\left| \P\big( R(\xi_1,\dots,\xi_n) \in A\,|\, M_n= x_n\big)
- \P\big( (\xi_1,\dots,\xi_{n-1}) \in A \big) \right| \\
=\bigg| \frac{\P\big( R(\xi_1,\dots,\xi_n) \in A,  M_n = x_n\big)}{\P(M_n = x_n)}
- \P\big( (\xi_1,\dots,\xi_{n-1}) \in A \big) \bigg|
\end{multline*}

by the sum of three terms $\tilde P_1, \tilde P_2, \tilde P_3$, that we now define and treat separately.
The first term is

\[
\tilde P_1 := \frac{\P(\exists \, i \neq j \le n \, : \, \xi_i, \xi_j =  x_n)}{\P(M_n =  x_n)} \leq  \frac{n^2 \P(\xi =  x_n)^2}{\P(M_n = x_n)} \xrightarrow[n\to\infty]{} 0 \,,
\]

using (<ref>). The second term $\tilde P_2$ is

\begin{multline*}
\!\!\! \bigg| \frac{\P\big( R(\xi_1,\dots,\xi_n) \in A \cap (-\infty, x_n)^{n-1}, M_n = x_n\big)}{\P(M_n = x_n)}
- \P\big( (\xi_1,\dots,\xi_{n-1}) \in A \cap (-\infty, x_n)^{n-1} \big) \bigg| \\
=  \left| \frac{n\P(\xi = x_n)}{\P(M_n = x_n)} - 1 \right|
\P\big( (\xi_1,\dots,\xi_{n-1}) \in A \cap (-\infty, x_n)^{n-1} \big)   \xrightarrow[n\to\infty]{} 0 \,,
\end{multline*}

using again (<ref>).
The last term is $\tilde P_3 := \P(M_{n-1}\geq  x_n)$, which goes to $0$, thanks to (<ref>).
This concludes the proof of (<ref>).

§ DISCUSSIONS ON ROZOVSKII'S THEOREM

In this section, we make some comments on Rozovskii's result, in several directions:

* We discuss the condition under which Theorem <ref> is true: Rozovskii states its result in terms of the function $q(\cdot)$, recall the definition (<ref>); we show in Section <ref> that it is equivalent to (<ref>), which is in turn equivalent to $\overline{F}$ being extended
regularly varying.

* We discuss how one may find equivalent (or sufficient) conditions to have (<ref>): our goal is to obtain a more tractable condition in order to get the large deviation asymptotics (<ref>); this is the purpose of Proposition <ref> below.

* We provide some examples of distributions for which Rozovskii's result can be applied, some for which it cannot be applied.

Again, for simplicity of notation, we assume that $\mu=\E[\xi]=0$ in this appendix.

§.§ About the condition (<ref>) and extended/intermediate regular variation

In <cit.>, Rozovskii assumes that the function $q$ defined in (<ref>) by $q(x) = x^2\overline{F}(x)/\mm(x)$
verifies that there exists some $c>0$ such that $x^c q(x)$ is asymptotically equivalent to a non-decreasing
We now state some equivalent statements and in particular we show that it is equivalent to $\overline{F}$ being extended regularly varying, see (<ref>).

Assume that $x\mapsto\mm(x)$ is slowly varying at $\infty$. The following statements are equivalent:

(i) $\exists$ $c>0$ such that $x^c q(x)$ is  equivalent to a non-decreasing function;

(ii) $\exists$ $c>0$ such that $x^c \overline F(x)$ is equivalent to a non-decreasing function, i.e. (<ref>);

(iii) there is a constant $c>0$ and a function $y\mapsto \delta_y$ such that $\lim_{y\to\infty} \delta_y =0$, for which one has

\begin{equation}
\label{cond:F}
\overline F(x) \leq \overline F (y) \leq (1+\delta_y)   \Big(\frac{x}{y}\Big)^c \, \overline F(x) \,\qquad \forall x\geq y \,;
\end{equation}

(iv) The function $\overline{F}$ is extended regularly varying at infinity (see (<ref>)).

$(i) \Rightarrow (ii)$.
If $x^c q(x)$ is asymptotically equivalent to a non-decreasing function $v(x)$,

\[
x^{c+2} \overline F(x) = x^{c} \mm(x) q(x) = (1+o(1)) \mm(x) v(x),
\]

with both $\mm(x)$ and $v(x)$ non-decreasing. Thus condition (<ref>) follows.

$(ii) \Rightarrow (i)$.
If $x^c \overline F(x)$ is  asymptotically equivalent to a non-decreasing function $v(x)$ (i.e.  assuming (<ref>)), then $x^c q(x) = (1+o(1)) v(x) x^2/\mm(x)$.
Note that $x^2/\mm(x)$ is asymptotically equivalent to $x^2/\overline\sigma^2(x)$ (recall that $\overline \sigma(x) := \E[(|\xi|\wedge x)^2]$) which is differentiable with derivative $\frac{2x}{\overline\sigma^2(x)} (1- \frac{x^2\P(|\xi|>x)}{\overline\sigma^2(x)}) \geq 0$ for $x$ large. Hence $x^c q(x)$ is asymptotic to a non-decreasing function.

$(ii) \Rightarrow (iii)$. If $x^c \overline F(x)$ is  asymptotically equivalent to an increasing function $v(x)$, we can write $x^c \overline F(x) = (1+\gep_x) v(x)$, with $\lim_{x\to \infty} \gep_x =0$. Hence, for $y\leq x$, using that $v$ is increasing, we have

\[
\frac{y^c \overline{F}(y)}{x^c \overline{F}(x)} = \frac{1+\gep_y}{1+\gep_x} \frac{v(y)}{v(x)} \leq  \frac{1+|\gep_y|}{1-\inf_{x\geq y} |\gep_x|} \to1,\,\text{ as } y\to\infty,
\]

which proves the upper bound in (<ref>) (the lower bound is trivial).

$(iii) \Rightarrow (ii)$.
Let $v(y)= y^c \overline{F}(y)$ and $\tilde v(y) = \inf_{x\geq y} v(x)$.
Then, clearly, $\tilde v$ is non-decreasing and by (<ref>) we have that
$ (1+\delta_y)^{-1} v(y) \leq \tilde v(y) \leq v(y)$, so $v(y)$ and $\tilde v(y)$ are asymptotically equivalent. Hence $(ii)$ is verified.

$(iii) \Rightarrow (iv)$ is obvious from the definition of extended regular variation; note that (<ref>) gives the upper and lower Matuszewska indices are $c$ and $1$ respectively.

$(iv) \Rightarrow (iii)$ follows by Potter's bound for extended regularly varying functions, see <cit.>: for any $\delta>0$, there is some $x_0>0$ such that for any $\lambda <1$ and $x\geq x_0$
$ 1 \leq \frac{\overline{F}(\lambda x)}{\overline{F}(x)} \leq  (1+\delta) \lambda^{-c-1}$,
where $c$ is the upper Matuszewska index;
the lower bound simply follows from the fact that $\overline{F}$ is non-increasing.

The fact that function $\overline{F}$ is intermediate regularly varying (see (<ref>))
is equivalent to the following:
there exists some non-decreasing function $\kappa:[1,\infty) \to [1,\infty)$ such that

\begin{equation}
\label{cond:F2}
\overline F(x) \leq  \overline F (y) \leq   (1+\delta_y)\kappa\Big(\frac xy\Big)\, \overline F(x) \qquad \forall x\geq y \,,
\end{equation}

with $\kappa$ satisfying $\lim_{s\to1+}\kappa(s)=1$ and $\lim_{s\to\infty} \frac1s \log \kappa(s)=0$.
Additionally, there is a constant $c>0$ such that
\begin{equation}\label{eqn:kpoly}
\kappa(s)\leq c s^{c}, \text{ for all } s\geq 1.
\end{equation}

Note that (<ref>) readily implies intermediate regular variation by verifying the definition.
For the reverse implication, we proceed exactly as for Claim <ref>:
using Potter's bound <cit.> for intermediate regularly varying functions, we obtain that for all $\delta>0$ there is some $x_0>0$ such that for any $\lambda >1$ and $x\geq x_0$,
$1 \leq \frac{\overline{F}(\lambda x)}{\overline{F}(x)} \leq  (1+\delta) \kappa(\lambda)^{-1}$ for some $\kappa(\lambda)$ with $\lim_{\lambda\to 1} \kappa(\lambda)=1$.

The fact that $\kappa$ grows at most polynomially follows directly from the representation theorem for intermediate regularly varying functions, see <cit.>.

§.§ About the condition (<ref>) in Theorem <ref>

We now find equivalent conditions to the condition (<ref>) in Theorem <ref>.
Recall that we defined

\[
\overline \sigma^2(x) =\E[(|\xi| \wedge x)^2]= 2 \int_0^x t \P(|\xi|>t) dt\,,
\]

and let $(a_n)_{n\geq 1}$ be a sequence defined by the relation (<ref>). Then, noticing that
\begin{equation}
\label{sigmabartosigma}
\overline \sigma^2(x)=\sigma^2(x)+x^2\P(|\xi|> x),
\end{equation}

we have $\lim_{y\to\infty} \frac{y^2\P(|\xi|> y)}{\sigma^2(y)}=0$ thanks to (<ref>), so

\begin{equation}\label{ddbar}
\overline \sigma^2(y)\sim \sigma^2(y).
\end{equation}

This shows that $(a_n)_{n\geq 1}$ defined by (<ref>), i.e. $a_n^2= n \overline \sigma(a_n)$, is a valid normalising sequence since it verifies (<ref>) (with $\alpha=2$).

Let $\alpha=2$, let $(a_n)_{n\geq 1}$ be defined by (<ref>), and assume that $\overline{F}$ is intermediate regularly varying.
Then the relation (<ref>) is equivalent to (<ref>), that is to

\begin{equation}
\label{final2}
\sigma^2(x\sqrt{|\log q(x)|})- \sigma^2(x)=o\left(\frac{ \sigma^2(x)}{|\log q(x)|}\right)\qquad \text{ as } x\to\infty \,.
\end{equation}

The rest of Section <ref> will be devoted to the proof of this proposition.

