# Emergence of Multivariate Extremes in Multilayer Inhomogeneous Random Graphs

Daniel<EMAIL_ADDRESS>[ Tiandong
<EMAIL_ADDRESS>[ Daren B.H.
<EMAIL_ADDRESS>[ Department of Statistics, Texas A&M
University presep=, ]e1,e3  Shanghai Center for Mathematical Sciences, Fudan
University presep=, ]e2

###### Abstract

In this paper, we propose a multilayer inhomogeneous random graph model
(MIRG), whose layers may consist of both single-edge and multi-edge graphs. In
the single layer case, it has been shown that the regular variation of the
weight distribution underlying the inhomogeneous random graph implies the
regular variation of the typical degree distribution. We extend this
correspondence to the multilayer case by showing that the multivariate regular
variation of the weight distribution implies the multivariate regular
variation of the asymptotic degree distribution. Furthermore, in certain
circumstances, the extremal dependence structure present in the weight
distribution will be adopted by the asymptotic degree distribution. By
considering the asymptotic degree distribution, a wider class of Chung-Lu and
Norros-Reittu graphs may be incorporated into the MIRG layers. Additionally,
we prove consistency of the Hill estimator when applied to degrees of the MIRG
that have a tail index greater than 1. Simulation results indicate that, in
practice, hidden regular variation may be consistently detected from an
observed MIRG.

05C80,

60G70,

05C82,

60F05,

Multilayer Networks,

Inhomogeneous Random Graphs,

Multivariate Regular Variation,

Tail Estimation,

###### keywords:

[class=MSC]

###### keywords:

, and

##  1 Introduction

The scale-free phenomenon, or the notion that many real-world networks tend to
exhibit power-law degree distributions, is a fundamental hypothesis in network
science that is supported by significant amount of empirical evidence [30,
41]. In fact, the prominence of scale-free networks is so well-established
that estimates of the power-law tail index regularly appear as summary
statistics for networks listed in network repositories such as KONECT [27].
Such findings have led researchers to develop network models that imitate
scale-free behavior. Prominent examples include the preferential attachment
model and the inhomogeneous random graph [3, 31, 7, 39].

Often research in network science makes inquires regarding large degree-degree
relationships when edges are of differing types. For example, in a X (formerly
Twitter) network, does a user having a large number of followers indicate that
they may receive have a large number of likes or retweets? Similarly, in
forum-like websites such as Reddit, is a large number of links in one sub-
forum necessarily associated with a large number of links in another sub-
forum? How does this relationship vary by sub-forum type? Such questions are
naturally analyzed in a multilayer network setting.

We study large degree-degree relationships in a multilayer inhomogeneous
random graph (MIRG) model through the lens of multivariate regular variation
(MRV). Through MRV, we are able to describe the extremal dependence structure
between the layer-wise degrees for a given node; see [35, 26] for more details
on MRV. For single-layer inhomogeneous random graphs, a latent weight is
associated to each node that confers its connectivity in the network. [6]
prove that, for a large class of inhomogeneous random graphs, if the weight
distribution is univariate reguarly varying, then so is the typical degree
distribution. We extend this correspondence to the multilayer setting. Here,
each node is associated a weight vector describing its connectivity in each
layer of the network. We show that if the distribution of the weight vectors
are multivariate regularly varying, then so is the multivariate asymptotic
degree distribution for a specified class of Norros-Reittu and Chung-Lu
models. Furthermore, if the weight vector distribution has hidden regular
variation (HRV), a second-order notion of MRV, then for cases where the HRV
index falls within a certain range, the degree distribution also has the HRV
property; see [14, 16, 17] for detailed discussion on HRV. In essence, for
large networks, the extremal dependence structure exhibited by the weights may
also be shared by the layer-wise degrees under certain circumstances.

As alluded to previously, our approach differs in two key ways from [6].
First, we restrict our study of inhomogeneous graphs to ones that are similar
to the Norros-Reittu and Chung-Lu graphs [32, 9, 10]. In particular, we allow
the layers of the MIRG to consist of both single-edge and multi-edge graphs.
This is an especially important consideration; most multilayer networks of
interest consist of both single-edge and multi-edge layers. Returning to our X
example, a user may only follow another user one time, while they may retweet
or like another user multiple times. Hence, the follower layer is a single-
edge graph, whereas the like and retweet layers allow multiple edges between
two nodes. The class of Norros-Reittu and Chung-Lu graphs considered herein
also extend the single layer results in [6] to graphs such as the generalized
random graph [8]. Secondly, in order to consider this extended class of
Norros-Reittu and Chung-Lu graphs, we study multivariate regular variation of
the asymptotic degree distribution rather than the typical degree
distribution. Modern problems in network science typically involve analyzing
networks with millions or billions of nodes, hence the study of the limiting
degree distribution should accurately describe the extremal behavior of
observed networks [31].

Of particular interest for scale-free networks is the estimation of the power-
law tail index. In addition to being an important object of study on its own,
inference on the tail index can further inform more complex statistical
procedures for scale-free networks [11, 12]. Hence, we additionally consider
consistency of the well-known Hill estimator for the tail index in the MIRG
model. Consistency of the Hill estimator is well-established in the iid and
some time series settings [35, 26]. In the network setting, however, verifying
consistency of the Hill estimator is a considerably more difficult task due to
the degree dependence present in most datasets. In both preferential
attachment networks and single-layer inhomogeneous random graphs, however,
consistency of the Hill estimator has been shown [43, 6]. In both cases,
consistency is obtained by approximating the degree sequence by a sequence of
independent processes; branching proccesses in the case of preferential
attachment graphs and latent weights for inhomogeneous random graphs [see 43,
6, for more details]. We follow a similar, though in some cases more general
approach for the MIRG model. For the Chung-Lu random graph, [6] prove
consistency of the Hill estimator when the tail index is strictly larger than
$2$. By imposing a reasonable restriction on the number of order statistics
used in the Hill estimator, we are able to extend the consistency results in
the Chung-Lu model, among others, for tail indices strictly larger than $1$.

Section 1.2 introduces the MIRG, along with the asymptotic properties of its
degrees. Section 2 provides a short introduction to MRV and HRV, then proves a
correspondence between MRV/HRV of the latent weight vectors and MRV/HRV of the
degree vectors. The Hill estimator is introduced and its consistency in the
MIRG model is considered in Section 3. Section 4 provides a short simulation
study exhibiting that MRV and HRV are detectable in practically-sized networks
generated from the MIRG model, as well as evidence suggesting that a
restriction of the number of order statistics used in the Hill estimator is
necessary to achieve consistency. Section 5 discusses future research
questions and potential extensions raised by the theory developed herein.
Finally, in Section 6 we introduce tools from extreme value theory and random
networks used to develop the main results along with their proofs.

