additional following assumption.

The law of the vector $\frac{G}{\sqrt{n}}$ satisfies a large deviation principle in the scale $n$ with a good rate function that we denote by $I$.

If Assumptions 2.1, 2.2, 2.5 and 2.6 hold, the law of the $m+p^{+}$ largest eigenvalues of $\widetilde{X_{n}}$ satisfies a large deviation principle with a good rate function $L^{o}$.

Again, Theorem 2.7 is true for both i.i.d. perturbation model and orthonormalized perturbation model, but the rate function is not the same for both models. A precise definition of $L^{o}$ will be given in Theorem 9.1.

Before going any further, let us discuss Assumption 2.6. On one side, let us give some natural examples for which the assumtion is fulfilled.

If $G=(g_{1},\ldots,g_{r})$ are i.i.d standard Gaussian variables, Assumption 2.6 holds with $I(v)=\frac{1}{2}\|v\|_{2}^{2}$.

Proof. The first result can be seens as a direct consequence of Schilder’s theorem. For the second, it is enough to notice by Tchebychev’s inequality that for all $L,\delta>0$,

so that taking the large $n$ limit and then $L$ going to infinity yields for any $\delta>0$

On the other side, we want to emphasize that in the case with outliers, the individual LDP stated in Assumption 2.6 will be crucial. To understand more deeply this phenomenon, we refer the interested reader to some couterexamples when this assumption is not fulfilled that are studied in [30, Section 2.3] and a related discussion in the introduction of .

5. Large deviations for the largest eigenvalues of perturbed matrix models

We apply hereafter the results above to study the large deviations of the law of the extreme eigenvalues of perturbations of randomly chosen matrices $X_{n}$ distributed according to the Gibbs measure

Let us first recall a few facts about the non-perturbed model. It is well known that if $X_{n}$ is distributed according to $\mu^{n}_{\beta},$ the law of the eigenvalues of $X_{n}$ is given by

We will make on the potential $V$ the following assumptions :

exists and is denoted by $\alpha_{V,\beta}^{p}$.

Under Assumption 2.9, the law of the $p$ largest eigenvalues $(\lambda_{1}^{n}>\cdots>\lambda_{p}^{n})$ of $X_{n}$ satisfies a large deviation principle in the scale $n$ and with good rate function given by

with $J_{V}(x)=V(x)-\beta\int\log|x-y|d\mu_{V}(y)$.

Note that in the case of the GOE and the GUE (see ),

Let us now go to the perturbed model. An important remark is that, due to the rotational invariance of the law of $X_{n},$ one can in fact consider very general orthonormal perturbations. We make the following

With these considerations in mind, we can state the large deviation principle for the extreme eigenvalues of $\widetilde{X_{n}}$. We recall that $b_{V}$ is the rightmost point of the support of $\mu_{V}$.

With $V$ satisfying Assumption 2.9, we consider the orthonormalized perturbation model under Assumption 2.12. Then, for any integer $k,$ the law of the $k$ largest eigenvalues $(\widetilde{\lambda}_{1}^{n},\cdots,\widetilde{\lambda}_{k}^{n})$ of $\widetilde{X_{n}}$ satisfies a large deviation principle in the scale $n$ and with good rate function given by

Scheme of the proofs

The strategy of the proof will be quite similar in both cases (with or without outliers), so, for the sake of simplicity, we will outline it in the present section only in the case without outliers (both the i.i.d. perturbation model and the orthonormalized perturbation model will be treated simultaneously).

The cornerstone is a nice representation, already crucially used in many papers on finite rank deformations (see e.g. ), of the eigenvalues $(\widetilde{\lambda}_{1}^{n},\ldots,\widetilde{\lambda}_{m}^{n})$ as zeroes of a fixed deterministic polynomial in the entries of matrices of size $r$ depending only on the resolvent of $X_{n}$ and the random vectors $(G^{n}_{i})_{1\leq i\leq r}$.

Indeed, if $V$ is the $n\times r$ matrix with column vectors $\begin{bmatrix}U_{1}^{n}\cdots U_{r}^{n}\end{bmatrix}$ in the orthonormalized perturbation model and $\begin{bmatrix}G_{1}^{n}\cdots G_{r}^{n}\end{bmatrix}$ in the i.i.d. perturbation model, $\Theta$ the matrix $\operatorname{diag}(\theta_{1},\ldots,\theta_{r})$ and $I_{n}$ the identity in $n\times n$ matrices, the characteristic polynomial of $\widetilde{X_{n}}$ reads

It means that the eigenvalues of $\widetilde{X_{n}}$ that are not We show in section 11.2 that the spectra of $X_{n}$ and $\widetilde{X_{n}}$ are disjoint in generic situation. eigenvalues of $X_{n}$ are the zeroes of $\det(I_{r}-V^{*}(zI_{n}-X_{n})^{-1}V\Theta),$ which is the determinant of a matrix whose size is independent of $n$.