§.§.§ Preliminary observations

Let us first give some consequences of the intermediate regular variation of $\overline{F}$, which we recall is equivalent to (<ref>).

Assume that $x\mapsto\mm(x)$ is slowly varying at infinity and that $\overline{F}$ is intermediate regularly varying.  Let $x\geq y>0.$ Then for $y$ large enough

\begin{equation}\label{qx/qy}
\left(\frac{y}{x}\right)^2\leq \frac{q(y)}{q(x)}\leq (1+\delta_y)\kappa\Big(\frac{x}{y}\Big)\frac{y}{x},
\end{equation}

where $\delta_y$ and $\kappa(\cdot)$ are the same as in (<ref>).

Note that by (<ref>), we have
$1\leq \frac{\overline F(y)}{\overline F(x)}\leq (1+\delta_y)\kappa (\frac{x}{y})$ for all $x\geq y>0$. Then we can use the definition (<ref>) of $q$ to obtain

\[
\frac{\sigma^2(x)}{\sigma^2(y)}\left(\frac{y}{x}\right)^2\leq \frac{q(y)}{q(x)}\leq (1+\delta_y)\kappa\left(\frac{x}{y}\right)\frac{\sigma^2(x)}{\sigma^2(y)}\left(\frac{y}{x}\right)^2.
\]

Note that $\sigma^{2}(x)\geq \sigma^2(y)$ for $x\geq y>0$ using monotonicity, and $\frac{\sigma^2(x)}{\sigma^2(y)}\leq \frac{x}{y}$ for $y$ large enough using Potter's bound for slowly varying function. Then we obtain (<ref>).

Assume that $x\mapsto\mm(x)$ is slowly varying at infinity and that $\overline{F}$ is intermediate regularly varying. Then uniformly for $\frac{x}{C |\log q(x)|} \leq y \leq  x$,
we have

\[
\label{asymp}
|\log q(x)|\sim |\log q(y)|\qquad \text{as } x\to\infty.
\]

In particular, recalling that $\omega_n := \frac{a_n}{\sqrt{|\log q(a_n)|}}$, we have $|\log q(\omega_n)|\sim |\log q(a_n)|$ as $n\to\infty$.

Similarly, the above display also holds uniformly for $x\leq y\leq C x|\log q(x)|$.

We will prove only the first part as the second part follows similar lines of reasoning. Recall that $\lim_{x\to\infty}q(x)=0$, see (<ref>).
Let us assume that $x$ is large enough so that $\log q(x)<0$ and let us set $h:=x/y$, which verifies $1\leq h \leq C|\log q(x)|$.
Using the second inequality in (<ref>), we have

$$\frac{q(y)}{q(x)}\leq (1+\delta_y)\frac{\kappa(h)}{h}\leq 2\frac{\kappa(h)}{h}, \quad \text{ for } x \text{ large enough}.$$

$\log q(x)\geq \log q(y)-\log 2\kappa(h) + \log h$, so that

\[
|\log q(x)|\leq |\log q(y)|+\log 2\kappa(h) \leq  |\log q(y)|+\log 2\kappa( C|\log q(x)|) \,,
\]

using that $\kappa$ is non-decreasing.
On the other hand, the first inequality in (<ref>) shows that  $\frac{q(y)}{q(x)}\geq  h^{-2}$,
which implies

\[
|\log q(x)|\geq |\log q(y)|-2\log h \geq |\log q(y)|-2\log\kappa(C |\log q(x)|).
\]

Since $\lim_{x\to\infty}q(x)=0$ and recalling that $\lim_{s\to\infty} \frac 1s \log \kappa(s) =0$, see (<ref>), this concludes the proof.

Thanks to Lemma <ref>, if $x_n\sim  c a_n \sqrt{|\log q(a_n)|}$ for some $c>0$, we get that
$x_n \sim  c a_n \sqrt{|\log q(x_n)|}$.
Hence, using also that $\frac{x_n}{a_n} \sim \frac{c}{\sqrt{|\log q(a_n)|} }= q(x_n)^{o(1)}$, we obtain

\[
\overline{\Phi}\Big(\frac{x_n}{a_n}\Big) \sim \frac{a_n}{x_n\sqrt{2\pi}} e^{-\frac{x_n^2}{2 a_n^2}} =  q(x_n)^{c^2/2+o(1)} \,.
\]

On the other hand, recalling the definition (<ref>) of $a_n$,

\[
n \overline{F}(x_n) = \frac{n \mm(x_n)}{x_n^2} q(x_n) \sim \frac{ a_n^2 \mm(x_n)}{x_n^2 \mm(a_n)} q(x_n) =q(x_n)^{1+o(1)} \,,
\]

where we used Potter's bound since $\sigma(\cdot)$ is slowly varying.
In conclusion, a sufficient condition to have that $\overline{\Phi}(\frac{x_n}{a_n}) = o(n\overline{F}(x_n))$ is $c>\sqrt{2}$.

§.§.§ Proof of Proposition <ref>

Recall that $(a_n)_{n\geq 1}$ is defined by (<ref>) and that we assumed that $\overline{F}$ is intermediate regularly varying.

Let us recall the definition $\omega_n = \frac{a_n}{\sqrt{|\log q(a_n)|}}$ with $q(x) = \frac{x^2}{\mm(x)} \overline{F}(x)$ defined in (<ref>).

The condition (<ref>) is equivalent to

\begin{align}\label{part1}
\lim_{n\to\infty} nF(-\omega_n)&= 0,\\
\text{ and } \qquad \overline \sigma^2(a_n)- \sigma^2(\omega_n)& =o\left(\frac{ \overline \sigma^2(a_n)}{\left|\log q(a_n)\right|}\right).
\label{bob}
\end{align}

The above conditions (<ref>)-(<ref>) are further equivalent to

\begin{equation}
\label{boo} \sigma^2(a_n)- \sigma^2(\omega_n)=o\left(\frac{ \sigma^2(\omega_n)}{\left|\log q(\omega_n)\right|}\right).
\end{equation}

Note that (<ref>) is equivalent to (<ref>) and
$\lim_{n\to\infty} \big|\frac{n}{\omega_n^2}\sigma^2(\omega_n)-\frac{a_n^2}{\omega_n^2}\big| = 0$.
Using the definition (<ref>) of $a_n$, we can write
\frac{\overline \sigma^2(a_n)}{|\log q(a_n)|}= n^{-1} \omega_n^2,
so we obtain that (<ref>) is equivalent to (<ref>) and
(<ref>). Next, we show that (<ref>) is equivalent to (<ref>)-(<ref>).

Step 1. We first show that (<ref>) implies (<ref>); we actually prove that (<ref>) implies

\begin{equation}\label{nomegan>0}
\lim_{n\to\infty} n\P(|\xi|\geq \omega_n)= 0.
\end{equation}

Note that we have

\begin{equation}\label{proba-sigma}
n\P(\omega_n<|\xi|\leq a_n) = \frac{n}{\omega_n^2} \omega_n^2\P(\omega_n<|\xi|\leq a_n) \leq  \frac{n}{\omega_n^2} (\sigma^2(a_n)-\sigma^2(\omega_n)) \,.
\end{equation}

If (<ref>) holds, then we have

\begin{equation}\label{3sigma}
\overline \sigma^2(a_n) \sim \sigma^2(a_n)\sim  \sigma^2(\omega_n).
\end{equation}

Using Lemma <ref>, we therefore get that

\begin{equation}\label{remainder}
n^{-1}\omega_n^2 =\frac{\overline \sigma^2(a_n)}{|\log q(a_n)|}\sim \frac{\sigma^2(\omega_n)}{|\log q(\omega_n)|} \,.
\end{equation}

Hence (<ref>) implies that $\sigma^2(a_n)- \sigma^2(\omega_n)=o(n^{-1}\omega_n^2)$. Then using (<ref>), we obtain $\lim_{n\to\infty} n\P(\omega_n<|\xi|\leq a_n) =0$.

It remains to show that $\lim_{n\to\infty} n\P(|\xi|> a_n)=0$.
But this is simply due to the fact that $n = \frac{a_n^2}{\overline\sigma^2(a_n)}$ (see  definition (<ref>)) and (<ref>).
Thus we have shown that (<ref>) implies (<ref>).

Step 2. We now show that (<ref>) implies (<ref>).
Thanks to (<ref>) and (<ref>), we only have to show that

\[
\overline \sigma^2(a_n) - \mm(a_n) = a_n \P(|\xi|>a_n) = o\left(\frac{ \overline \sigma^2(a_n)}{\left|\log q(a_n)\right|}\right) \,.
\]

Recall that (<ref>) implies (<ref>). Using that $a_n^2= n\overline\sigma^2(a_n)$, we get that (<ref>) is equivalent to
\omega_n^2\P(|\xi|\geq \omega_n)=o\big(\frac{ \overline\sigma^2(a_n)}{|\log q(a_n)|}\big)$, as $n\to\infty$,
which is further equivalent to the following, using (<ref>),

\[
\omega_n^2\P(|\xi|\geq \omega_n)=o\left(\frac{ \sigma^2(\omega_n)}{\left|\log q(\omega_n)\right|}\right),\qquad \text{as } n\to\infty \,.
\]

This concludes the proof since $\omega_n$ can be replaced by any sequence growing to infinity (one can use the argument that $\omega_{\lfloor cn\rfloor}\sim \sqrt{c}\omega_{n}$ for any fixed $c>0,$ which can be proved  using Lemma <ref> and the fact that $a_n$ also satisfies this property).

Step 3. Finally we show that (<ref>) implies (<ref>). Note that (<ref>) implies (<ref>).
Applying Lemma <ref>, we find that (<ref>) holds. We also observe that

\[
0\leq \sigma^2(a_n)- \sigma^2(\omega_n) \leq \overline \sigma^2(a_n)- \sigma^2(\omega_n) = o\left(\frac{ \overline \sigma^2(a_n)}{\left|\log q(a_n)\right|}\right) \,,
\]

where the last step is due to (<ref>) and (<ref>). This completes the proof, thanks to (<ref>).