###  1.1 Notation

For $n\in\mathbb{N}$, we use the notation $[n]$ to denote the collection
$\\{1,2,\dots,n\\}$. Vectors will be denoted in boldface. If $\mathbf{z}$ is a
random vector in $\mathbb{R}^{p}$, we let the math-italics $z_{i}$ denote the
value of its $i$-th entry, $i\in[p]$. The zero vector in $\mathbb{R}^{p}$ is
denoted as $\mathbf{0}$ regardless of the dimension $p$. Let
$\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{n}$ be vectors and
$\mathcal{I}=\\{i_{1},i_{2},\dots,i_{r}\\}\subset[n]$, $r\leq n$. The notation
$\mathbf{x}_{\mathcal{I}}$ is then used to denote the collection of vectors
$\\{\mathbf{x}_{i_{1}},\mathbf{x}_{i_{2}},\dots,\mathbf{x}_{i_{r}}\\}$. For
real numbers $a$ and $b$, let $a\wedge b$ and $a\vee b$ denote the minimum and
maximum of $a$ and $b$, respectively.

###  1.2 Multilayer inhomogeneous random graph model

In this section we introduce the multilayer inhomogeneous random graph (MIRG)
model. Suppose $\mathbf{A}(n)=\\{A_{ijl}\\}_{ijl}$ is an $n\times n\times L$
adjacency cube of $n$ nodes and $L$ layers. That is, $A_{ijl}$ represents the
number of edges between nodes $i$ and $j$ in layer $l$. For every $l\in[L]$,
$\\{A_{ijl}\\}_{ij}$ is a symmetric matrix; each layer of the MIRG is an
undirected (multi)graph that allows self-loops. We let
$\mathcal{L}_{1}\subset[L]$ collect the layers of $\mathbf{A}(n)$ that are
multigraphs (i.e. allow multiple edges) and let
$\mathcal{L}_{2}=[L]\setminus\mathcal{L}_{1}$ refer to the layers which are
graphs (i.e. allow at most one edge between two nodes).

Let $\mathbf{W}_{[n]}=\\{\mathbf{W}_{i}\\}_{i=1}^{n}$ be i.i.d. random vectors
in $\mathbb{R}_{+}^{L}$ with a common continuous distribution. We often refer
to the $\mathbf{W}_{[n]}$ as weight vectors. We may think of $W_{il}$ as some
latent attribute that conveys the connectivity of node $i$ in layer $l$. In
most applications, a larger value of $W_{il}$ indicates that node $i$ will
attract more edges in layer $l$, though the MIRG model permits one to augment
the extent to which that is true. For each layer $l\in[L]$, we let
$T_{l}(n)=\sum_{i=1}^{n}W_{il}$ denote the sum of the weights in layer $l$.

Conditional on the latent weight vectors $\mathbf{W}_{[n]}$, the MIRG models
the multigraph and graph layers similarly. For $l\in\mathcal{L}_{1}$, we
assume

$\displaystyle
A_{ijl}\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Poisson}\left(g_{l}(W_{il}W_{jl}/{T_{l}(n)})\right)\qquad\text{for
}1\leq i\leq j\leq n,$ (1)

and for $l\in\mathcal{L}_{2}$ we have

$\displaystyle
A_{ijl}\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Bernoulli}\left(g_{l}(W_{il}W_{jl}/{T_{l}(n)})\right)\qquad\text{for
}1\leq i\leq j\leq n,$ (2)

Here, $g_{l}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ allows for heterogeneity
in the layer-wise connection probabilities. For $l\in\mathcal{L}_{2}$, we
naturally restrict the range of $g_{l}$ to the interval $[0,1]$. Typically
$g_{l}$ is chosen to satisfy $g_{l}(0)=0$. Practical choices of $g_{l}$ are
often monotonically increasing and concave, indicating that a larger weight
contributes to more connections, although with diminishing returns. When
$g_{l}$ is the identity function for $l\in\mathcal{L}_{1}$, the multigraph
layers are easily recognized as the Norros-Reittu (NR) random multigraph [32].
For $l\in\mathcal{L}_{2}$, $g_{l}(x)=x\wedge 1$ corresponds to the Chung-Lu
random graph and $g_{l}(x)=x/(1+x)$ corresponds to a version of the
generalized random graph (GRG) [8, 9, 10, 38]. The theory presented herein
covers these choices of $g_{l}$, and more general models as well.

Define the degree of node $i$ in layer $l$ to be
$D_{il}(n)=\sum_{j=1}^{n}A_{ijl}$. From (1) and the conditional independence
of the $A_{ijl}$, it is easily seen that for $l\in\mathcal{L}_{1}$

$\displaystyle
D_{il}(n)\mid\mathbf{W}_{[n]}\sim\text{Poisson}\left(\sum_{j=1}^{n}g_{l}(W_{il}W_{jl}/{T_{l}(n)})\right),\qquad\text{for
}i=1,\dots,n,$ (3)

and for $l\in\mathcal{L}_{2}$

$\displaystyle D_{il}(n)\mid\mathbf{W}_{[n]}\sim\text{Poisson-
binomial}\left(g_{l}(W_{il}W_{1l}/{T_{l}(n)}),\dots,g_{l}(W_{il}W_{nl}/{T_{l}(n)})\right),$
(4)

for $i=1,\dots,n$ where the Poisson-binomial distribution is classically
defined as the sum of independent Bernoulli random variables [37]. For
understanding properties of the degree distribution, it is far easier to work
with (3) due to the availability of results regarding the Poisson
distribution. Results derived from (3) are then typically translated to
results for (4) using the maximal coupling of Poisson and Bernoulli random
variables [see Section 2.2 of 40]. We follow a similar strategy, though in
some cases we must work with (4) alone.

We now introduce the only non-trivial condition on the functions
$\\{g_{l}\\}_{l\in[L]}$ in the MIRG model. A similar assumption is made in
[38] when studying the distances in finite-variance inohomogeneous random
graphs. Throughout, we denote the three main conditions we build upon as (C1),
(C2) and (C3), the latter two of which will be introduced as needed.

  * (C1)

Each $g_{l}$ satisfies

$\displaystyle g_{l}(x)=c_{l}x+O(x^{1+\nu})\qquad\text{as }x\downarrow
0,\text{ for some }\nu>0,$

for constants $c_{l}>0$.

From (C1), we may find constants $M,\delta>0$, not depending on $l$ such that

$\displaystyle|g_{l}(x)-c_{l}x|\leq Mx^{1+\nu}\qquad\text{for all
}x<\delta\text{ and }l\in[L],$ (5)

a fact we repeatedly use. Condition (C1) states for every $l\in[L]$,
$g_{l}(x)$ can be approximated by a scalar multiple of $x$ when $x$ is small.
This, combined with a finite mean assumption on the weights, allows us to
approximate

$\displaystyle\sum_{j=1}^{n}g_{l}(W_{il}W_{1l}/{T_{l}(n)})\approx\sum_{j=1}^{n}c_{l}W_{il}W_{1l}/{T_{l}(n)}=c_{l}W_{il},$
(6)

which combined with (3) suggests that for large networks, we may be able to
approximate the degree of node $i$ in layer $l$ by its weight. Furthermore,
(6) suggests that the degrees may be approximately independent, a key
ingredient in proving the consistency of the Hill estimator in the MIRG model.