Because of the relation between $V$ and the random vectors $G_{1}^{n},\ldots,G^{n}_{r}$, it is not hard to check that, if we let, for $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\},$ $K^{n}(z)$ and $C^{n}$ be the elements of the set $\mathsf{H}_{r}$ of $r\times r$ Hermitian matrices given, for $1\leq i\leq j\leq r$, by

In both i.i.d and orthonormalized perturbation models, there exists a function $P_{\Theta,r}$ defined on ${\mathsf{H}_{r}}\times{\mathsf{H}_{r}}$ which is polynomial in the entries of its arguments and depends only on the matrix $\Theta,$ such that any $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\}$ is an eigenvalue of $\widetilde{X_{n}}$ if and only if

The law of $((K^{n}(z))_{z\in\mathcal{K}},C^{n})$ on $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ equipped with the uniform topology, satisfies a large deviation principle in the scale $n$ and with good rate function ${\bf I}$.

By the contraction principle, we therefore deduce

with $P_{\Theta,r}$ the polynomial function of Proposition 3.1.

The organisation of the paper will follow the scheme we have just described: in the next section, we detail the orthonormalization procedure and prove Proposition 3.1. Section 5 and Section 6 will then deal more specifically with the case without outliers. In Section 5, we establish the functional large deviation principles for $(K^{n}(\cdot),C^{n})$ and $H^{n}$, whereas Section 6 is devoted to the proof of our main results in this case, namely the large deviation principle for the largest eigenvalues of $\widetilde{X_{n}}$ and the almost sure convergence to the minimisers of the rate function. In Section 7, we will see that the rate function can be studied further in the special case when $X_{n}=0.$ We then turn to the case with outliers in Sections 8 and 9. Therein, the proofs will be less detailed, but we will insist on the points that differ from the previous case. The extension to random matrices $X_{n}$ given by classical matrix models is presented in Section 10. To make the core of the paper easier to read, we gather some technical results in Section 11.

The goal of this section is to prove Proposition 3.1. As will be seen further, the proof of this proposition is straightforward in the i.i.d. perturbation model but more involved in the orthonormalized perturbation model and we first detail the orthonormalization procedure.

and the lower triangular matrix $A=[A_{ij}]_{1\leq j\leq i\leq r}$ as follows : for all $1\leq j<i\leq r$,

Note that by linear independence of the $G_{i}$’s, none of the $q_{i}$’s is zero so that the matrix $A$ is well defined.

Then the vectors $W_{1},\ldots,W_{r}$ defined, for $i=1,\ldots,r$, by

are orthogonal and the $U_{i}$’s, defined, for $i=1,\ldots,r$, by

are orthonormal. They are said to be the Gram-Schmidt orthonormalized vectors from $(G_{1},\ldots,G_{r}).$ The following proposition, which can be easily deduced from the definitions we have just introduced, will be useful in the sequel.

For each $i_{0}=1,\ldots,r$, there is a real function $P_{i_{0}}$, defined on ${\mathsf{H}_{r}}$, polynomial in the entries of the matrix, not depending on $n$ and nor on the $G_{i}$’s, such that

Moreover, the polynomial function $P_{i_{0}}$ is positive on the set of positive definite matrices.

As explained in Section 3, a crucial observation (see [11, Proposition 5.1]) is that the eigenvalues of $\widetilde{X_{n}}$ can be characterized as the zeroes of a polynomial function of matrices of size $r\times r.$ This was stated in Proposition 3.1 which we prove below.

Proof of Proposition 3.1. We first recall (3), that is for $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\},$

Hence any $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\}$ is an eigenvalue of $\widetilde{X_{n}}$ if and only if

We denote by $\mathbf{G}$ the $n\times r$ matrix with column vectors $(G_{i}^{n})_{1\leq i\leq r},$ so that $K^{n}(z)=\frac{1}{n}\mathbf{G}^{*}(zI_{n}-X_{n})^{-1}\mathbf{G}.$

In the i.i.d. perturbation model, as $V=\mathbf{G},$ Proposition 3.1 follows immediately with

which is actually a polynomial, depending on $\Theta$, in the entries of $K^{n}(z)$.

In the orthonormalized perturbation model, the Gram-Schmidt procedure makes things a bit more involved.

If we denote by $D$ the $r\times r$ diagonal matrix given by $D=\operatorname{diag}(\|W_{1}^{n}\|_{2},\ldots,\|W_{r}^{n}\|_{2})$ and $\Sigma=(A^{n})^{T},$ then $V$ is equal to $n^{-1/2}\mathbf{G}\Sigma D^{-1}$ and we deduce that

Now, if we define $Q=\operatorname{diag}(q_{1}^{n},\ldots,q_{r}^{n})$ (recall (6)), $E=DQ$, $F=\Sigma Q$ and $H^{n}(z):=\det(E^{*}\Theta^{-1}E-F^{*}K_{n}(z)F)$ then on one hand, one can check that

so that any $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\}$ is an eigenvalue of $\widetilde{X_{n}}$ if and only if it is a zero of $H^{n}.$ On the other hand, $H^{n}(z)$ is obviously a polynomial (depending only on the matrix $\Theta$) of the entries of $K^{n}(z)$, $E^{*}\Theta^{-1}E$ and $F$. Furthermore, $E^{*}\Theta^{-1}E$ is a diagonal matrix whose $i$-th entry is given by $(E^{*}\Theta^{-1}E)_{i}=\theta_{i}^{-1}\|q_{i}^{n}W_{i}^{n}\|^{2}_{2}=\theta_{i}^{-1}P_{i}(C^{n})$ (by Property 4.1) and $F_{ij}=\det[\gamma_{k,l}^{j}]_{k,l=1}^{i-1}$ with $\gamma_{k,l}^{j}$ defined in (7). This concludes the proof. $\square$

We assume throughout this section that Assumptions 2.1, 2.2 and 2.3 hold.