By Lemma <ref>, setting $y:=x/\sqrt{|\log q(x)|}$, the condition (<ref>) is equivalent to

\begin{equation}\label{oneb}
\sigma^2(x)- \sigma^2(y)=o\left(\frac{ \sigma^2(y)}{|\log q(y)|}\right)\,, \qquad \text{ as } x\to\infty \,.
\end{equation}

This is because (<ref>) still holds if we replace $(a_n)_{n\geq 1}$ by an arbitrary increasing sequence. We only need  to show that (<ref>) is equivalent to (<ref>). To this purpose, it suffices to show that

\begin{equation*}\label{ylogtox}
\mm(y\sqrt{|\log q(y)|})-\mm(x)=o\left(\frac{\sigma^2(y)}{|\log q(y)|}\right)\,,
\end{equation*}

under either the assumption of (<ref>) or (<ref>). This is implied by the following claim, using that $y\sqrt{|\log q(y)|}\sim x$ thanks to Lemma <ref> (and the fact that $\mm$ is non-decreasing).

If (<ref>) or if (<ref>) holds, then
$\mm(x)-\mm(\frac12 x) =o\left(\frac{ \sigma^2(x)}{|\log q(x)|}\right)$; similarly we have $\mm(2x)-\mm(x) =o\left(\frac{ \sigma^2(x)}{|\log q(x)|}\right)$.

Assume that (<ref>) holds; the proof is identical if we assume that (<ref>) holds instead.
First, since $y=x/\sqrt{|\log q(x)|}$,  we have that $y\leq \frac12 x$ for $x$ large. Hence,

\[
0\leq \mm(x)-\mm(\tfrac12 x) \leq \mm(x)-\mm(y) =o\left(\frac{ \sigma^2(x)}{|\log q(x)|}\right) \,,
\]

by (<ref>) and $\frac{ \sigma^2(x)}{|\log q(x)|}\sim \frac{ \sigma^2(y)}{|\log q(y)|}.$ Indeed the later holds due to (<ref>) and Lemma <ref>.
Moreover, we can replace $x$ by $2x$ in the above display: indeed, we have $\mm(2x)\sim \mm(x)$ (since $\mm(\cdot)$ is slowly varying) and $|\log q(2x)|\sim |\log q(x)|$, see Lemma <ref>.

Assume that Proposition <ref>, i.e. (<ref>), holds. If $x_n\sim ca_n\sqrt{|\log q(a_n)|}$ for some constant $c>0$, then recalling the asymptotics (<ref>) of $r_n$ and using (<ref>), we have $r_n\sim\frac{a_n}{c\sqrt{|\log q(a_n)|}}$ and $\sigma^2(r_n)\sim \sigma^2(a_n)$ (using also that $\sigma^2(\cdot)$ is slowly varying).
Therefore, we have $\sigma^2(r_n)\sim \sigma^2(a_n)$ uniformly for $a_n\leq x_n\leq ca_n\sqrt{|\log q(a_n)|}$, using that $\sigma^2(\cdot)$ is an increasing function (we also clearly have $r_n\sim a_n/c$ and so $\sigma^2(r_n)\sim \sigma^2(a_n)$, if $x_n\sim c a_n$).

§.§ A few examples

First, let us give a generic example where Rozovskii's condition is verified.

Assume that $\overline F(x)\sim L(x)x^{-\beta}$ as $x\to\infty$, for some slowly varying function $L(\cdot)$ and some $\beta\geq 2$.
Assume also that the left tail is “dominated” by the right tail, in the sense that $F(-x)\sim c\overline{F}(x)$ for some $c\geq 0$; with $F(-x) = o(\overline{F}(x))$ if $c=0$.
This is for instance the case if $\xi =X-\E[X]$ for some non-negative random variable $X$.

If $\beta>2$, then we have $\E[|\xi|^{2+\delta}]<\infty$ for some $\delta >0$, so Nagaev's condition is verified.
We will therefore focus on the case $\beta=2$: Nagaev's condition is not verified, but we will see that Rozovskii's condition (<ref>) is verified for $a_n$ defined by $a_n^2 = n \overline\sigma^2(a_n)$.

If $\beta=2$, then we have $\overline\sigma^2(x) =2 \int_0^{x} t \P(|\xi|>t) d t$, with $\P(|\xi|>t) \sim (1+c)L(t)t^{-2}$ as $t\to\infty$. Denote $\ell(t) = 2 t^2 \P(|\xi|>t)$, so that

\begin{equation}\label{sigmaint}
\overline \sigma(x)^2 = \int_0^x \ell(t) t^{-1} \d t
\end{equation}

and note that $\ell(x)\sim  2(1+c) L(x)$ as $x\to\infty$.
Then we have that $q(x) \sim  c' \ell(x)/\mm(x)$, with $\ell(x)/\mm(x)$ which is known to verify $\lim_{x\to\infty} \ell(x)/\mm(x)=0$, see <cit.>.

Then, thanks to Proposition <ref>, we simply need to verify that

\begin{equation}
\label{rozov:goal}
\mm(x\sqrt{|\log q(x)|}) -\mm(x) = o\left( \frac{\mm(x)}{|\log q(x)|} \right) \,.
\end{equation}

Since $\overline \sigma^2(x) -\mm(x) = x^2 \P(|\xi|>x) = \ell(x)$
and since $\ell(x) \sim 2(1+c) \mm(x) q(x) =o\left( \frac{\mm(x)}{|\log q(x)|} \right)$, it suffices to prove the above display with $\mm(\cdot)$ replaced by $\overline \sigma^2(\cdot)$ on the l.h.s. of (<ref>), thanks to Lemma <ref>.
We can use Claim <ref> below to get that

\[
\begin{split}
0\leq \overline\sigma^2(x\sqrt{|\log q(x)|}) -\overline\sigma^2(x)
& \leq \ell(x) \Big(\tfrac12\log |\log q(x)| + \gep_x \sqrt{|\log q(x)|}\Big) \\
&  \leq C \mm(x) q(x)  \sqrt{|\log q(x)|}  = o\left( \frac{\mm(x)}{|\log q(x)|} \right) \,,
\end{split}
\]

which concludes the proof.

Let $\ell$ be a non-negative slowly varying function.
Then there exists $\gep_x$ that verifies $\lim_{x\to\infty} \gep_x =0$ such that for any $z\ge x$

\[
\int_x^z \frac{\ell(t)}{t} \d t \leq \ell(x) \Big(\log \Big(\frac zx\Big) + \gep_x \, \Big(\frac{z}{x}-1\Big)\Big) \,, \qquad \text{ as }x\to\infty \,.
\]

Setting $g(y) :=\ell(y) y^{-1}$, we get by a change of variable that

\[
\int_x^{z} \ell(t)t^{-1} \d t =  \ell(x) \int_1^{z/x} \frac{g(ux)}{g(x)} \d u \,.
\]

Now, using that $g$ is regularly varying of index $-1$, we have that uniformly for $u\geq 1$,  $g(ux)/g(x) = u^{-1} + \gep_x$ with $\lim_{x\to\infty}\gep_x = 0$ (this is the uniform convergence theorem, see <cit.>).
We end up with

\[
0\leq \int_x^{z} \ell(t)t^{-1} \d t \leq  \ell(x)\Big( \log \Big( \frac zx \Big)  + \gep_x\, \Big(\frac{z}{x}-1\Big) \Big) \,,
\]

which concludes the proof.

We now present an example where the left tail is heavier than the right tail: we give some conditions for (<ref>) to hold; this will also  stress the importance of the choice of the normalising sequence $(a_n)_{n\geq 1}$.

Assume that $\xi$ is centered and that $\sigma^2 =\mathrm{Var}(\xi)<\infty$. We will focus on a case where Nagaev's condition $\E[|\xi|^{2+\delta}]<\infty$ for some $\delta>0$ is not satisfied.
We consider a case where $|\log q(x)|\asymp \log x$ as $x\to\infty$; this is ensured for instance if the right tail verifies $x^{-\beta_+} \leq \overline F(x) \leq x^{-\beta_-}$  for some $\beta_+>\beta_->2$ (for $x$ large).

Our goal is to discuss the conditions imposed by (<ref>) on the left tail of the distribution. We assume that $F(-x) \sim \ell(x) x^{-2}$ for some slowly varying function $\ell(\cdot)$; think about taking $\ell(x) = (\log x)^{-a}$ for some $a>1$ (so that $\sigma^2<\infty$).
We now compare two possible choices for the normalising sequence $(a_n)_{n\geq 1}$.

If one defines $a_n=\sigma \sqrt{n}$, then recalling (<ref>), (<ref>) and (<ref>),  (<ref>) is equivalent to

\[
\int_x^{\infty} \frac{\ell(t)}{t} \d t  \sim \E[\xi^2 \ind_{\{|\xi|>x\}}] = o\left(\frac{1}{\log x}\right) \,,
\]

(recall also that $|\log q(x)|\asymp \log x$).
Since $\ell(x) = (\log x)^{-a}$ for some $a>1$, this amounts to having $a>2$ for this particular choice of $a_n$.

If on the other hand one defines $a_n$ by $a_n^2=n \overline{\sigma}(a_n)$, then Proposition <ref> tells that (<ref>) is equivalent to

\[
\E[\xi^2 \ind_{\{x< |\xi|<x\sqrt{|\log q(x)|}\}}] \, \sim
\, \int_x^{x\sqrt{|\log q(x)}|} \frac{\ell(t)}{t} \d t = o\left(\frac{1}{\log x}\right)
\]

By Claim <ref>, we have that the integral is $o(\ell(x) \sqrt{\log x})$ (using that $|\log q(x)|\asymp \log x$), so it is enough to have $\ell(x) \leq (\log x)^{-3/2}$ (or $a>3/2$).
If $\ell(x)$ is non-increasing, then the integral is bounded by a constant times $\ell(x) \log \log x$, so a sufficient condition becomes $\ell(x) = o( \frac{1}{\log x \log \log x} )$; this is verified if $\ell(x) = (\log x)^{-a}$ with any $a>1$.