We now describe the behavior of the degree distribution in the MIRG model. Let
$\mathbf{D}_{i}(n)$ be a random vector in $\mathbb{R}^{+}_{L}$ collecting the
degrees of node $i$ in each layer. From (3) and (4), it is seen that the
degrees across nodes are identically distributed. Thus, throughout we only
consider the distribution of $\mathbf{D}_{1}(n)$. The asymptotic distribution
of $\mathbf{D}_{1}(n)$ is presented in Lemma 1.1. In order to adequately
describe the asymptotic distribution, we define the random variable
$\boldsymbol{\mathcal{D}}_{1}\in\mathbb{R}_{+}^{L}$ which, when conditioned on
$\mathbf{W}_{1}$, has conditionally independent Poisson entries with element-
wise means $(c_{1}W_{11},\dots,c_{L}W_{1L})$. Results similar to Lemma 1.1
have been proved in the single-layer network case (see Theorem 3.13 of [7] or
Corollary 6.11 of [40], for example), though we present the result and its
proof for completeness. Lemma 1.1 is proven in Section 6.3.1.

######  Lemma 1.1.

Suppose $\mathbb{E}[\|\mathbf{W}_{1}\|_{1}]<\infty$ and $\mathbf{D}_{1}(n)$ is
generated from a MIRG satisfying (C1). Then for each
$\mathbf{m}\in\mathbb{Z}_{+}^{L}$

$\displaystyle\mathbb{P}\left(\mathbf{D}_{1}(n)=\mathbf{m}\right)\rightarrow\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}=\mathbf{m}\right)\equiv\mathbb{E}\left[\prod_{l=1}^{L}\frac{\left(c_{l}W_{1l}\right)^{m_{l}}e^{-c_{l}W_{1l}}}{m_{l}!}\right],$

as $n\rightarrow\infty$.

##  2 Regular variation of the MIRG model

###  2.1 Preliminaries on MRV and HRV

In this section we provide a short overview of the multivariate regular
variation. These are important tools to characterize the extremal dependence
structure between elements of a random vector. Suppose we are working in the
metric space $(\mathbb{R}_{+}^{L},d_{p})$, $L\geq 1$, where
$d_{p}(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|_{p}$ is the $\ell_{p}$
norm for any $p\geq 1$ and $\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{L}$. Let
$\mathbb{C}_{0}\subset\mathbb{C}$, be two closed cones in
$\mathbb{R}_{+}^{L}$. The theoretical foundation of regularly varying measures
is given by $\mathbb{M}$-convergence on $\mathbb{C}\setminus\mathbb{C}_{0}$,
which we provide in Definition 2.1 [see 5, 24, 26, 28, for more on regular
variation].

######  Definition 2.1.

Let $\mathbb{M}(\mathbb{C}\setminus\mathbb{C}_{0})$ be the set of Borel
measures on $\mathbb{C}\setminus\mathbb{C}_{0}$ which are finite on sets
bounded away from $\mathbb{C}_{0}$. Let
$\mathcal{C}(\mathbb{C}\setminus\mathbb{C}_{0})$ denote the set of all
continuous, bounded and non-negative functions on
$\mathbb{C}\setminus\mathbb{C}_{0}$ whose supports are bounded away from
$\mathbb{C}_{0}$. Then for
$\mu_{n},\mu\in\mathbb{M}(\mathbb{C}\setminus\mathbb{C}_{0})$, we say that
that $\mu_{n}\rightarrow\mu$ in
$\mathbb{M}(\mathbb{C}\setminus\mathbb{C}_{0})$ if

$\displaystyle\int fd\mu_{n}\rightarrow\int fd\mu,$

for all $f\in\mathcal{C}(\mathbb{C}\setminus\mathbb{C}_{0})$.

Without loss of generality, we may take the functions in
$\mathcal{C}(\mathbb{C}\setminus\mathbb{C}_{0})$ to be uniformly continuous as
well. Here, $f$ is uniformly continuous if the modulus of continuity

$\displaystyle\Delta_{f}^{p}(\delta)=\sup_{\mathbf{x},\mathbf{y}\in\mathbb{C}\setminus\mathbb{C}_{0}}\left\\{\left|f(\mathbf{x})-f(\mathbf{y})\right|:\|\mathbf{x}-\mathbf{y}\|_{p}<\delta\right\\},$

is such that $\Delta_{f}^{p}(\delta)\rightarrow 0$ as $\delta\downarrow 0$.
Note that Definition 2.1 is equivalent to requiring that
$\lim_{n\rightarrow\infty}\mu_{n}(A)=\mu(A)$ for all $\mu$-continuity Borel
sets $A$ bounded away from $\mathbb{C}_{0}$ [see 28, Theorem 2.1]. With
$\mathbb{M}$-convergence in hand, we may now formally define a regularly
varying distribution function in the case of $\mathbb{C}=\mathbb{R}_{+}^{d}$
and $\mathbb{C}_{0}=\\{\mathbf{0}\\}$.

######  Definition 2.2.

We say that the distribution $\mathbb{P}\left(\mathbf{Z}\in\cdot\right)$ of a
random vector $\mathbf{Z}$ on $\mathbb{R}_{+}^{L}$, $L\geq 1$, is (standard)
regularly varying on $\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\}$ with index
$\alpha>0$ if there exists some regularly varying scaling function $b(t)$ with
index $1/\alpha$ and a limit measure
$\nu\in\mathbb{M}(\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$ such that as
$t\rightarrow\infty$

$\displaystyle
t\mathbb{P}\left(\mathbf{Z}/b(t)\in\cdot\right)\rightarrow\nu(\cdot),\qquad\text{in
}\mathbb{M}(\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\}).$ (7)

If $\mathbb{P}\left(\mathbf{Z}\in\cdot\right)$ is regularly varying, we may
write
$\mathbb{P}\left(\mathbf{Z}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$.

In applications, extreme values often occur in more than one risk region.
Moreover, these risk regions may exhibit subtle behavior that may be
overlooked when estimating tail probabilities by assuming just one degree of
regular variation. When the limit measure $\nu$ concentrates on a subcone of
$\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\}$, we may seek additional regular
variation outside of the subcone by employing hidden regular variation as in
Definition 2.3.

######  Definition 2.3.

We say that the regularly varying distribution
$\mathbb{P}\left(\mathbf{Z}\in\cdot\right)$ on
$\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\}$ has hidden regular variation on
$\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}$ is there exists
$0<\alpha\leq\alpha_{0}$, scaling functions $b(t)\in\text{RV}_{1/\alpha}$,
$b_{0}(t)\in\text{RV}_{1/\alpha_{0}}$ with $b(t)/b_{0}(t)\rightarrow\infty$
and limit measures $\nu$, $\nu_{0}$ such that

$\displaystyle\mathbb{P}\left(\mathbf{Z}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})\cap\text{MRV}(\alpha_{0},b_{0}(t),\nu_{0},\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}),$
(8)

or, in other words,
$\mathbb{P}\left(\mathbf{Z}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$
and
$\mathbb{P}\left(\mathbf{Z}\in\cdot\right)\in\text{MRV}(\alpha_{0},b_{0}(t),\nu_{0},\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$.