In the sequel, $\mathcal{K}$ will denote any compact interval included in $(b,\infty),$ and we denote by $z^{*}$ its upper bound. We equip $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ with the uniform topology which is given by the distance $d$ defined, for $(K_{1},C_{1}),(K_{2},C_{2})\in\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ by

where $\|M\|_{2}=\sqrt{\operatorname{Tr}(M^{2})}$ for all $M\in{\mathsf{H}_{r}}.$

With $G=(g_{1},\ldots,g_{r})$ satisfying Assumption 2.2, we define $Z$ a matrix in ${\mathsf{H}_{r}}$ such that, for $i\leq j,$ $Z_{ij}=\overline{g_{i}}g_{j}$ and $\Lambda$ given, for any $H\in{\mathsf{H}_{r}}$ by

The goal of this section is to show the following theorem.

The law of $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$, viewed as an element of the space $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ equipped with the uniform topology, satisfies a large deviation principle in the scale $n$ and with good rate function $\bf I$ which is infinite if $K$ is not Lipschitz continuous and otherwise defined, for $K\in\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})$ and $C\in{\mathsf{H}_{r}}$, by

and the supremum is taken over piecewise constant functions $P$ with values in ${\mathsf{H}_{r}}$ and $X,Y$ in ${\mathsf{H}_{r}}.$

with $P_{\Theta,r}$ the polynomial function of Proposition 3.1.

The reminder of the section will be devoted to the proof of the first part of the theorem and the study of the properties of the rate function $\bf I,$ in particular its minimisers.

2. Proof of Theorem 5.1.

The strategy will be to establish a LDP for finite dimensional marginals of the process $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ based on [30, Theorem 2.2] (see also and ). From that, we will establish a LDP in the topology of pointwise convergence via the Dawson-Gärtner theorem. As $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ will be shown to be exponentially tight for the uniform topology, the LDP will also hold in this latter topology.

We start with the exponential tightness, stated in the following lemma. As $\mathcal{K}$ is a compact subset of $(b,\infty)$ and the largest eigenvalue $\lambda_{1}^{n}$ tends to $b,$ there exists $0<\varepsilon<1$ (depending only on $\mathcal{K}$) such that for $n$ large enough, for any $z\in\mathcal{K}$ and $1\leq i\leq n,$ $z-\lambda_{i}^{n}>\varepsilon.$ We fix hereafter such an $\varepsilon.$

In particular, the law of $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ is exponentially tight for the uniform topology on $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}.$

whereas since $|C^{n}_{ij}|^{2}\leq C^{n}_{ii}C^{n}_{jj}$, $\|C^{n}\|_{2}\leq r\max_{1\leq i\leq r}C^{n}_{ii}$ and $\|K^{n}(z)\|_{2}\leq\frac{1}{\varepsilon}r\max_{1\leq i\leq r}C^{n}_{ii}.$

where the last inequality holds for $n$ and $L$ large enough. This gives

By the Arzela-Ascoli theorem, $\mathcal{C}_{\mathcal{K},L}$ is a compact subset of $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ for any $L>0$, from which we get immediately the second part of the lemma. $\square$

2.2. Large deviation principle for finite dimensional marginals

We now study the finite dimensional marginals of our process. More precisely, we intend to show the following:

Let $M$ be a positive integer and $b<z_{1}<z_{2}<\cdots<z_{M}.$ The law of $\left((K^{n}(z_{i}))_{1\leq i\leq M},C^{n}\right)$ viewed as an element of $\mathsf{H}_{r}^{M+1}$ satisfies a large deviation principle in the scale $n$ with good rate function $I_{M}^{z_{1},\ldots,z_{M}}$ defined, for $K_{1},\ldots,K_{M},C\in{\mathsf{H}_{r}}$ by

with $\Gamma_{M}(\Xi_{1},\ldots,\Xi_{M},Y)$ defined by the formula

Proof. The proof of the proposition is a direct consequence of Theorem 2.2 of . Indeed, let $Z_{1}$ be the ${\mathsf{H}_{r}}$-valued random variable such that for all $1\leq i,j\leq r,$