Let us consider the case where $\xi$ is centered and where the left and right tails verify $\overline F(x) \sim L(x) x^{-2}$ and $F(-x)\sim \ell(x) x^{-2}$, with the left tail being heavier, that is $L(x)=o(\ell(x))$ as $x\to\infty$.
In this case, $q(x)$ is slowly varying and the interplay between the slowly varying functions $L,\ell,\mm$ is subtle.
For simplicity, let us consider the case where $\ell(x) \sim (\log x)^{-a}$ for some $a\in \mathbb R$.
We will find conditions on the right tail for (<ref>) to be verified.

If $\sigma^2 <\infty$ (i.e. $a>1$), let us set for now $a_n=\sigma \sqrt{n}$.
We have that

\[
\E[\xi^2\ind_{\{|\xi|>x\}}] \sim \int_x^\infty \frac{\ell(t)}{t} \d t \sim  (\log x)^{1-a} \,,
\]

so (<ref>) is equivalent to $|\log q(x)| = o( (\log x)^{a-1})$.
If $a\geq 2$, the fact that $q$ is slowly varying ensures that $|\log q(x)|=o(\log x)$, thus (<ref>) holds. If $a\in (1,2)$, we have to check the right tail: for instance, if $L(x)\asymp e^{(\log x)^b}$ for some $b\in (0,1)$, then $|\log q(x)| \asymp (\log x)^b$ and the condition (<ref>) is verified provided that $a-1>b$.

If we define $a_n$ by the relation $a_n^2 = n \overline \sigma^2(a_n)$, we need to consider

\[
\E\big[\xi^2\ind_{\{x<|\xi|<x \sqrt{|\log q(x)|}\}} \big]
\sim \int_x^{x \sqrt{|\log q(x)|}} \frac{\ell(t)}{t} \d t
\sim \frac12 (\log x)^{-a} \log |\log q(x)| \,.
\]

Therefore, (<ref>) is equivalent to
$|\log q(x)| \log (|\log q(x)|) = o\left( \mm(x) (\log x)^a\right)$.
If $a>1$, this is automatic, since we have $|\log q(x)| = o(\log x)$ as noticed above.
If $a=1$, this is also verified thanks to the same observation, using also that $\mm(x)\sim \log\log x$.
If $a<1$, then $\mm(x) \asymp (\log x)^{1-a}$, so the condition becomes $|\log q(x)| \log |\log q(x)|= o(\log x)$;
this is verified for instance if in the right tail we have $L(x) \asymp e^{(\log x)^b} $ for some $b\in (0,1)$, but is not verified if one takes for instance $L(x) \asymp \exp( \log x/\log\log x)$.

We now provide an example of a distribution in the domain of attraction of the normal law where one cannot find any normalising sequence $(a_n)_{n\geq 1}$ that verifies Rozovskii's condition (<ref>).

The example is constructed to make the left tail too heavy compared to the right tail. Take some function $q(x)$ and suppose that  $\sigma^2(x) = \exp\big(\log x /\log^{(3)}q(x) \big)$,  for $x$ large enough with $\log^{(3)} t =\log \log \log \frac1t$. Let $(a_n)_{n\geq1}$ be any normalising sequence, i.e. verifying (<ref>).

Since $\omega_n =a_n /\sqrt{|\log q(a_n)|}$, we have that

\[
\frac{\log \omega_n}{\log^{(3)} q(\omega_n)} = \frac{\log a_n}{ \log^{(3)} q(\omega_n)}  - \frac{\frac12\log |\log q(a_n)|}{ \log^{(3)} q(\omega_n)}.
\]

So if we assume that $\log^{(3)} q(\omega_n) -\log^{(3)} q(a_n)  = o(\log^{(3)} q(a_n)/\log a_n) $ (take for instance $q(y)=y^{-1}$ or $q(y)=1/\log y$), we get that

\[
\frac{\log \omega_n}{\log^{(3)} q(\omega_n)} = \frac{\log a_n}{ \log^{(3)} q(a_n)}  - \frac12 \frac{\log |\log q(a_n)|}{ \log^{(3)} q(a_n)} +o(1)\,.
\]

\[
\sigma^2(\omega_n) \sim  e^{-\frac12 \frac{\log |\log q(a_n)|}{ \log^{(3)} q(a_n)}} \sigma^2(a_n) \,
\]

\[
\left|\frac{n}{w_n^2}\sigma^2(\omega_n)-\frac{a_n^2}{\omega_n^2}\right|
= \frac{a_n^2}{\omega_n^2} \left| \frac{n \sigma^2(\omega_n)}{a_n^2}-1\right| \,,
\]

with $\frac{a_n^2}{\omega_n^2} \to \infty$ and

\[
\frac{n \sigma^2(\omega_n)}{a_n^2}
\sim  e^{-\frac12 \frac{\log |\log q(a_n)|}{ \log^{(3)} q(a_n)} }  \frac{n \sigma^2(a_n)}{a_n^2} \to 0.
\]

This shows that
$\left|\frac{n}{w_n^2}\sigma^2(\omega_n)-\frac{a_n^2}{\omega_n^2}\right|\to\infty$, which tells that (<ref>) does not hold.

§ ONE-BIG-JUMP PHENOMENON WHEN $\ALPHA\IN(0,2)$

In this section we prove Corollaries <ref> and <ref>.
We begin with the following lemma.

For any $\xi$ that is non-degenerate (i.e. whose law is not concentrated on a single point), we have that

\begin{equation}\label{dtvnot0}d_{\mathrm{TV}}\Big( \mathscr{L}\big( R(\xi_1,\dots,\xi_n) \big), \,
\big(\mathscr{L}(\xi)\big)^{\otimes (n-1)} \Big)\text{ does not converge to }0 \text{ as } n\to\infty.
\end{equation}

Let us denote

\[
\]

and $N_n$ be the second largest variable in $\xi_1,\ldots,\xi_n$, so  $N_n=\max \{\eta_1,\ldots, \eta_{n-1} \}$.
Define $T:=\sup\{t:\P(\xi>t)>0\}$ and let us distinguish two cases.

If $\P(\xi=T)>0$ it is easy to see that

\[
\P(\eta_1=T)=\P(\xi_1=\xi_2=T)=\P(\xi=T)^2<\P(\xi_1=T).
\]

Thus (<ref>) holds in this case.

If $\P(\xi=T)=0,$ let $x$ converge to $T$ from the left and put $n=n(x)=\lfloor \frac{1}{\P(\xi >  x)}\rfloor.$ Then

\[
\lim_{n\to\infty}\P(M_{n-1}\leq  x)=\lim_{n\to\infty} (1-\P(\xi>x))^{n-1}=e^{-1},
\]

\[
\lim_{n\to\infty}\P(N_n\leq  x)=\lim_{n\to\infty}\P(M_{n}\leq  x)+\lim_{n\to\infty}n\P(\xi>x)\P(M_{n-1}\leq x)=2e^{-1}.
\]

\[
\lim_{n\to\infty}\P(N_n\leq  x)=e^{-1}\neq 2e^{-1} = \lim_{n\to\infty}\P(M_{n-1}\leq  x).
\]

Hence (<ref>) holds in this case as well.

§.§ Proof of Corollary <ref>

The statement (<ref>) is already contained in the proof of <cit.>, so we focus on the proof of (<ref>).

The fact that $\lim_{n\to\infty} \frac{x_n}{a_n}=\infty$ is sufficient to obtain (<ref>)
can be seen by using (<ref>) and the statement (<ref>) in Proposition <ref>. We will prove here the necessity.

Let $(x_n)_{n\geq 1}$ be such that

\begin{equation}
\label{xan}
\lim_{n\to\infty}\frac{x_n}{a_n} = l \in [-\infty, \infty) \,.
\end{equation}

If $l=-\infty,$ then

\begin{equation}\label{prelemma}
\lim_{n\to\infty} d_{\mathrm{TV}}\Big( \mathscr{L}\big( R(\xi_1,\dots,\xi_n) \, \big| \, S_n-b_n \geq  x_n \big), \, \mathscr{L}\big( R(\xi_1,\dots,\xi_n) \big) \Big) = 0,
\end{equation}

since $\lim_{n\to\infty}\P(S_n-b_n\geq x_n)=1$. By Lemma <ref>, (<ref>) does not hold. Next we assume $l\in(-\infty,\infty)$ and we prove that (<ref>) does not hold.
We split the proof according to several cases: recall our assumption (<ref>), which ensures the convergence of $S_n$ (properly centered and normalised) to an $\alpha$-stable random variable $\mathcal S_{\alpha}$, recall (<ref>).

(i) If $\alpha\in(0,1)$,  $p=1$, and $l\in (-\infty,0]$.
The limiting $\alpha$-stable random variable $\mathcal S_{\alpha}$ is then supported on $[0,\infty)$ (see <cit.> and the following remark). Then we have that
$\lim_{n\to\infty}\P(S_n-b_n>x_n)=1$ and we deduce (<ref>). By the same argument as in the case $l=-\infty$, we conclude that (<ref>) does not hold.