For purposes of statistical inference, it is often convenient to express the
limit measure $\nu_{0}$ in terms of generalized polar coordinates [14, 28].
For the unit sphere with respect to the forbidden zone $\mathbb{C}_{0}$
defined by
$\aleph_{\mathbb{C}_{0}}=\left\\{\mathbf{x}\in\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}:d_{p}(\mathbf{x},\mathbb{C}_{0})=1\right\\}$,
we may further define the generalized polar coordinate transform GPOLAR:
$\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}\mapsto(0,\infty)\times\aleph_{\mathbb{C}_{0}}$
by

$\displaystyle\text{GPOLAR}(\mathbf{x})=\left(d_{p}(\mathbf{x},\mathbb{C}_{0}),\frac{\mathbf{x}}{d_{p}(\mathbf{x},\mathbb{C}_{0})}\right).$

With the GPOLAR transformation in hand, we may rewrite (8) as

$\displaystyle
t\mathbb{P}\left(\left(\frac{d_{p}(\mathbf{Z},\mathbb{C}_{0})}{b_{0}(t)},\frac{\mathbf{Z}}{d_{p}(\mathbf{Z},\mathbb{C}_{0})}\right)\in\cdot\right)\rightarrow(\nu_{\alpha_{0}}\times
S_{0})(\cdot)\qquad\text{in
}\mathbb{M}((0,\infty)\times\aleph_{\mathbb{C}_{0}}),$ (9)

where $\nu_{\alpha_{0}}(x,\infty)=x^{-\alpha_{0}}$ for $x>0$ and $S_{0}$ is a
probability measure on $\aleph_{\mathbb{C}_{0}}$. The convergence (9)
immediately implies that $d_{p}(\mathbf{Z},\mathbb{C}_{0})$ has a regularly
varying distribution function with tail index $\alpha_{0}$.

###  2.2 MRV and HRV in the MIRG model

In this section we relate regular variation of the weights to regular
variation of the degrees in the MIRG model. For a large class of inhomogeneous
random graphs (i.e. single-layer), [6] find that regular variation of the
weight distribution implies regular variation of the typical degree
distribution. It is important to note that this is a finite-sample result;
regular variation of the typical degree distribution is achieved irrespective
of the network size. Modern network science, however, often deals with
networks that have an extremely large number of vertices [31]. Thus, it is
also reasonable to instead consider properties of the asymptotic degree
distribution. Inspection of the asymptotic degree distribution also permits
the study of a range of models beyond the Chung-Lu or Norros-Reittu random
graphs as mentioned in Section 1.2.

As elucidated in Theorem 2.4, we find that multivariate regular variation of
the weight distribution implies multivariate regular variation of the
asymptotic degree distribution. Furthermore, the asymptotic degree
distribution inherits the same scaling function and limit measure as the
(scaled) weight distribution. Here, we define
$\boldsymbol{\mathcal{W}}_{1}=(\mathcal{W}_{11},\dots,\mathcal{W}_{1L})\equiv(c_{1}W_{11},\dots,c_{L}W_{1L})$
and $\boldsymbol{\mathcal{D}}_{1}$ as in Lemma 1.1. In addition, from
Definition 2.3, it is clear that hidden regular variation of the weight
distribution also implies hidden regular variation of the degree distribution,
as long as the hidden regular variation is not too subtle. Theorem 2.4 is
proven in Section 6.3.2.

######  Theorem 2.4.

Let $\alpha>0$ and $\alpha_{0}\in[\alpha,2\alpha)$.

  * (a)

If
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$,
for some scaling function $b(t)\in RV_{1/\alpha}$ and limit measure $\nu$,
then
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$.

  * (b)

If
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})\cap\text{MRV}(\alpha_{0},b_{0}(t),\nu_{0},\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$,
for scaling functions $b(t)\in RV_{1/\alpha}$, $b_{0}(t)\in RV_{1/\alpha_{0}}$
with $b(t)/b_{0}(t)\rightarrow\infty$ as $t\rightarrow\infty$, limit measures
$\nu$, $\nu_{0}$, and a closed cone $\mathbb{C}_{0}$, then
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})\cap\text{MRV}(\alpha_{0},b_{0}(t),\nu_{0},\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$.

For part (b), note that we require $\alpha_{0}\in[\alpha,2\alpha)$. Seemingly,
even if regular variation of
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)$ is confined to a
subcone $\mathbb{C}_{0}$,
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)$ will produce
lighter tailed behavior on $\mathbb{R}_{+}^{L}\setminus\ \mathbb{C}_{0}$ due
to the Poisson dispersion of $\boldsymbol{\mathcal{D}}_{1}$ off of
$\mathbb{C}_{0}$. As evidenced in simple example to follow, the behavior off
of $\mathbb{C}_{0}$ is not necessarily hidden regular variation. The example
also shows that degree distances from $\mathbb{C}_{0}$, however, can be
regularly varying with tail index $2\alpha$. Thus, in order for hidden regular
variation of $\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)$ to
confer to $\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)$, we
require that the hidden tail behavior of
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)$ dominates the
Poisson dispersion. Otherwise, the hidden regular variation from
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)$ may be corrupted
by degrees that spread from $\mathbb{C}_{0}$.

######  Example.

To explore the tail behavior of
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)$ on
$\mathbb{R}_{+}^{L}\setminus\ \mathbb{C}_{0}$, we provide a simple example in
$(\mathbb{R}_{+}^{2},d_{2})$. Suppose $W_{i1}$ is regularly varying with tail
index $\alpha>0$ and let $W_{i2}=W_{i1}$ for $i\in[n]$. It is easily seen that
$\mathbf{W}_{1}$ is multivariate regularly varying and satisfies full
asymptotic dependence [17]. That is,
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$
and the limit measure $\nu$ only place mass on the ray
$\mathbb{C}_{0}=\\{\mathbf{x}\in\mathbb{R}_{+}^{2}:x_{1}=x_{2}\\}$. Here we
take $b(t)$ to be the $1-1/t$ quantile of $W_{11}$, $t\geq 1$. Let

$\displaystyle A_{ij1}\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}$
$\displaystyle\text{Poisson}(W_{i1}W_{j1}/T_{1}(n))\qquad\text{for }1\leq
i\leq j\leq n,$ $\displaystyle
A_{ij2}\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}$
$\displaystyle\text{Poisson}(W_{i2}W_{j2}/T_{2}(n))\qquad\text{for }1\leq
i\leq j\leq n,$

so that $\mathbf{D}_{1}(n)$ and $\boldsymbol{\mathcal{D}}_{1}$ have the same
conditional distribution given $\mathbf{W}_{[n]}$, for all $n\in\mathbb{N}$.
In order to determine whether hidden regular variation exists off of
$\mathbb{C}_{0}$, we evaluate the joint distribution of
$d_{2}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})=|\mathcal{D}_{11}-\mathcal{D}_{12}|/\sqrt{2}$
and
$\boldsymbol{\mathcal{D}}_{1}/d_{2}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})$.
Let $b_{0}(t)$ be the $1-1/t$ quantile of $\sqrt{W_{11}}$. In Section 6.3.3,
we prove that for any $u,v\geq 0$

$\displaystyle\begin{split}t\mathbb{P}\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{b_{0}(t)}\left(\frac{\sqrt{\pi}}{\Gamma(\alpha+\frac{1}{2})}\right)^{\frac{1}{2\alpha}}>u,\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}>v\right)&\\\
\rightarrow
u^{-2\alpha}\cdot\frac{2^{1-\alpha}\sqrt{\pi}}{\Gamma(\alpha+\frac{1}{2})}&\int_{0}^{1/v}z^{2\alpha}\phi(z)dz,\end{split}$
(10)

where $\phi(\cdot)$ is the standard normal density function. Statement (10)
immediately gives that the marginal distribution of
$d_{2}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})$ is regularly varying with
tail index $2\alpha$. Thus, as a practical concern, one is unable to detect
potential hidden regular variation conferred from
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)$ unless the
associated tail index, $\alpha_{0}$, is smaller than $2\alpha$ (see Section
4.1 for more details). In addition, (10) suggests that
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)$ is not hidden
regularly varying off of $\mathbb{C}_{0}$. In essence, the lack of hidden
regular variation is due to a mismatch of orders. Conditional on large
$W_{11}$, the distance of $\boldsymbol{\mathcal{D}}_{1}$ from the diagonal is
on the order of $\sqrt{W_{11}}$ while its component-wise means are both
$W_{11}$. Hence, in order for the distribution of
$\boldsymbol{\mathcal{D}}_{1}/d_{2}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})$
to stabilize and result in a limiting product measure, one must correct this
mismatch by scaling the distances by $\sqrt{W_{11}}$.