Now, if $(Z_{k})_{1\leq k\leq n}$ are iid copies of $Z_{1},$ we denote by

A slight problem is that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\lambda_{i}^{n}}$ do not fulfill Assumption A.1 in in the sense that this assumption requires that for all $i,$ $\lambda_{i}^{n}$ belongs to the support of the limiting measure $\mu.$ Nevertheless, it is easy to construct (as was done in the proof of Theorem 3.2 in ) a sequence $\bar{\lambda}_{i}^{n}$ such that $\frac{1}{n}\sum_{i=1}^{n}\delta_{\bar{\lambda}_{i}^{n}}$ fulfills Assumption A.1 in and $\bar{L}_{n}:=\frac{1}{n}\sum_{k=1}^{n}f(\bar{\lambda}_{k}^{n})$ is exponentially equivalent to $L_{n}.$ Then from Theorem 2.2 of , we get that $L_{n}$ satisfies an LDP in the scale $n$ with good rate function $I_{M}^{z_{1},\ldots,z_{M}}.$ $\square$

The next step is to establish a LDP for the law of $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ associated with the topology of pointwise convergence. The following proposition will be a straightforward application of the Dawson-Gärtner theorem on projective limits.

The law of $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ as an element of $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ equipped with the topology of pointwise convergence satisfies a LDP in the scale $n$ with good rate function $\bf J$ defined as follows : for $K\in\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})$ and $C\in{\mathsf{H}_{r}},$

Moreover $\mathbf{J}$ equals the rate function $\mathbf{I}$ given in Theorem 5.1.(1).

Proof. Let $\mathcal{J}$ be the collection of all finite subsets of $\mathcal{K}$ ordered by inclusion. For $j=\{z_{1},\ldots,z_{|j|}\}\in\mathcal{J}$ and $f$ a measurable function from $\mathcal{K}$ to ${\mathsf{H}_{r}},$ $p_{j}(f)=(f(z_{1}),\ldots,f(z_{|j|}))\in\mathsf{H}_{r}^{|j|}$. We know from Proposition 5.3 that the law of $(p_{j}(K^{n}),C^{n})$ satisfies a LDP with good rate function $I_{|j|}^{z_{1},\ldots,z_{|j|}}.$ Moreover, one can check that the projective limit of the family $\mathsf{H}_{r}^{|j|}\times{\mathsf{H}_{r}}$ is $\mathsf{H}_{r}^{\mathcal{K}}\times{\mathsf{H}_{r}}$ equipped with the topology of pointwise convergence. Therefore, the Dawson-Gärtner theorem [16, Theorem 4.6.1] proves the LDP with rate function ${\mathbf{J}}$. The identification of ${\mathbf{J}}$ as $\mathbf{I}$ is straightforward as by a simple change of variables, $\mathbf{J}$ is the supremum of

over the choices of $\Xi,M,z$. We may assume without loss of generality that $z_{M}=z^{*}$. Putting $P(z)=\sum_{l=1}^{M-1}\Xi(z_{l})\mathds{1}_{[z_{l},z_{l+1}]}$ and $X=\Xi(z_{M})$, we identify ${\mathbf{J}}$ and $\mathbf{I}.$ Thus the proof of the proposition is complete. $\square$

To complete the proof of Theorem 5.1(1), we now need to show that the LDP is also true for the uniform topology. From Proposition 5.4 and Lemma 5.2, and as the topology of uniform convergence is finer than the topology of pointwise convergence, we can apply [16, Corollary 4.2.6] and get that the law of $\left((K^{n}(z))_{z\in\mathcal{K}},C^{n}\right)$ as an element of $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ equipped with the uniform topology satisfies a LDP in the scale $n$ with good rate function $\bf J.$

3. Properties of the rate function

To finish the proof of Theorem 5.1(1), the last thing to check is that $\mathbf{I}(K(\cdot),C)$ is infinite whenever $K$ is not Lipschitz continuous. This is the object of this subsection (see Lemma 5.5.(6)), together with providing further information on the functions $(K,C)$ with finite $\mathbf{I}$ that will be useful in the sequel.

$H\mapsto\Lambda(H)$ is increasing, $\Lambda(-H)\leq 0$ if $H\geq 0$.

If we assume moreover that $G$ satisfies the first part of Assumption 2.2 (existence of some exponential moments), we have the following properties.

If ${\bf I}(K(\cdot),C)$ is finite, $C\geq 0$ and $K(z)\geq 0$, for any $z\in\mathcal{K}.$ Moreover, for all $L$, there exists a finite constant $M_{L}$ so that on $\{{\bf I}\leq L\}$, we have

If ${\bf I}(K(\cdot),C)$ or ${\bf J}(K(\cdot),C)$ are finite, then $z{\rightarrow}K(z)$ is non increasing.

For all $L$, there exists a finite constant $M_{L}$ so that on $\{{\bf I}\leq L\}$, we have

In particular, $K^{\prime}$ exists almost surely and is bounded by $M_{L}$.

If we assume now that $G$ satisfies both parts of Assumption 2.2 (the law of $G$ does not put mass on hyperplanes), we then have the following additionnal properties.

For all non null positive semi-definite $H\in\mathsf{H}_{r}$,

If ${\bf I}(K(\cdot),C)$ is finite, then $C>0$ and $K(z)>0$ for any $z\in\mathcal{K}.$ Moreover, for almost any $z\in\mathcal{K}$ and for any non zero vector $e$, there is no interval with non-empty interior on which the function $\langle e,K^{\prime}(.)e\rangle$ vanishes everywhere.