(ii) In the rest, we treat all the following cases in the same way (so we will not distinguish them in the following analysis):

\begin{equation}
\label{hardwork}
\begin{cases}
\alpha \in (0,1),\   p=1, \   l\in (0,\infty), \\[2pt]
\alpha\in(0,1),\   p\in (0,1), \  l\in(-\infty,\infty),\\[2pt]
\alpha\in [1,2), \  p\in (0,1], \  l\in (-\infty,\infty).
\end{cases}
\end{equation}

Let $\Lambda$ be the Lévy measure associated to the $\alpha$-stable random variable $\mathcal S_{\alpha}$, given by $\Lambda(\dd x) = \frac{2-\alpha}{\alpha} |x|^{-\alpha-1} (p\ind_{\{x>0\}} +q\ind_{\{x<0\}}) \dd x$;
the characteristic function of $\mathcal S_{\alpha}$ is given by:

\begin{equation}
\label{chara}
\phi(t):=\E[e^{it\mathcal S_{\alpha}}]=\exp\left(\int_{-\infty}^{\infty}(e^{itx}-1-it\sin(x))d\Lambda(x)\right), \quad t\in\R \,.
\end{equation}

Let $Y=(Y_u)_{u\geq 0}$ be a Lévy process with Lévy measure $\Lambda$ with $Y_0=0$, and denote by $\zeta_1,\zeta_2$ the largest and second largest jump sizes of the Lévy process $Y$ in the interval $[0,1]$. Let $N_n$ be the second largest value of $\xi_1,\ldots,\xi_n$. Then by (<ref>) and the stable functional central limit theorem <cit.>, we have for $y\in \mathbb R$

\begin{equation}\label{pp1}
\lim_{n\to\infty}\P\big( a_n^{-1} N_n <y \,\big|\, S_n-b_n\geq x_n \big)=\P\big(\zeta_2<y \,\big|\, Y_1\geq l\big)
\end{equation}

\begin{equation}\label{pp2}
\lim_{n\to\infty}\P\big(a_n^{-1} M_n < y\big)=\P\big(\zeta_1
< y\big)=\exp(-\Lambda_y),
\end{equation}

where $\Lambda_y=\Lambda([y,\infty)).$
To prove that (<ref>) does not hold, it suffices to show that there exists some $y>0$ such that
$\lim_{n\to\infty}\P( a_n^{-1} N_n <y \,|\, S_n-b_n\geq x_n )\neq \lim_{n\to\infty}\P(a_n^{-1} M_n < y)$,
or in other words that there is some $y>0$ such that

\begin{equation}\label{c}
\P(\zeta_2<y \, |\, Y_1\geq l )\neq \P(\zeta_1<y)=\exp(-\Lambda_y).
\end{equation}

Considering jumps that occur on $[0,1]$, let $Y_1^{(1)}$ be the sum of all jump sizes that are not smaller than $y$;
let also $Y_1^{(2)}=Y_1-Y_1^{(1)}$. Then $Y_1^{(1)}$ and $Y_1^{(2)}$ are independent, with respective characteristic functions

\[
\E\Big[e^{itY_1^{(1)}}\Big]=\exp\left(\int_{y}^{\infty}(e^{itx}-1)d\Lambda(x)\right), \quad t\in \R
\]

\[
\E\Big[e^{itY_1^{(2)}}\Big]=\exp\left(\int_{y}^{\infty}-it\sin(x)d\Lambda(x)+\int_{-\infty}^{y}(e^{itx}-1-it\sin(x))d\Lambda(x)\right), \quad t\in \R.
\]

Let $V$ be a random variable with distribution $\Lambda\1_{[y,\infty)}/\Lambda_y$ which is independent of $Y_1^{(2)}$. Note that if $\zeta_1<y$ then $Y_1^{(1)}=0$: using this fact,  we obtain

\begin{align*}\P(\zeta_2<y \, | \, Y_1\geq l)&=\frac{\P(\zeta_1<y, Y_1^{(2)}\geq l)}{\P(Y_1\geq l)}+\frac{\P(\zeta_2< y \leq  \zeta_1, \zeta_1+Y_1^{(2)}\geq l)}{\P(Y_1\geq l)}\\
&=\P(K=0)\frac{\P(Y_1^{(2)}\geq l)}{\P(Y_1\geq l)}+\P(K=1)\frac{\P(V+Y_1^{(2)}\geq l)}{\P(Y_1\geq l)},\end{align*}

where $K$ denotes the number of jumps with size not smaller than $y$. Note that $K$ follows the Poisson distribution $\mathrm{Poi}(\Lambda_y)$ with parameter $\Lambda_y$ and is independent of $Y_1^{(2)}$.
Then we have

\begin{equation}
\label{2compare}
\begin{split}
\P(\zeta_1<y)&-\P(\zeta_2<y \, | \, Y_1\geq l)\\
&  =\P(K=0)- \P(K=0)\frac{\P(Y_1^{(2)}\geq l)}{\P(Y_1\geq l)}-\P(K=1)\frac{\P(V+Y_1^{(2)}\geq l)}{\P(Y_1\geq l)}\\
&  =\frac{\exp(-\Lambda_y)}{\P(Y_1\geq l)}\left(\P(Y_1^{(2)}< l, \,Y_1^{(1)}+Y_1^{(2)}\geq l)-\Lambda_y\P(V+Y_1^{(2)}\geq l)\right)\\
& \leq \frac{\exp(-\Lambda_y)}{\P(Y_1\geq l)}\left(\P(K>0)\P(Y_1^{(2)}<l)-\Lambda_y\P(Y_1^{(2)}\geq l-y)\right),
\end{split}
\end{equation}

where we have used the fact that $V\geq y$.  Note that, as $y\to\infty$, we have that
$Y_1^{(2)}$ converges in distribution to $Y_1$
and $\P(Y_1^{(2)}\geq l-y)$ goes to $1$.
Since $\Lambda_y\to 0$, as $y\to\infty$, we use the above display to obtain that, as $y\to\infty$,

\begin{align*}
\P(K>0)\P(Y_1^{(2)}<l)-\Lambda_y\P(Y_1^{(2)}\geq l-y) &=\big(1-e^{-\Lambda_y} \big)\P(Y_1^{(2)}<l)-\Lambda_y\P(Y_1^{(2)}\geq l-y)\\
&  =-\Lambda_y\P(Y_1\geq l)+o(\Lambda_y) \,.
\end{align*}

Therefore we have that
$\P(K>0)\P(Y_1^{(2)}<l)-\Lambda_y\P(Y_1^{(2)}\geq l-y)<0$
for $y$ large enough.
Going back to (<ref>), we end up with

\[
\P(\zeta_1<y)-\P(\zeta_2<y \, | \, Y_1\geq l)<0
\]

for $y$ large enough, which proves (<ref>).
This completes the proof for (<ref>).

§.§ Proof of Corollary <ref>

In the following, to simplify notation, we assume that $b_n=0$.
The proof that $\lim_{n\to\infty} \frac{x_n}{a_n} =\infty$ is a sufficient condition to get (<ref>) follows the same line of proof as that of Theorem <ref>, see Section <ref> (following ideas of <cit.>), so we skip it.

We now show that $\lim_{n\to\infty} \frac{x_n}{a_n} =\infty$ is necessary. Assume that that (<ref>) holds, i.e. that $\lim_{n\to\infty} \frac{x_n}{a_n} = l \in [-\infty,\infty)$. We will distinguish two set of cases (i) and (ii).

(i) Consider first the cases

\begin{equation}\label{easypart}
\begin{cases}
\alpha\in (0,1),\quad  p=1, \quad  l\in [-\infty,0], \\[2pt]
\alpha\in (0,1),\quad  p\in (0,1), \quad l=-\infty, \\[2pt]
\alpha\in[1,2), \quad p\in (0,1], \quad l=-\infty.\end{cases}
\end{equation}

If (<ref>) holds, due to (<ref>), we have that

\[
\LL\Big(\frac{S_n-M_n-b_n}{a_n}\;\Big|\; S_n- b_n =x_n\Big)\stackrel{d}{\longrightarrow}\mathcal S_{\alpha} ,
\quad \text{ so } \quad  \LL\Big(\frac{M_n}{a_n}\;\Big|\; S_n- b_n =x_n\Big)\stackrel{d}{\longrightarrow} l-\mathcal S_{\alpha} \,.
\]

Note that $l-\mathcal S_{\alpha}<0$ for the first case in (<ref>) (recall that $\mathcal S_{\alpha}$ is supported on $[0,\infty)$) and that $l-\mathcal S_{\alpha} =-\infty$ in the other cases. Notice also that $\frac{b_n}{n a_n}$ and $\frac{x_n}{na_n}$ both go to $0$ as $n\to\infty$ (indeed $b_n=0$ if $\alpha \in (0,1)$ and $b_n\leq (1+o(1)) n \E[|\xi|\ind_{|\xi|\leq a_n}]$ if $\alpha\in [1,2)$, with $\E[|\xi|\ind_{|\xi|\leq a_n}]$ slowly varying):
the above display therefore implies that

\[
\lim_{n\to\infty}\P\Big(M_n < - \frac{b_n+x_n}{n}  \, \Big| \, S_n- b_n  =x_n \Big) =1 \,.
\]

But we clearly have that $\P(M_n< \frac{1}{n}(b_n+x_n), S_n =b_n+x_n) =0$, which brings a contradiction.
We conclude that (<ref>) is not true for the cases in (<ref>).

(ii) Next we consider the cases in (<ref>). Let $g_{\alpha}$ be the density function of $\mathcal S_{\alpha}$: by Stone's local limit theorem <cit.>, we have

\[
\P(S_n- b_n  =x_n)=\frac{1}{a_n} \big( g_{\alpha}(l) +o(1)\big), \qquad \text{ as } n\to\infty.
\]

Here $g_{\alpha}(l)>0$ for any $l$ in the ranges given in (<ref>) in different cases.
Choose any $\epsilon \in (0,1)$ and any $t\in \R$ such that $\P(\xi>t)>\epsilon$. Then $\P(M_n\leq t)<(1-\epsilon)^n$.
Combining with the above, we get that

\[
\P(M_n\leq t\,\, |\,\, S_n- b_n  =x_n) \leq \frac{\P(M_n\leq t)}{\P(S_n- b_n  =x_n)} \leq c (1-\epsilon)^n a_n \,,
\]

which goes to $0$ as $n$ goes to $\infty$, since $a_n$ is regularly varying.
We therefore obtain that
$\lim_{n\to\infty}\P(M_n>t\, |\, S_n- b_n  =x_n)=1$,
which in turn implies that

\[
\lim_{n\to\infty}\P\big(S_n-M_n- b_n <x_n-t\, \big|\,S_n- b_n  =x_n\big)=1.
\]

However if (<ref>) holds, given that $S_n-M_n$ is the sum of $R(\xi_1,\ldots,\xi_n)$, we would have

\[
\lim_{n\to\infty}\P\big( S_n-M_n- b_n <x_n-t\,\big |\, S_n- b_n =x_n\big)=\P(\mathcal S_{\alpha}<l)\in(0,1),
\]

which is a contradiction. This proves that (<ref>) is not true in cases (<ref>).
This concludes the proof.