Nevertheless, Theorem 2.4 provides a flexible framework for generating
multilayer networks with multivariate regularly varying degree distributions.
Furthermore, full control is exerted on the extremal dependence demonstrated
by the degrees. This is in stark contrast with the currently available avenues
of modeling degree dependence in network science. For example, in single-layer
directed preferential attachment networks, the state of the art only allows
extreme out/in-degrees to concentrate on lines through the origin [42].
Compared to the direct simulation of networks with multivariate regularly
varying degree distributions, generating weights in $\mathbb{R}_{+}^{L}$ with
a given extremal dependence structure is a much simpler task [see 16, for more
on generating data with MRV]. Additionally, said dependence structure is
immediately conferred to the degrees. Such properties, along with the
flexibility allotted between layers in both edge counts and node affinity make
the MIRG model a formidable choice for modeling multilayer networks with
extremal degree behavior.

##  3 Tail index estimation in the MIRG model

###  3.1 Preliminaries on the Hill estimator

In this section, we formally introduce the Hill estimator and provide
definitions of associated tools commonly used to prove its consistency.
Suppose $X_{1},X_{2},\dots,X_{n}$ is a sequence of iid regularly varying
random variables with tail index $\alpha>0$. In addition, let their order
statistics be given by $X_{(1)}\geq X_{(2)}\geq\dots\geq X_{(n)}$. Then, the
Hill estimator of the inverse of the tail index is given by

$\displaystyle
H_{k,n}=\frac{1}{k}\sum_{i=1}^{k}\log\frac{X_{(i)}}{X_{(k+1)}},$ (11)

where $k\geq 1$ denotes the number of order statistics used in in the
estimation of $\alpha^{-1}$ [23]. When $k=k_{n}\rightarrow\infty$ and
$k_{n}/n\rightarrow\infty$ as $n\rightarrow\infty$, we have that

$\displaystyle H^{-1}_{k,n}\xrightarrow{p}\alpha.$

Note that the Hill estimator has been shown to be consistent in a wide variety
of data contexts beyond the iid setting. Such instances include stationary
time series and network datasets [29, 34, 43]. For iid data, a minmum distance
procedure has been proposed to choose the optimal number of order statistics
used in the estimation of $\alpha$ [13, 20]. In the network setting, the Hill
estimator has received considerable attention in applications due to the
effect the tail index has on other properties of the degree distribution [18].
[41] use the Hill estimator, among other tail index estimators, to verify that
a significant portion of networks have power law degree distributions. In
addition, [41] discuss the practical considerations when employing the Hill
estimator to network data, including the difficulty faced in threshold
selection in the presence of discrete power-law data.

Often, consistency of the Hill estimator is derived through analysis of the
tail empirical measure

$\displaystyle\nu_{n}(\cdot)=\frac{1}{k_{n}}\sum_{i=1}^{n}\epsilon_{X_{i}/b(n/k_{n})}(\cdot),$
(12)

where $b(t)$ satisfies $\mathbb{P}\left(X_{i}>b(t)\right)\sim 1/t$ as
$t\rightarrow\infty$. Here, $\epsilon_{x}(A)$ places point mass on
$A\subset(0,\infty]$ if $x\in A$ and is $0$ otherwise. Denote the set of non-
negative Radon measures on $(0,\infty]$ by $M_{+}((0,\infty])$ and define the
measure $\nu_{\alpha}$ by $\nu_{\alpha}(y,\infty]=y^{-\alpha}$ for $y>0$. By
seeking weak convergence of the random measure $\nu_{n}$ to $\nu_{\alpha}$ in
$M_{+}((0,\infty])$, one may derive consistency of the Hill estimator through
continuous mapping arguments. Such methods are reviewed in Section 6.5 in the
MIRG model setting. See Theorem 3.1 for more details.

###  3.2 Hill estimation in the MIRG model

In this section we prove consistency of the Hill estimator in the MIRG model
when the weight vectors are multivariate regularly varying with tail index
$\alpha>1$. Consistency of the Hill estimator in the single layer case is
proven for the Norros-Reittu model for $\alpha>0$ and Chung-Lu model for
$\alpha>2$ in [6]. Though one could consider applying the Hill estimator to
each individual layer-wise degree sequence $\\{D_{il}(n)\\}_{i=1}^{n}$,
$l\in[L]$, in the MIRG model we consider a procedure that simultaneously
employs all of $\\{\mathbf{D}_{i}(n)\\}_{i=1}^{n}$ in the estimation, i.e.
$\|\mathbf{D}_{i}(n)\|_{p}$. This is also a common strategy when exploring the
dependence structure of multivariate extremes. In addition, we extend
consistency of the Hill estimator to the case of $\alpha\in(1,2]$ for the
Chung-Lu model, a case not covered in [6], among other inhomogenous random
graphs. The case where $\alpha\in(1,2]$ is of distinct importance since many
networks of interest have a tail index within that range [1, 2, 4, 31].

Thus, in order to meaningfully estimate $\alpha$ from the degrees of the MIRG
model, we introduce our second assumption:

  * (C2)

The random vector $\boldsymbol{\mathcal{W}}_{1}$ is such that
$\mathbb{P}\left(\boldsymbol{\mathcal{W}}_{1}\in\cdot\right)\in\text{MRV}(\alpha,b(t),\nu,\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\})$
for $\alpha>1$.