The first point is just based on the fact that almost surely, $\operatorname{Tr}(HZ)\geq 0$ if $H\geq 0$.

The second point follows from Jensen’s inequality.

The third point is due to the fact that $\operatorname{Tr}(HZ)\leq\|H\|_{\infty}\sum_{i=1}^{r}|g_{i}|^{2}$ so that by Hölder’s inequality,

which is finite by Assumption 2.2 if $\|H\|_{\infty}r\leq\alpha.$

To prove the fourth point let $(C,K)\in\{{\mathbf{I}}\leq L\}$. We first show that $C\geq 0$. We take $P,X\equiv 0$ to get

for all vector $u$ with norm one, that is $\|C\|_{\infty}\leq\gamma^{-1}(L+B).$ Similar considerations hold for the bound over $\|K(z)\|_{\infty}.$

by (1) of this lemma. Thus for all $t>0$,

It follows that $u^{*}(K(z_{2})-K(z_{1}))u$ is non positive by letting $t$ going to infinity, which completes the proof of this point.

where we used that $Y$ is bounded by $\varepsilon^{-2}$. This provides the expected bound by the fourth point.

Consider $\eta>0$ and a non vanishing orthogonal projector $p\in\mathsf{H}_{r}$ such that $H\geq\eta p$. For all $t>0$, we have

(where we used Assumption 2.2 in the last equality), we have

which goes to infinity as $t$ goes to infinity by the previous consideration. Thus, this is not possible. As we have already seen that $K(a)\geq 0$ for all $a\in\mathcal{K}$, we see that $K(a^{\prime})>0$ for $a^{\prime}<a$ unless there exists $e$ so that $\langle e,(a-a^{\prime})^{-1}(K(a)-K(a^{\prime}))e\rangle$ vanishes, which is impossible by the above.

4. Study of the minimisers of 𝐈𝐈\mathbf{I}

We characterise the minima of ${\mathbf{I}}$ as follows :

For any compact set $\mathcal{K}$ of $(b,\infty)$, the unique minimizer of ${\mathbf{I}}$ on $\mathcal{C}(\mathcal{K},{\mathsf{H}_{r}})\times{\mathsf{H}_{r}}$ is the pair $(K^{*},C^{*})$ given, for $1\leq i,j\leq r$, by

Proof. ${\mathbf{I}}$ vanishes at its minimisers (as a good rate function) and therefore a minimizer $(K,C)$ satisfies for all $P,X,Y$,

Now, for any fixed $(P,X,Y),$ there exists $\varepsilon_{0}>0$ such that for any $0<\varepsilon<\varepsilon_{0},$ for any $x$ in the support of $\mu$ we have

with $\alpha$ given by Assumption 2.2. Therefore, there exists a constant $L$ such that for any $x$ in the support of $\mu$

As a consequence, for any minimizer $(K,C),$ we find after replacing $(P,X,Y)$ by $\varepsilon(P,X,Y)$, using (12) and letting $\varepsilon$ going to zero, that

Changing $(P,X,Y)$ in $-(P,X,Y)$ gives the equality. This implies that

and therefore $(K,C)=(K^{*},C^{*}).$ $\square$

Large deviations for the largest eigenvalues in the case without outliers

We again assume throughout this section that Assumptions 2.1, 2.2 and 2.3 hold.

Note that in the latter product, the $\alpha_{i}$’s appear with multiplicity. $S_{\emptyset,\gamma}^{\varepsilon}$ will denote the set of functions as above but with no zeroes on $\mathcal{K}_{\varepsilon}$. We have the following theorem.

is increasing, so that its limits as $\varepsilon$ decreases to zero exists.

Note that $J_{\mathcal{K}_{\varepsilon}}(f)$ is infinite if $f$ has more than $r$ zeroes greater than $b$. Indeed, by definition, if $J_{\mathcal{K}_{\varepsilon}}(f)$ is finite,

The minimisers are described by the following result.

so that we recover [11, Theorem 2.1] or [10, Theorem 1.3].

2. Preliminary remarks and strategy of the proof

Let us first notice that at most $m$ eigenvalues of $\widetilde{X_{n}}$ can deviate from the bulk since by Weyl’s interlacing inequalities (see e.g. [27, Section 4.3])

which converges to $b$ as $n$ goes to infinity.

Secondly, let us state the following lemma.

The law of the sequence $(\widetilde{\lambda}_{1}^{n},\ldots,\widetilde{\lambda}_{m}^{n})$ of the $m$ largest eigenvalues of $\widetilde{X_{n}}$ is exponentially tight in the scale $n$.