§ ACKNOWLEDGEMENT

The authors thank Jochen Blath, Denis Denisov, Sergey Foss, Dalia Terhesiu and Vitali Wachtel for their friendly comments and pointers to the literature.
We also thank an anonymous referee whose insightful comments helped us improve the presentation.

Missing 'biblatex' package
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Cambridge university press

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# How to measure entanglement at finite temperatures?

Cheolhee Han  School of Physics and Astronomy, Tel Aviv University, Tel Aviv
6997801, Israel  Yigal Meir  Department of Physics, Ben-Gurion University of
the Negev, Beer-Sheva, 84105 Israel  Department of Physics, Princeton
University, Princeton, NJ 08540  Eran Sela  School of Physics and Astronomy,
Tel Aviv University, Tel Aviv 6997801, Israel

###### Abstract

It is desirable to relate entanglement of many-body systems to measurable
observables. In systems with a conserved charge, it was recently shown that
the number entanglement entropy (NEE) - i.e. the entropy change due to an
unselective subsystem charge measurement - is an entanglement monotone. Here
we derive finite-temperature equilibrium relations between Rényi moments of
the NEE, and multi-point charge correlations. These relations are exemplified
in quantum dot systems where the desired charge correlations can be measured
via a nearby quantum point contact. In quantum dots recently realizing the
multi-channel Kondo effect we show that the NEE has a nontrivial universal
temperature dependence which is now accessible using the proposed methods.

Entanglement is a key concept in quantum mechanics, and its measures are being
repeatedly used in many areas of physics, including basic quantum theory,
quantum information and many-body physics. For systems at zero temperature or
those described by a pure state, one can quantify the entanglement by the
system’s entanglement entropy. However, the task of measuring entanglement
entropy in a many-body system is daunting as, in principle, one needs to
measure the full density matrix, requiring access to all degrees of freedom
(see, however, Ref. Bartkiewicz et al. (2016); Zhou et al. (2021)), and as a
result only very few experiments have been able to measure the entanglement
entropy, and only in specific systems Islam et al. (2015); Brydges et al.
(2019); Elben et al. (2020); Lukin et al. (2019). Thus the measurement of
entanglement entropy remains one of the outstanding challenges in many-body
physics. The task of experimental quantification of entanglement at finite
temperatures is even more challenging, as one cannot use the entanglement
entropy measure in this case.

An important step forward was the recognition Samuelsson et al. (2003);
Beenakker et al. (2003); Klich et al. (2006); Klich and Levitov (2009); Song
et al. (2010, 2011); Rachel et al. (2012) of a relationship between
entanglement entropy and fluctuations in condensed matter systems. For
example, for a system with a conserved total number of particles
$N=N_{A}+N_{B}$ a ground state with fluctuating number of particles in
subsystem $A$ directly implies a nonlocal quantum correlation with subsystem
$B$. These works focused mainly on generating Samuelsson et al. (2003);
Beenakker et al. (2003), measuring Klich et al. (2006); Klich and Levitov
(2009), and understanding of the scaling of many-body quantum entanglement
Song et al. (2010, 2011); Rachel et al. (2012). In fact, for non-interacting
fermions an exact relation between current moments and entanglement entropy
was found Klich and Levitov (2009). (Note, however, that in general, such
current correlations do not fully capture the charge-neutral contributions,
often referred to as configuration entropy Wiseman and Vaccaro (2003);
Barghathi et al. (2019); Goldstein and Sela (2018); Xavier et al. (2018);
Bonsignori et al. (2019) and thus the number entanglement entropy deduced from
charge or current fluctuations only yields a lower bound for the entanglement
entropy Klich et al. (2006)).

Yet, while all these previous works considered pure states, general condensed
matter systems are described by mixed states, e.g. due to finite temperature.
Then, as mentioned above, the entanglement entropy has to be replaced by
generalized entanglement measures, such as negativity Peres (1996); Vidal and
Werner (2001) or number entanglement entropy (NEE) Macieszczak et al. (2019);
Ma et al. (2022). However, unlike the case of a pure state, mentioned above,
relations between mixed state entanglement quantifiers and charge/current
correlations are not known, which makes the task of measuring entanglement at
finite temperatures even harder. Previous successful attempts to quantify
entanglement for mixed states relied on random unitary measurements Elben et
al. (2020); Zhou et al. (2020); Neven et al. (2021); Vitale et al. (2022);
Rath et al. (2022), which are hard to apply in general condensed matter
systems. Here, we provide exact thermodynamic relations between moments of the
NEE and measurable subsystem charge fluctuations. Consequently, measuring
these quantities will facilitate obtaining information about entanglement in
many-body condensed matter systems at finite temperatures.

The NEE $\Delta S$ is defined as the entropy change of a density matrix $\rho$
upon an unselective measurement of $N_{A}$ Ma et al. (2022),

$\displaystyle\Delta
S=-\text{Tr}[\rho_{\hat{N}_{A}}\log\rho_{\hat{N}_{A}}]+\text{Tr}[\rho\log\rho],$
(1)

where $\rho_{\hat{N}_{A}}=\sum_{N_{A}}\Pi_{N_{A}}\rho\Pi_{N_{A}}$ and
$\Pi_{N_{A}}$ is a projector to a subspace with subsystem charge $N_{A}$. A
similar quantity in terms of the reduced density matrix was studied in Ref.
Ares et al., 2022. Equivalently, $\rho_{\hat{N}_{A}}$ is obtained from $\rho$
by annihilating all off-diagonal matrix elements with respect to $N_{A}$. When
$N_{A}$ uniquely specifies the subsystem’s state, $\Delta S$ coincides with
the entanglement entropy for a pure state (one can easily generalize NEE to
include spin and other globally conserved quantum numbers). Note that even for
a mixed state, $\Delta S$ does not depend on which of the subsystems one
traces over, unlike the standard entanglement entropy.

The difficult task of directly measuring the entanglement entropy or
negativity in many-body systems was circumvented in the literature by
suggestions to measure instead its first Rényi moments Abanin and Demler
(2012); Daley et al. (2012); Pichler et al. (2013); Gray et al. (2018); Elben
et al. (2018, 2020). We will apply a similar strategy for the NEE. The Rényi
moments of $\Delta S$ are defined by

$\displaystyle\Delta
S^{(\ell)}=\frac{1}{1-\ell}\log\Big{(}\frac{\text{Tr}[\rho_{\hat{N}_{A}}^{\ell}]}{\text{Tr}[\rho^{\ell}]}\Big{)}.$
(2)

One can then extract the NEE from its moments via $\Delta S=\lim_{\ell\to
1}\Delta S^{(\ell)}$ (see also Ref. D’Hoker et al. (2021)). Hereafter we write
the NEE as $\Delta S^{(1)}$ for convenience. For pure states
$\text{Tr}[\rho^{\ell}]=1$ and $\Delta
S^{(\ell)}=\frac{1}{1-\ell}\log\sum_{N_{A}}(P(N_{A}))^{\ell}$ becomes the
Rényi entropy of the subsystem charge distribution $P(N_{A})$. When the
subsystem charge $N_{A}$ uniquely specifies its state, $\Delta S^{(\ell)}$
coincides with the usual Rényi entropy. For general mixed states, where
$\text{Tr}[\rho^{\ell}]\neq 1$, it can be shown that $\Delta S^{(\ell)}>0$ is
a sufficient condition for quantum entanglement. The second Rényi moment of
the NEE can be used, for example, to construct a computable mixed-state
entanglement monotone that reduces to known measures of entanglement in the
pure state limit Carisch and Zilberberg (2022).

Typically, Rényi moments are obtained using multiple copy methods Abanin and
Demler (2012); Daley et al. (2012); Pichler et al. (2013); Islam et al.
(2015); Gray et al. (2018); Cornfeld et al. (2018, 2019); Azses et al. (2020),
which are quite difficult to generate experimentally, especially in condensed
matter systems. Here, focusing on general thermal states, rather than
physically implementing $\ell$ copies of the system, we demonstrate that the
$\ell-$th Rényi moment of the NEE of a thermal state at temperature $T$ is
directly related to a correlation function of the same system at temperature
$T/\ell$.