If the weights $\mathbf{W}_{[n]}$ were accessible, condition (C2) implies the
tail index $\alpha$ could be estimated by using Hill estimator computed on the
weight radii $\\{\|\boldsymbol{\mathcal{W}}_{i}\|_{p}\\}_{i\in[n]}$ for some
$p\in\mathbb{N}$. That is, one would estimate $\alpha^{-1}$ via the statistic

$\displaystyle
H^{\star}_{k,n}=\frac{1}{k}\sum_{i=1}^{k}\log\frac{R^{\star}_{(i)}(n)}{R^{\star}_{(k+1)}(n)},$

where we define $R^{\star}_{i}(n)=\|\boldsymbol{\mathcal{W}}_{i}\|_{p}$ for
$i\in[n]$. Since the weights are inaccessible, however, it behooves us to
instead consider employing the estimator based on the degrees

$\displaystyle
H_{k,n}=\frac{1}{k}\sum_{i=1}^{k}\log\frac{R_{(i)}(n)}{R_{(k+1)}(n)},$

where $R_{i}(n)=\|\mathbf{D}_{i}(n)\|_{p}$. From, (3), (4) and the
approximation (6), we may expect that $R^{\star}_{(i)}$ is close to $R_{(i)}$
for $i\in[n]$. This approximation is more viable when $i$ is small since we
expect strong concentration of $R_{(i)}$ around $R^{\star}_{(i)}$ given the
mixed-Poisson nature of the asymptotic degree distribution in Lemma 1.1 and
the fact that Poisson random variables with large rates experience strong
concentration. Hence, we expect that $H_{k,n}$ should approximate
$H^{\star}_{k,n}$, and therefore $H^{-1}_{k_{n},n}$ should be a consistent
estimator of $\alpha$ for a carefully specified sequence $k_{n}$.

In order to make these notions rigorous, we introduce the tail empirical
measure based on the sequence $\\{R^{\star}_{i}\\}_{i=1}^{n}$

$\displaystyle\nu^{\star}_{n}(\cdot)=\frac{1}{k_{n}}\sum_{i=1}^{n}\epsilon_{R^{\star}_{i}(n)/b(n/k_{n})}(\cdot),$
(13)

where we take $b(t)$ to be the $1-1/t$ quantile of the distribution of
$R^{\star}_{1}$ for $t\geq 1$. Since $R^{\star}_{1}$ has a regularly varying
distribution function with tail index $\alpha$, then $b(t)$ is reguarly
varying with index $1/\alpha$ [see Lemma 3.3 of 6]. Under (C2), it is well
known that

$\displaystyle\nu^{\star}_{n}\Rightarrow\nu_{\alpha}.$ (14)

in $M_{+}((0,\infty])$ as $n\rightarrow\infty$, $k_{n}\rightarrow\infty$ and
$k_{n}/n\rightarrow 0$ [see Theorem 4.1 in 35]. We seek an analogous weak
convergence result for the tail empirical measure based on the degrees

$\displaystyle\nu_{n}(\cdot)=\frac{1}{k_{n}}\sum_{i=1}^{n}\epsilon_{R_{i}(n)/b(n/k_{n})}(\cdot),$
(15)

As in [6], this weak convergence is obtained via an approximation by the tail
empirical measure based on the independent weights, $\nu^{\star}_{n}$. Weak
convergence of $\nu_{n}$ is presented in Theorem 3.1. In order to prove
Theorem 3.1, we require our third and final condition.

  * (C3)

Suppose $n^{1/\alpha}/k_{n}=O(n^{-\kappa})$ for some
$\kappa\in(0,(\alpha-1)/\alpha)$.

Condition (C3) implies that in order to achieve the weak convergence of the
tail empirical measure, and thus consistency of the Hill estimator in the MIRG
model, $k_{n}$ must be chosen to grow at a slightly faster rate than
$n^{1/\alpha}$. That is, as the tail becomes heavier, more order statistics
must be employed to obtain consistency of the Hill estimator. The assumption
(C3) emerges in the approximation (6) which is valid when the ratio of the
weights to the sum of the weights are small. As $\alpha$ approaches $1$ from
above, however, we may expect a portion of the large weights to be so large
that their associated ratios are near $1$ and the approximation no longer
holds. In order to counteract this effect, we must ensure that the Hill
estimator is majorly compromised of smaller order statistics that satisfy the
approximation (6).

######  Theorem 3.1.

Let $\\{\mathbf{D}_{i}(n)\\}_{i=1}^{n}$ be a degree sequence from a MIRG
satisfying (C1) and (C2). Suppose $k_{n}$ is an intermediate sequence
satisfying $k_{n}\rightarrow\infty$ and $k_{n}/n\rightarrow 0$ as
$n\rightarrow\infty$. Additionally assume that $k_{n}$ satisfies (C3). Then

$\displaystyle\nu_{n}\Rightarrow\nu_{\alpha},$ (16)

in $M_{+}((0,\infty])$ as $n\rightarrow\infty$.

Theorem 3.1 is proven in Section 6.4. In order to obtain consistency of the
Hill estimator from Theorem 3.1, an intermediate step is to use standard
continuous mapping arguments reviewed in Section 6.5 to derive weak
convergence of

$\displaystyle\hat{\nu}_{n}(\cdot)=\frac{1}{k_{n}}\sum_{i=1}^{n}\epsilon_{R_{i}(n)/R_{(k_{n})}(n)}(\cdot),$
(17)

where the unknown $b(n/k_{n})$ in (15) is estimated by $R_{(k_{n})}(n)$. That
is, we obtain

$\displaystyle\hat{\nu}_{n}\Rightarrow\nu_{\alpha},$ (18)

in $M_{+}((0,\infty])$ as $n\rightarrow\infty$. Such steps are outlined in
Section 6.5. With (18) in hand, we may now present the consistency of the Hill
estimator for the MIRG model.

######  Theorem 3.2.

Let $\\{\mathbf{D}_{i}(n)\\}_{i=1}^{n}$ be a degree sequence from a MIRG
satisfying (C1) and (C2). Suppose $k_{n}$ is an intermediate sequence
satisfying $k_{n}\rightarrow\infty$ and $k_{n}/n\rightarrow 0$ as
$n\rightarrow\infty$. Additionally assume that $k_{n}$ satisfies (C3). Then as
$n\rightarrow\infty$

$\displaystyle H_{k_{n},n}\xrightarrow{p}1/\alpha,$

in $\mathbb{R}$.

The proof of Theorem 3.2 is provided in Section 6.5.

##  4 Simulation evidence

In this section, we provide some simulations that elucidate the phenomena
presented in Theorems 2.4 and 3.2. In particular, for Theorem 2.4, we present
empirical evidence supporting the claim that hidden regular variation can be
detected in the MIRG model. For Theorem 3.2, we perform simulations that
inspect the neccessity of assumption (C3) for achieving consistency of the
Hill estimator in the MIRG model.

###  4.1 Hidden regular variation in the MIRG

In this section, we present an experiment that indicates that hidden regular
variation may be detected in the MIRG model. Following a similar procedure as
in [17], Example 2, for generating hidden regular variation, suppose
$V_{1}\sim\text{Pareto}(\alpha)$ and $V_{2}\sim\text{Pareto}(\alpha_{0})$
independently, where $\alpha=1.1$ and $\alpha_{0}=1.3$. Suppose
$\Theta_{1}\sim\text{Beta}(5,5,0.4,0.6)$ and
$\Theta_{2}\sim\text{Uniform}(0,0.4)$ independently of each other, $V_{1}$ and
$V_{2}$. Here, $Y\sim\text{Beta}(b_{1},b_{2},c_{1},c_{2})$ if
$Y=(c_{2}-c_{1})X+c_{1}$ for $X\sim\text{Beta}(b_{1},b_{2})$ and
$b_{1},b_{2}>0$, $c_{2}>c_{1}\geq 0$. Then let