Proof. Let us define $R_{n}:=\widetilde{X_{n}}-X_{n}$ and denote by $\|R_{n}\|_{\infty}$ the operator norm of the perturbation matrix $R_{n}$. Note that for all $k$,

Since for any fixed $k$, the non random sequence $\lambda_{k}^{n}$ converges to $b$ as $n$ tends to infinity, it suffices to prove that

For the orthonormalized perturbation model, since $\|R_{n}\|_{\infty}=\max\{\theta_{1},-\theta_{r}\}$, (13) is clear. In the i.i.d. perturbation model, we have, for $\theta:=\max_{1\leq i\leq r}|\theta_{i}|$,

It implies, by Tchebychev’s inequality, that

which allows to conclude by Assumption 2.2. $\square$

As the law of $(\widetilde{\lambda}_{1}^{n},\ldots,\widetilde{\lambda}_{m}^{n})$ is exponentially tight, the proof of Theorem 6.1 reduces to establishing a weak LDP. In virtue of [16, Theorem 4.1.11] (see also [1, Corollary D.6]), this weak LDP (and the fact that $L$ is a rate function) will be a direct consequence of Equation (18) and Lemma 6.9 below. The fact that $L$ is a good rate function is then implied by exponential tightness [16, Lemma 1.2.18].

From Proposition 3.1, we know that the $\widetilde{\lambda}^{n}_{i}$’s are essentially the zeroes of $H^{n}.$ However, $H^{n}$ could a priori have other zeroes than these eigenvalues or take arbitrary small values. To control this point, we need to understand better the structure of $H^{n}$. Let

For any $\varepsilon>0$ small enough, there exists a positive integer $n_{0}(\varepsilon)$, $L(\varepsilon)>0$ and a sequence of random functions $(g_{n})$ such that for any $z\in\mathcal{K}_{\varepsilon}$ and $n\geq n_{0}(\varepsilon),$

Going back to the proof of Proposition 3.1, one can easily see that, for any $z\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\},$

We can rewrite the above as $H^{n}(z)=c_{n}g_{n}(z)\,\,\prod_{i=1}^{m}(z-\widetilde{\lambda}_{i}^{n})$ with

Now, for $\varepsilon>0$ fixed, we shall bound $g_{n}$ and its Lipschitz constant on $\mathcal{K}_{\varepsilon}$.

As $\mathcal{K}_{\varepsilon}$ is compact and the $\lambda_{i}^{n}$ belong to a fixed compact, for $\varepsilon>0$ small enough, for any $i$ and $z\in\mathcal{K}_{\varepsilon}$ we have $z-\lambda_{i}^{n}\leq\frac{2}{\varepsilon}$ and $|\lambda_{i}^{n}|\leq\frac{2}{\varepsilon}$ so that

We choose $n_{0}(\varepsilon)$ such that for $n\geq n_{0}(\varepsilon)$ and any $i$ and $z\in\mathcal{K}_{\varepsilon}$ we have $\frac{\varepsilon}{2}\leq z-\lambda_{i}^{n}$ so that as $z-\lambda_{i}^{n}\leq\frac{2}{\varepsilon}$,

Now, using Weyl’s interlacing properties, we have for any $i\geq m+1,$

For $0\leq x\leq 1-\frac{\varepsilon^{2}}{4},$ $\log(1-x)\geq-\frac{4}{\varepsilon^{2}}x,$ so that we finally get by (17),

By very similar arguments (using $\log(1+x)\leq x$ for $x\geq 0$), one can also check that for any $n\geq n_{0}(\varepsilon),$

The proof of the uniform equicontinuity of $g_{n}$ on $\mathcal{K}_{\varepsilon}$ is left to the reader as the arguments are very similar since $z{\rightarrow}(z-\lambda_{i}^{n})^{-1}$ is uniformly continuous on $\mathcal{K}_{\varepsilon}$ for $n$ large enough.

The main application of the previous Lemma will be the following continuity properties of the zeroes of functions in $C^{\varepsilon}_{\gamma}$.

4. Core of the proof

The weak LDP will then be a direct consequence of the following lemma, with $k$ the numbers of eigenvalues going to $b,$

with the obvious convention that $\bigcap_{m-k+1\leq i\leq m}\{\widetilde{\lambda}^{n}_{i}\leq b+\varepsilon\}=\Omega$ if $k=0.$

Proof. Let $\delta$ and $\varepsilon$ be positive small enough constants so that $\alpha_{m-k}-\delta\geq b+2\varepsilon.$ In particular, $\cap_{i=1}^{m-k}[\alpha_{i}-\delta,\alpha_{i}+\delta]\subset\mathcal{K}_{\varepsilon}.$ On the set $\bigcap_{1\leq i\leq m-k}\{|\widetilde{\lambda}^{n}_{i}-\alpha_{i}|\leq\delta\}\bigcap_{m-k+1\leq i\leq m}\{\widetilde{\lambda}^{n}_{i}\leq b+\varepsilon\}$, for all $i\leq m-k,$ $\widetilde{\lambda}^{n}_{i}$ is in $\mathcal{K}_{\varepsilon}.$ On the other hand, for $n$ large enough, $\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\}\cap\mathcal{K}_{\varepsilon}=\emptyset.$ Therefore, $\widetilde{\lambda}^{n}_{i}\notin\{\lambda_{1}^{n},\ldots,\lambda_{n}^{n}\}$ for $i\in\{1,\ldots,m-k\}$ and, by Proposition 3.1, is a zero of $H^{n}.$