Substituting $\rho(T)=Z(T)^{-1}\sum_{i}e^{-E_{i}/T}|i\rangle\langle i|$ where
$Z(T)=\sum_{i}e^{-E_{i}/T}$, and $|i\rangle$ are the eigenstates of $H$,
$H|i\rangle=E_{i}|i\rangle$, for all the factors of $\rho$ in the the
numerator in Eq. (2), we find

$\displaystyle\text{Tr}[(\rho_{\hat{N}_{A}})^{\ell}]$
$\displaystyle=\sum_{N_{A}}\text{Tr}[(\rho\Pi_{N_{A}})^{\ell}]=\sum_{j_{1},...j_{\ell}}\frac{e^{-(E_{j_{1}}+...+E_{j_{\ell}})/T}}{Z(T)^{\ell}}$
$\displaystyle\times$ $\displaystyle\langle
j_{1}|\Pi_{N_{A}}|j_{\ell}\rangle\langle
j_{\ell}|\Pi_{N_{A}}|j_{\ell-1}\rangle...\langle
j_{2}|\Pi_{N_{A}}|j_{1}\rangle.$ (3)

We now note that this expression is equivalent to the $\ell-$point imaginary
time Green’s function of projection operators,

$\displaystyle\langle\Pi_{N_{A}}(-i\tau_{\ell-1})\Pi_{N_{A}}(-i\tau_{\ell-2})\cdots\Pi_{N_{A}}(-i\tau_{1})\Pi_{N_{A}}(0)\rangle_{T^{\prime}}$
$\displaystyle=$
$\displaystyle\sum_{j_{1},...j_{\ell}}\frac{e^{-E_{j_{1}}/T^{\prime}}}{Z(T^{\prime})}\langle
j_{1}|\Pi_{N_{A}}|j_{\ell}\rangle\cdots\langle j_{2}|\Pi_{N_{A}}|j_{1}\rangle$
(4) $\displaystyle\times
e^{(E_{j_{1}}-E_{j_{\ell}})\tau_{\ell-1}}e^{(E_{j_{\ell}}-E_{j_{\ell-1}})\tau_{\ell-2}}\cdots
e^{(E_{j_{3}}-E_{j_{2}})\tau_{1}},$

by setting the temperature to $T^{\prime}=T/\ell$ and $\tau_{j}=j/T$.
Combining this with the denominator of Eq. (2) we obtain

$\displaystyle\Delta S^{(\ell)}=\frac{1}{1-\ell}\log\bigg{[}\sum_{N_{A}}$
$\displaystyle\Big{\langle}\Pi_{N_{A}}\Big{(}\frac{\ell-1}{iT}\Big{)}\Pi_{N_{A}}\Big{(}\frac{\ell-2}{iT}\Big{)}\cdots$
$\displaystyle\times\Pi_{N_{A}}\Big{(}\frac{1}{iT}\Big{)}\Pi_{N_{A}}(0)\Big{\rangle}_{T/\ell}\bigg{]}.$
(5)

This equation, which is the main result of this paper, expresses the Rényi NEE
at temperature $T$ in terms of a correlation function at temperature $T/\ell$.

In order to demonstrate the power of this relation, let us focus on a many
body system in the Coulomb blockade regime, such as a quantum dot (QD) or a
metallic grain with a large charging energy. In this case, the subsystem
supports only two charge states, and we provide an explicit protocol to
extract the 2nd and 3rd Rényi moments solely from the $2-$point charge
correlation function of the QD. These relations allow to measure an
entanglement quantifier in mesoscopic systems at finite temperatures.

Consider a subsystem $A$ with only two charge states, denoted for simplicity
$N_{A}=0,1$. In this case the projection operators become linear in $N_{A}$,
$\Pi_{N_{A}=1}=\hat{N}_{A}$, $\Pi_{N_{A}=0}=1-\hat{N}_{A}$. Thus Eq. (5)
becomes a sum of $q-$point ($q\leq\ell$) charge correlators.

Notably, the 2nd and 3rd Rényi moments of the NEE in Eq. (5) can be written
explicitly in terms of $\langle\hat{N}_{A}\rangle_{T/\ell}$, the occupation
number of $A$ and
$\langle\hat{N}_{A}(-i(\ell-1)/T)\hat{N}_{A}(0)\rangle_{T/\ell}$, the 2-point
imaginary time charge correlator. Furthermore, at thermal equilibrium the
imaginary time correlator
$\langle\hat{N}_{A}(-i\tau)\hat{N}_{A}(0)\rangle_{T}$ can be related with the
Fourier transform of the appropriate real-time correlator, defined as

$\displaystyle\chi(\omega,T)=\int dte^{i\omega
t}\langle\hat{N}_{A}(t)\hat{N}_{A}(0)\rangle_{T}.$ (6)

This is the charge noise of the QD. As a result Eq. (5) becomes SM

$\begin{split}\Delta
S^{(2)}=&-\log\Big{[}1-2\langle\hat{N}_{A}\rangle_{T/2}+2\int\frac{d\omega}{2\pi}\chi\big{(}\omega,T/2\big{)}e^{-\frac{\omega}{T}}\Big{]},\\\
\Delta
S^{(3)}=&-\frac{1}{2}\log\Big{[}1-3\langle\hat{N}_{A}\rangle_{T/3}+3\int\frac{d\omega}{2\pi}\chi\big{(}\omega,T/3)e^{-\frac{2\omega}{T}}\Big{]}.\end{split}$
(7)

This is our main result for Coulomb blockaded systems. It allows one to
measure Rényi moments of the NEE in thermal states at temperature $T$, from
charge correlations measured at temperature $T/\ell$. The advantage of our
approach is that these charge correlations have been repeatedly measured in
mesoscopic systems using charge sensing techniques (see, e.g., Fujisawa et al.
(2006); Gustavsson et al. (2006a, b); Dial et al. (2013); Wilen et al.
(2021)). In the supplementary material SM  we relate explicitly
$\chi(\omega,T)$ to the voltage-dependent noise Blanter and Büttiker (2000) in
a quantum-point-contact charge detector electrostatically coupled to the QD as
depicted in Fig. 1(a).

Eqs. (7) apply generally to Coulomb blockaded systems. The simplest possible
example is a spinless double dot system. The entanglement of $\log 2$ due to
coherent hopping of an electron between the two dots can be directly computed
from the density matrix, or equivalently from the charge-charge correlation
via Eq. (7), as demonstrated in the supplementary material SM . However, the
power of the relations (7) lies in the prospect of applying them to strongly
correlated systems, such as magnetic impurities embedded in a continuum, which
can lead to a multitude of Kondo effects and to a competition between spin
correlations and screening in multiple-impurity systems (for a review see,
e.g., Grobis et al. (2007)). Here we exemplify the usefulness of these
relations for multi-channel Kondo systems.

Figure 1: (a) The charge-multi-channel-Kondo setup Matveev (1995); Furusaki
and Matveev (1995); Iftikhar et al. (2015, 2018), shown here for two channels
(2CK). In each of the channels ($\alpha=1,2$) an electron can hop from the
metallic dot to its respective lead with matrix elements $J_{\alpha}$ and thus
change the dot occupancy by unity, $N_{d}=0\leftrightarrow 1$, effectively
flipping the “impurity spin”. The gate detuning from charge degeneracy $\Delta
E$ serves as an effective magnetic field. The asymmetry
$T_{-}\propto(J_{1}-J_{2})^{2}$ allows to study the crossover between two and
one-channel Kondo models. We added an additional quantum point contact coupled
electrostatically to the dot, which allows measurements of the dot charge and
its fluctuations. (b) The charge $N_{d}(T)$ and (c) its fluctuations
$\chi(\omega)$, required for the evaluation of the Rényi moments $\Delta
S^{(\ell)}$ of the number entanglement entropy (NEE). $N_{d}(T)$ is shown for
$\Delta E=T_{K}/2$ (yellow), $0$ (red), $-T_{K}$ (blue) with $T_{-}=0$. And
$\chi(\omega)$ is shown for $T=0.01T_{K}$ (solid lines) and $T=0.5T_{K}$
(dashed line) for $T_{-}=\Delta E=0$ (2CK), $T_{-}=T_{K}$ (resonant level
model - RLM) with $\Delta E=0$, and for $\Delta E=T_{K}/2$ with $T_{-}=0$. In
the plot, $\delta$ function peaks SM  are not drawn. (d) The Rényi moment of
NEE $\Delta S^{(2)}$ obtained from $N_{d}$ and $\chi(\omega)$ using Eq. (7)
(solid lines). The NEE $\Delta S^{(1)}$ is also shown for comparison (dashed
lines). We compare the 2CK case, the RLM case, and also the 2CK with finite
level detuning $\Delta E$. (e) $\log-\log$ plot of $\log(2)-\Delta
S^{(\ell)}(T)$ for the 2CK and RLM cases, demonstrating distinct power law
behaviour; up (down) triangles refer to $\ell=1$ ($\ell=2$). (f)
$\log(2)-\Delta S^{(\ell)}(T)$ for a small channel anisotropy scale
$T_{-}/T_{K}=0.1$, displaying a crossover from the 2CK behavior for $T>T_{-}$
(blue) to 1CK behavior for $T<T_{-}$ (red).

The multi-channel Kondo (MCK) model describes an impurity spin-1/2 interacting
antiferromagnetically with $M$ spinfull channels with continuous density of
states. For $M=1$ the impurity is perfectly screened at $T=0$, while for $M>1$
the ground state is a non-Fermi liquid, due to overscreening of the impurity
by the $M$ channels Nozières and Bladin (1980). In both cases, though, the
spin-screening cloud [see, e.g, Barzykin and Affleck, 1998; Affleck et al.,
2009] leads to a zero-temperature entanglement entropy of $\log 2$. Recently,
finite-temperature entanglement measures, such as the entanglement of
formation (EoF) and negativity (${\cal N}$), have been applied to this model
Lee et al. (2015); Kim et al. (2021), demonstrating, for both quantities, a
low-temperature scaling behavior of

${\rm{EoF}},{\cal N}\simeq\log 2-a_{M}T^{2\Delta_{M}},$ (8)

where $a_{M}$ is an $M$-dependent numerical coefficient and $\Delta_{M}$ is
the scaling dimension of the impurity spin operator in the $M$-channel
problem. As determined by the conformal field theory solution Affleck and
Ludwig (1991), $\Delta_{M}=2/(2+M)$ for $M\geq 2$, and for the single channel
case $\Delta_{1}=1$. Unfortunately, these entanglement measures, while
calculable, are practically impossible to measure (as they require
measurements of all elements of the density matrix). Below we demonstrate that
the entanglement measure NEE for the charge-Kondo model, and its Rényi
moments, which can be directly measured by applying Eqs. (7), obey the same
scaling relation as Eq. (8), and thus allow direct measurement of the finite-
temperature entanglement and the scaling dimensions of MCK systems.