$\displaystyle\mathbf{W}=(W_{1},W_{2})=\begin{cases}(V_{1}\Theta_{1},V_{1}(1-\Theta_{1}))&\text{with
probability }1/2\\\ (V_{2}\Theta_{2},V_{2}(1-\Theta_{2}))&\text{with
probability }1/2\end{cases}.$ (19)

In (19), we have employed the inverse $\ell_{1}$ polar coordinate transform to
produce a random variable $\mathbf{W}$ that possesses MRV on
$\mathbb{R}^{2}_{+}\setminus\\{\mathbf{0}\\}$ with tail index $\alpha=1.1$. In
particular, the limit measure concentrates mass on the cone

$\displaystyle\mathbb{C}_{0}=\left\\{\mathbf{x}\in\mathbb{R}_{+}^{2}:\frac{2}{3}x_{1}\leq
x_{2}\leq\frac{3}{2}x_{1}\right\\}.$

We may seek further regular variation off of $\mathbb{C}_{0}$. For simplicity,
(19) was designed so that the limit measure $\nu_{0}$ only places mass above
$\mathbb{C}_{0}$, though hidden regular variation below $\mathbb{C}_{0}$ may
also be considered. Using the Euclidean metric,

$\displaystyle
d_{2}(\mathbf{x},\mathbb{C}_{0})=\frac{x_{2}-1.5x_{1}}{\sqrt{1+1.5^{2}}},\qquad\text{for
}x_{2}>1.5x_{1},$

and $\aleph_{\mathbb{C}_{0}}$ is given by

$\displaystyle\aleph_{>\mathbb{C}_{0}}\cup\aleph_{<\mathbb{C}_{0}}$
$\displaystyle=\left\\{(v,1.5v+\sqrt{1+1.5^{2}}):v\geq
0\right\\}\cup\left\\{(v,(2/3)v-\sqrt{1+(2/3)^{2}}):v\geq
1.5\sqrt{1+(2/3)^{2}}\right\\}.$

Hence, using the GPOLAR transformation, we may rewrite (8)

$\displaystyle
t\mathbb{P}\left(\left(\frac{W_{2}-1.5W_{1}}{\sqrt{1+1.5^{2}}b_{0}(t)},\frac{\sqrt{1+1.5^{2}}\mathbf{W}}{W_{2}-1.5W_{1}}\right)\in\cdot\right)\rightarrow(\nu_{\alpha_{0}}\times
S_{0})(\cdot)\qquad\text{in
}\mathbb{M}((0,\infty)\times\aleph_{>\mathbb{C}_{0}}).$ (20)

Since $S_{0}$ places no mass on $\aleph_{<\mathbb{C}_{0}}$, we omit the
trivial convergence in $\mathbb{M}((0,\infty)\times\aleph_{<\mathbb{C}_{0}})$.
Following [17], when analyzing the hidden regular variation in (20), it
suffices to consider the pair $(W_{2}-1.5W_{1},W_{2}/W_{1})$ since for $y>1.5$

$\displaystyle t\mathbb{P}$
$\displaystyle\left(\frac{W_{2}-1.5W_{1}}{b_{0}(t)}>x,\frac{W_{2}}{W_{1}}\leq
y\right)$
$\displaystyle\rightarrow\left(1+1.5^{2}\right)^{-\alpha_{0}/2}x^{-\alpha_{0}}S_{0}\left\\{(u_{1},1.5u_{1}+\sqrt{1+1.5^{2}});u_{1}\geq\frac{\sqrt{1+1.5^{2}}}{y-1.5}\right\\}.$

We consider $n=2{,}000{,}000$ replicates of $\mathbf{W}$, $\mathbf{W}_{[n]}$.
Generate a $L=2$ layer MIRG with adjacency cube $\mathbf{A}(n)$ as follows.
Let

$\displaystyle\begin{split}A_{ij1}&\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Poisson}\left(W_{i1}W_{j1}/T_{1}(n)\right),\qquad\text{for
}1\leq i\leq j\leq n,\\\
A_{ij2}&\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Bernoulli}\left(1-\exp\left\\{-W_{i2}W_{j2}/T_{2}(n)\right\\}\right),\qquad\text{for
}1\leq i\leq j\leq n.\end{split}$ (21)

Borrowing nomenclature from Section 1.2, we have that $g_{1}(x)=x$ and
$g_{2}(x)=1-e^{-x}$ for $x>0$. That is, the first layer of the MIRG consists
on a Norros-Reittu random graph and the second layer consists of what is
commonly known as the Poissonian random graph [38]. It is clear that $g_{1}$
and $g_{2}$ satisfy assumption (C1). According to Theorem 2.4, statement (20)
should approximately hold with $\mathbf{D}_{1}(n)$ in place of $\mathbf{W}$,
assuming that $n$ is large enough so that
$\mathbb{P}(\mathbf{D}_{1}(n)\in\cdot)$ is close to
$\mathbb{P}(\boldsymbol{\mathcal{D}}_{1}\in\cdot)$.

In order to detect hidden regular variation of $\mathbf{D}_{1}(n)$, we employ
the Hillish estimator [15]. Theorem 2.4, along with (20) suggests that the
pairs $(\xi_{i},\eta_{i})\equiv(D_{i2}(n)-1.5D_{i1}(n),D_{i2}(n)/D_{i1}(n))$,
$i\in[n]$, satisfy a conditional extreme value (CEV) model. The Hillish
estimator aims to detect CEV models by defining
$\xi_{(1)}\geq\dots\geq\xi_{(n)}$ and letting $\eta^{\star}_{i}$ be the
$\eta$-variable corresponding to $\xi_{(i)}$, also known as the concomitant of
$\eta^{\star}_{i}$. If $N_{i}^{k}$ is the rank of $\eta^{\star}_{i}$ among
$\eta^{\star}_{1},\dots,\eta^{\star}_{k}$, the Hillish estimator for $k\in[n]$
is defined as

$\displaystyle\text{Hilllish}_{k,n}((\xi,\eta)_{[n]}):=\frac{1}{k}\sum_{i=1}^{k}\log\left(\frac{k}{i}\right)\log\left(\frac{k}{N_{i}^{k}}\right).$
(22)

It is shown in [15] that if $(\xi,\eta)_{[n]}$ are independent pairs
satisfying a CEV model, then $\text{Hilllish}_{k_{n},n}((\xi,\eta)_{[n]})$ has
an in probability limit as $n\rightarrow\infty$ and $k_{n}/n\rightarrow 0$.
Additionally, the limiting probability measure associated with the CEV model
is a product measure if and only if

$\displaystyle\text{Hilllish}_{k_{n},n}((\xi,\eta)_{[n]})\xrightarrow{p}1\text{
and }\text{Hilllish}_{k_{n},n}((\xi,-\eta)_{[n]})\xrightarrow{p}1,$ (23)

as $n\rightarrow\infty,k_{n}/n\rightarrow 0$. Although the theoretical
properties of the Hillish estimator have been analyzed only under the iid
assumption, [17] applies it to network data to detect the existence of HRV.
Hence, we follow a similar strategy here.