Let us next prove the large deviation upper bound and fix $\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{m-k}>b$. A function $f\in C_{\gamma,k}^{\varepsilon}$ which vanishes within a distance $\delta$ of $(\alpha_{i})_{1\leq i\leq m-k}$ with $\delta<\alpha_{m-k}-b$ belongs to the set

Since $J_{\mathcal{K}_{\varepsilon}}$ is a good rate function, $(B_{\alpha,\gamma,\delta}^{\varepsilon})_{\delta>0}$ is a nested family and $\cap_{\delta>0}B_{\alpha,\gamma,\delta}^{\varepsilon}=S_{(\alpha_{1},\ldots,\alpha_{m-k}),\gamma}^{\varepsilon}$, Theorem 5.1 gives with [16, Lemma 4.1.6] that

Taking $\gamma^{\prime}=\gamma^{\prime}_{0}$ small enough, (15) and (20) give for $\gamma/(\varepsilon^{\prime})^{m}<\gamma_{0}^{\prime}$,

We can finally take $\gamma=0$ (nothing depends on it anymore), $\varepsilon-\varepsilon^{\prime}$ going to zero, as the left hand side obviously decreases as $\varepsilon-\varepsilon^{\prime}$ decreases to and, as we already mentioned it in Remark 6.2, the right hand side increases as $\varepsilon$ decreases to $0.$

We turn to the lower bound, which is a bit more delicate. Let us again consider $\delta$ and $\varepsilon$ small enough so that $\cap_{i=1}^{m-k}[\alpha_{i}-\delta,\alpha_{i}+\delta]\subset\mathcal{K}_{\varepsilon}$. As $J_{\mathcal{K}_{\varepsilon}}$ is a good rate function and $S_{\alpha,\gamma}^{\varepsilon}$ is closed, for all $\gamma>0$, the infimum $\inf_{S_{(\alpha_{1},\ldots,\alpha_{m-k}),\gamma}^{\varepsilon}}J_{\mathcal{K}_{\varepsilon}}$ is achieved, say at $f_{\gamma}^{k,\varepsilon}.$ To complete the proof, we need the following lemma, based on the structure of $H_{n}$ and whose proof is a direct application of Lemma 6.8.

Let $\varepsilon,\gamma$ be fixed and small enough. There exists $\delta_{0}$ such that for any $\delta\leq\delta_{0}$, there exists $\delta^{\prime}>0$ such that for any $n,$

To prove the lower bound in Theorem 5.1, we may assume without loss of generality that

we can choose $\varepsilon,\gamma$ small enough so that $J_{\mathcal{K}_{\varepsilon}}(f_{\gamma}^{k,\varepsilon})\leq J+\eta$. By (16), there exists $L(\gamma,\varepsilon)$ going to infinity as $\gamma,\varepsilon$ go to zero so that for $n$ large enough,

We choose $\gamma,\varepsilon$ small enough so that $L(\varepsilon,\gamma)>J+2\eta$.

Lemma 6.10 implies, that for $\delta\leq\delta_{0},$ for $\delta^{\prime}$ small enough, $\eta>0$, for $n$ large enough,

the last inequality following from Theorem 5.1.(2). As $\eta$ can be chosen as small as we want, we conclude by taking first $n$ going to infinity, and then $\delta,\varepsilon,\eta$ to zero.

5. Identification of the minimizers

Large deviations for the eigenvalues of Wishart matrices

In this section, we study the i.i.d. perturbation model when $X_{n}=0.$ More precisely, we consider $G=(g_{1},\ldots,g_{r})$ satisfying Assumption 2.2, $n\times r$ matrices $G_{n}$ whose rows are i.i.d. copies of $G,$ a diagonal matrix $\Theta=\operatorname{diag}(\theta_{1},\ldots,\theta_{r})$ and we study the large deviations of Wishart matrices $W_{n}=\frac{1}{n}G_{n}\Theta G_{n}^{*}.$ This matrix has zero as an eiganvalue with muliplicity at least $n-r$ and we refer in the whole section to the $r$ eigenvalues of $W_{n}$ that can be non-zero as “the eigenvalues of $W_{n}$”. The large deviations for the largest and smallest such eigenvalues were already studied in in the case when $\Theta=(1,\ldots,1)$ and the $g_{i}$’s are i.i.d.

Assume that $G$ satisfies Assumption 2.2. Let $\Theta=\operatorname{diag}(\theta_{1},\theta_{2},\ldots,\theta_{r})$ be a diagonal matrix with positive entries. Then, the law of the eigenvalues of $W_{n}$ satisfies a large deviation principle in the scale $n$ with rate function which is infinite unless $\alpha_{1}\geq\cdots\geq\alpha_{r}\geq 0$ and in this case given by

Note that the previous proposition could also have been deduced directly from Cramér’s theorem and the contraction principle.