Unlike the commonly observed one-channel Kondo effect, signatures of a two-
channel spin-Kondo behavior in quantum dots have only been reported in Potok
et al. (2007); Keller et al. (2015). A major step forward has been recently
taken in Refs. Iftikhar et al., 2015, 2018 which utilized a mapping between
the two charge states of a metallic dot $N_{d}=0,1$ and spin $S_{z}=\pm 1/2$
Matveev (1995); Furusaki and Matveev (1995) to demonstrate scaling and
crossover between single and multi-channel $M=1,2,3$ Kondo effects [here the
number of channels is controlled by the number of one dimensional leads
attached to the dot, see Fig. 1(a)]. Since, in this case of a charge-Kondo
effect, the role of spin is played by the deviation of the occupation from
half-filling, then charge correlations in the charge-Kondo system map onto
spin correlation in spin-Kondo system, thus allowing us to extract the finite
temperature entanglement measure NEE for non-Fermi liquid MCK systems by
utilizing Eqs. (7). In this case the NEE reduces at $T=0$ to the full
entanglement entropy.

To be specific, the charge-MCK Hamiltonian is given by Matveev (1995);
Furusaki and Matveev (1995)

$\displaystyle H=$
$\displaystyle\sum_{\alpha=1}^{M}J_{\alpha}\sum_{k,k^{\prime}}(c_{\alpha\uparrow
k}^{\dagger}c_{\alpha\downarrow k^{\prime}}S_{-}+c_{\alpha\downarrow
k}^{\dagger}c_{\alpha\uparrow k^{\prime}}S_{+})$
$\displaystyle+\sum_{\alpha,k,\sigma}kc_{\alpha\sigma
k}^{\dagger}c_{\alpha\sigma k}+\Delta ES_{z}.$ (9)

The system is depicted in Fig. 1(a) for the case $M=2$. Here $J_{\alpha}$ is
the tunneling between the $\alpha$-th lead and the dot, $c_{\alpha\sigma k}$
and $c^{\dagger}_{\alpha\sigma k}$ denote annihilation and creation operators,
respectively, for electrons with momentum $k$ in the leads for
$\sigma=\uparrow$ or in the dot for $\sigma=\downarrow$. Here the spin flip
operator changes the charge state of the dot, $S_{+}|0\rangle=|1\rangle$, and
$S_{z}=(|1\rangle\langle 1|-|0\rangle\langle 0|)/2=\hat{N}_{d}-1/2$, where
$|0\rangle$ and $|1\rangle$ are the empty and occupied dot states, and
$\hat{N}_{d}$ is the dot number operator. $\Delta E$ is a gate voltage
detuning away from the charge degeneracy point, which corresponds to a
magnetic field in the conventional MCK Kondo model. The electrons are assumed
to be spinless, e.g. due to a large magnetic field. The Hamiltonian (9) is
identical to the spin-MCK Hamiltonian, where the role of the local spin is
played by the occupation states of the metallic dot.

Consider now the NEE between the QD and the leads. Using Eqs. (7), which
expresses the NEE as a correlation function, we find for the second Rényi NEE

$\Delta S^{(2)}=-\log\Big{[}\frac{1}{2}+2\langle
S_{z}(\frac{1}{iT})S_{z}(0)\rangle_{T/2}\Big{]}.$ (10)

The second term is a two-point correlation function of the impurity spin
operator in imaginary time. Using the general conformal field theory
correlation function
$\langle\mathcal{O}(-i\tau)\mathcal{O}(0)\rangle_{T}=\left(\frac{\pi
T}{\sin\pi\tau T}\right)^{2\Delta_{\mathcal{O}}}$ for an operator
$\mathcal{O}$ with scaling dimension $\Delta_{\mathcal{O}}$, we obtain SM

$\langle S_{z}(\frac{1}{iT})S_{z}(0)\rangle_{T/2}\propto\left(\frac{\pi
T/2}{\sin(\pi\frac{1}{T}T/2)}\right)^{2\Delta_{M}}\sim T^{2\Delta_{M}}.$ (11)

Furthermore, we find that at low temperature the NEE $\Delta S^{(1)}$ is also
dominated by the same two point correlator SM .

Accordingly the entanglement measure NEE and its Rényi moments obey the exact
same scaling behavior as EoF and ${\mathcal{N}}$ in Eq. (8). For example for
the 1CK case we obtain a quadratic temperature dependence, $\Delta
S^{(\ell)}_{1CK}\simeq\log 2-a_{1\ell}T^{2}$, whereas in the 2CK case the
dependence is linear, $\Delta S^{(\ell)}_{2CK}\simeq\log 2-a_{2\ell}T$, where
now the non-universal coefficients $a_{M\ell}$ depend also on the Rényi moment
$\ell$. As pointed out in Refs. Lee et al. (2015); Kim et al. (2021), this
linear behavior, corresponding to $\Delta_{2}=1/2$, is an indication of the
existence of a Majorana fermion in the ground state.

The temperature dependence obtained in Eq. (11) is based on a general scaling
argument. In order to make a more quantitative comparison with experiments for
the case of the 1CK and 2CK models, we can use the exact solution of the
corresponding low-temperature Hamiltonian. This allows us to obtain results
for the temperature behavior along the crossover between 1CK and 2CK
behaviors. Also, the same model allows us to provide quantitative results for
the simpler resonant level model.

After a renormalization process at low temperatures, the Hamiltonian for the
$M=2$ MCK model can be mapped into a Majorana resonant level model Emery and
Kivelson (1992); Zaránd and von Delft (2000)

$\begin{split}H=&\sum_{k}k\gamma_{x,-k}\gamma_{x,k}+k\eta_{x-k}\eta_{x,k}\\\
&+i\sqrt{\frac{2T_{K}}{L}}\sum_{k}\gamma_{k}\eta_{d}+i\sqrt{\frac{2T_{-}}{L}}\sum_{k}\eta_{k}\gamma_{d}+i\Delta
E\gamma_{d}\eta_{d}.\end{split}$ (12)

Here $T_{K}$ is the Kondo energy scale and $T_{-}\propto(J_{1}-J_{2})^{2}$
Mitchell et al. (2016) is an energy scale associated with channel asymmetry.
Here $\gamma_{d}$ and $\eta_{d}$ are local Majorana zero modes, and
$\gamma_{x,k}$,$\eta_{x,k}$ represent the mode expansion of a Majorana fields.
The charge occupation operator of the metallic dot is written in terms of the
local Majorana operators as
$\hat{N}_{d}=\frac{1}{2}+i\gamma_{d}\eta_{d}\equiv\hat{N}_{A}$.

This model describes different regimes. The 2CK state corresponds to $\Delta
E=T_{-}=0$. As long as ${\rm{max}}\\{\Delta E,T_{-}\\}\ll T_{K}$, this model
describes the vicinity of the 2CK fixed point. As $T_{-}$ increases, it
faithfully describes the crossover Pustilnik et al. (2004); Mitchell and Sela
(2012) from the 2CK to the 1CK fixed point. Additionally, for the special
value $T_{-}=T_{K}$, this model maps into the non-interacting resonant level
model (RLM), which can be realized in a single level spinless QD, with on site
energy $\Delta E$ and width $\Gamma\equiv T_{K}$.

This model can be solved exactly Emery and Kivelson (1992). The results for
the occupation number $N_{d}$ and $\chi(\omega)$, which can be measured using
the charge detector, were obtained by solving numerically the resulting
integral expressions SM  and are shown in Fig. 1(b,c) for selected model
parameters. For $\Delta E=0$ (either 2CK or RLM), by symmetry $N_{d}=1/2$, and
the NEE Rényi moments $\Delta S^{(\ell)}$ can be extracted using Eq. (7)
solely from the charge noise $\chi(\omega)$. We plot in Fig. 1(d) the
resulting $\Delta S^{(2)}$, and for comparison SM  the NEE ($=\Delta S^{(1)}$)
which is an entanglement monotone Ma et al. (2022), with very similar
behavior. Specifically, the resulting $\Delta S^{(\ell)}$ for the 2CK state
confirms our field theory scaling results as seen in the $\log-\log$ plot in
Fig. 1(e). For the RLM the temperature scaling is quadratic.

In Fig. 1(f) we show the crossover from 2CK to 1CK behavior which can be
observed for a small channel anisotropy $T_{-}$. While $\lim_{T\to 0}\Delta
S^{(\ell)}$ remains $\log 2$ for any $J_{-}$, since by symmetry $N_{d}=1/2$,
the temperature dependence becomes quadratic $\log 2-\Delta S^{(\ell)}\propto
T^{2}$ for $T<T_{-}$. For a finite gate voltage detuning from the fixed point
($\Delta E\neq 0$) we have that $\langle N_{d}\rangle\neq 1/2$ as seen in Fig.
1(b). As a result, $\lim_{T\to 0}\Delta S^{(\ell)}$ decreases below $\log 2$
as shown in Fig. 1(d). Since $\Delta E$ is a relevant perturbation, we observe
a non-monotonic temperature dependence for some range of values of $\Delta E$
as shown in Fig. 1(d) (finite $\Delta E$).

To conclude, we have proposed in this paper an experimental procedure to
measure entanglement at finite temperatures. In particular we have
demonstrated that an entanglement measure - the number entanglement entropy
(NEE) - can be obtained solely from the subsystems charge distribution
function and correlation functions. We formulated exact thermodynamic
relations between moments of the NEE and subsystem charge correlation
functions. A similar observation was made for the quantum Fisher information
which witnesses multipartite entanglement Hauke et al. (2016). In our
approach, the $\ell$-th moment of the NEE of a thermal state at temperature
$T$ is related to a charge correlation function of the same Hamiltonian at
temperature $T/\ell$. The setup we propose for measurement of the NEE in
quantum dot systems has already been utilized to measure the charge noise,
thus we expect that it can be readily extended to measure finite temperature
entanglement measures in various systems, including the frustrated multi-
channel Kondo system.

## Acknowledgments.

We acknowledge support from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme under grant
agreement No. 951541, ARO (W911NF-20-1-0013), the US-Israel Binational Science
Foundation (Grant No. 2016255) and the Israel Science Foundation, grant number
154/19.

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