The Gaussian case allows an exact computation, given by the following

Assume that $G=(g_{1},\ldots,g_{r})$ is a Gaussian vector with positive definite covariance matrix $R.$ Let $\Theta=\operatorname{diag}(\theta_{1},\ldots,\theta_{r})$ be a diagonal matrix with positive entries. We denote by $0<r_{1}(\Theta)\leq r_{2}(\Theta)\leq\ldots\leq r_{r}(\Theta)$ the eigenvalues of the matrix $\Theta^{-1/2}R^{-1}\Theta^{-1/2}$ in increasing order. Then, the law of the eigenvalues of $W_{n}$ satisfies a large deviation principle in the scale $n$ with rate function which is infinite unless $\alpha_{1}\geq\alpha_{2}\geq\ldots\geq\alpha_{r}>0$ and otherwise given by

In the particular case when the entries are i.i.d. standard normal, the above rate function can be rewritten

Now, by a straightforward use of the contraction principle, we can derive some results about the deviations of the largest eigenvalue. This problem was addressed in particular in . The following corollary holds for the Gaussian case.

Under the assumptions of Corollary 7.2, the law of the largest eigenvalue satisfies a LDP with good rate function

with the convention that $r_{r+1}(\Theta)=\infty.$

In particular, in the i.i.d. standard case when $\Theta=\operatorname{diag}(1,\ldots,1),$ we have

and this allows to retrieve [23, Corollary 2.1] (note that a direct proof based on the formula for the joint law of the eigenvalues is then also available). This is in agreement with the fact that as $r$ goes to infinity, we expect the deviations below one to be impossible in this scale.

From there, one can easily improve the upper bound on the probability of deviations of the largest eigenvalue of [23, Theorem 2.1] :

Assume that $G$ satisfies Assumption 2.2 and that the $g_{i}$’s are i.i.d. with mean 0 and variance 1. Let $\Theta=\operatorname{diag}(\theta_{1},\theta_{2},\ldots,\theta_{r})$ be a diagonal matrix with positive entries, with $\theta_{1}\geq\theta_{2}\geq\ldots\geq\theta_{r}.$ Then we have that, for $\alpha\geq\theta_{1},$

Note that when $\alpha\geq\theta_{1},$ $I_{r,\Theta}(\alpha)=\inf_{[\alpha,\infty)}L_{max}$ and in particular $I_{r,\Theta}$ is not necessarily lower semicontinuous. We refer to for more properties of $I_{r,\Theta}$, related results and conjectures.

Proof of Proposition 7.1. In the case where $X_{n}=0$, we can apply Theorem 6.1 with $P_{\Theta,r}(K(z),C)=\det(z-\Theta^{\frac{1}{2}}C\Theta^{\frac{1}{2}})$ and ${\bf I}(C)=J(C)$. Hence, for $\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{r}>0$, $L(\alpha)$ is the infimum of $J$ over the nonnegative Hermitian matrices $C$ such that $\Theta^{\frac{1}{2}}C\Theta^{\frac{1}{2}}$ has spectrum $(\alpha_{1},\ldots,\alpha_{r})$. ∎

We finally take the infimum over $C$ so that $\Theta^{\frac{1}{2}}C\Theta^{\frac{1}{2}}=\sum_{i=1}^{r}\alpha_{i}e_{i}e_{i}^{*}$ for some orthonormal basis (ONB) $(e_{i})_{1\leq i\leq r}$. This gives

Proof of Corollary 7.4. We only need to take, in the definition of $J(C)$, $Y=tvv^{*}$ if $C$ has eigenvector $v$ for its largest eigenvalue to get a lower bound on $J(C)$, and thus on $L$.∎

Proof of Corollary 7.5. The inequality in Corollary 7.4 gives the upper bound and the lower bound is obtained by the same proof as in , that is by noticing that

and that for fixed $x,$ $\langle x,W_{n}x\rangle=n^{-1}\sum_{j=1}^{n}(\langle x,\Theta^{\frac{1}{2}}G^{j}_{n}\rangle)^{2}$ is a sum of i.i.d. random variables so that Cramer’s theorem apply. By arguments as in , one can also check that $I_{r,\Theta}$ is increasing on $[\theta_{1},\infty),$ which concludes the proof.∎

We now go to the proof of the LDP in the presence of outliers, that will be stated in details in Theorem 9.1. The proof follows the same lines as in the case without outliers and starts therefore with the study of the deviations of $H^{n}.$

We assume that Assumptions 2.1, 2.2, 2.5 and 2.6 hold.

The law of $\left((K^{n}(z))_{z\in\mathcal{K}^{o}},C^{n}\right)$, viewed as an element of the space $\mathcal{C}(\mathcal{K}^{o},\mathsf{H}_{r})\times\mathsf{H}_{r}$ endowed with the uniform topology, satisfies a large deviation principle in the scale $n$ with rate function ${\bf I}^{o}$. For $K\in\mathcal{C}(\mathcal{K}^{o},\mathsf{H}_{r})$ and $C\in\mathsf{H}_{r}$, ${\bf I}^{o}(K(\cdot),C)$ is infinite if $z\rightarrow K(z)$ is not uniformly Lipschitz on $\mathcal{K}^{o}$. Otherwise, it is given by