Since the limit in (20) is a product measure, we test whether
$\mathbf{D}_{1}(n)$ has hidden regular variation by simulating $1{,}000$
replicates of the MIRG with $n=2{,}000{,}000$ nodes and plotting the Hillish
estimator for $k\in[4{,}000]$ over the $1{,}000$ replicates. Although the
Hillish estimator is designed with independent data in mind, the approximation
(6) suggests that the dependence between degrees may not overly inhibit its
ability to detect HRV. The left-hand panel of Figure 1 presents pointwise
$(10,90)$-th (light pink) and $(25,75)$-th (purple) quantiles for the Hillish
estimators across the $1{,}000$ iterations. The black line represents the
pointwise Hillish means across the $1{,}000$ iterations. The plots of both
Hillish estimators suggest that $\mathbf{D}_{1}(n)$ does indeed exhibit hidden
regular variation; the Hillish estimators stabilize around $1$ near
$k_{n}=500$. As $k_{n}$ becomes large however, the Hillish estimators drift
away from $1$, indicating that that $k_{n}$ has grown too large in comparison
to $n$ in order to satisfy the convergence in (23).

Figure 1: Pointwise means (black), $(10,90)$-th (light pink) and $(25,75)$-th
(purple) quantiles for the Hillish estimators based on $(\xi,\eta)_{[n]}$ and
$(\xi,-\eta)_{[n]}$ for $n=2{,}000{,}000$ and $k\in[4,000]$ over $1{,000}$
replicates of the MIRG model with $\alpha=1.1$ and
$\alpha_{0}\in\\{1.3,2.5\\}$.

In order to guarantee that hidden regular variation from
$\boldsymbol{\mathcal{W}}_{1}$ confers to $\boldsymbol{\mathcal{D}}_{1}$,
Theorem 2.4 requires that $\alpha_{0}\in[\alpha,2\alpha)$. To further assess
the necessity of this requirement, we simulate $1{,}000$ replicates of the
MIRG (21) with $n=2{,}000{,}000$ except now $\alpha_{0}=2.5$. In this setting,
$\alpha_{0}\notin[\alpha,2\alpha)$ and we do not expect that hidden regular
variation from the weights is detectable from the degrees as discussed in
Section 2.2. For each replicate, we compute the Hillish estimator over
$k\in[4{,}000]$. The right-hand panel of Figure 1 plots the $(10,90)$-th and
$(25,75)$-th quantiles as well as pointwise means for the Hillish estimators
across the $1{,}000$ iterations. The Hillish plots indicate that the observed
$\mathbf{D}_{1}(n)$ do not possess hidden regular variation; neither set of
Hillish estimators concentrate around $1$ for any value of $k\in[4{,}000]$.

###  4.2 Consistency of the Hill estimator in the MIRG

In this section we present a simulation that inspects the necessity of
condition (C3) to obtain consistency of the Hill estimator in the MIRG model
for $\alpha>1$. Let

$\displaystyle\mathbf{W}=(W_{1},W_{2})=(V\Theta,V(1-\Theta)),$

where $V\sim\text{Pareto}(\alpha)$ and $\Theta\sim\text{Beta}(5,5,0.4,0.6)$
independently for $\alpha>0$. By construction, $\mathbf{W}$ exhibits
multivariate regular variation with tail index $\alpha$. Consider
$n=1{,}000{,}000$ independent replicates of $\mathbf{W}$, $\mathbf{W}_{[n]}$,
and define a $L=2$ layer MIRG with adjacency cube $\mathbf{A}(n)$ as follows.
Let

$\displaystyle A_{ij1}$
$\displaystyle\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Poisson}\left(W_{i1}W_{j1}/T_{1}(n)\wedge
1\right),\qquad\text{for }1\leq i\leq j\leq n,$ $\displaystyle A_{ij2}$
$\displaystyle\mid\mathbf{W}_{[n]}\overset{\text{ind}}{\sim}\text{Bernoulli}\left(W_{i2}W_{j2}/T_{2}(n)/(1+W_{i2}W_{j2}/T_{2}(n))\right),\qquad\text{for
}1\leq i\leq j\leq n.$

Here, $g_{1}(x)=x\wedge 1$ and $g_{2}(x)=x/(1+x)$ for $x\geq 0$, both of which
satisfy assumption (C1). The first layer of the proposed MIRG is a multigraph
that limits the extent to which large weights encourage a higher number of
links. The second layer of the MIRG is a version of the generalized random
graph that behaves similarly to the first layer, except in a more continuous
fashion. In order to estimate $\alpha$ from the degrees of the MIRG model, we
naturally employ the inverse of the Hill estimator

$\displaystyle
H_{k_{n},n}=\frac{1}{k_{n}}\sum_{i=1}^{n}\log\left(R_{(i)}(n)/R_{(k_{n}+1)}(n)\right),$

where we set $R_{i}$ to be the $\ell_{1}$ norm of the degree vector
$(D_{i1}(n),D_{i2}(n))$, or the total degree across the layers. For each
$\alpha\in\\{1,1.2,1.4,1.6,1.8,2\\}$, we generate $1{,}000$ networks and for
each simulated network, we compute $H^{-1}_{k_{n},n}$ across
$k_{n}\in\\{100,200,500,10^{3},5\times 10^{3},10^{4},10^{5}\\}$. Table 1
presents the bias and MSE (in parentheses) for each $\alpha$ and $k_{n}$
combination across the $1{,}000$ networks. Bold values indicate that the
choice of $k_{n}$ minimizes the empirical absolute bias or MSE for the given
value of $\alpha$. Note that our theory does not cover the case $\alpha=1$.
Table 1 indicates that consistency of the Hill estimator actually may not hold
for $\alpha=1$ since the empirical biases are all over $0.1$, regardless of
the value of $k_{n}$. Note that as $\alpha$ increases, however, the value of
$k_{n}$ that minimizes the empirical asbolute bias tends to decrease. This
suggests than an assumption that controls the rate at which $k_{n}$ diminishes
as $\alpha$ increases may indeed be necessary. On the other hand, the value of
$k_{n}$ that minimizes the empirical MSE tends to stay relatively stable at
$k_{n}=5{,}000$.

$k_{n}$ | $\alpha$
---|---
| $1.0$ | $1.2$ | $1.4$ | $1.6$ | $1.8$ | $2.0$
$100$ | 0.157 (0.039) | 0.077 (0.021) | 0.036 (0.022) | 0.021 (0.028) | 0.029 (0.039) | 0.028 (0.043)
$200$ | 0.141 (0.027) | 0.062 (0.011) | 0.026 (0.011) | 0.014 (0.013) |  0.021 (0.019) |  0.026 (0.020)
$500$ | 0.127 (0.018) | 0.051 (0.005) | 0.020 (0.004) |  0.013 (0.005) | 0.022 (0.008) | 0.034 (0.010)
$10^{3}$ | 0.117 (0.015) | 0.044 (0.003) |  0.018 (0.002) | 0.015 (0.003) | 0.027 (0.005) | 0.045 (0.007)
$5\times 10^{3}$ | 0.103 (0.012) | 0.036 (0.002) | 0.020 (0.001) | 0.023 (0.002) | 0.078 (0.008) | 0.054 (0.004)
$10^{4}$ |  0.102 (0.014) | 0.036 (0.002) | 0.026 (0.002) | 0.072 (0.008) | 0.131 (0.018) | 0.134 (0.019)
$10^{5}$ | 0.117 (0.065) |  0.002 (0.007) | 0.164 (0.027) | 0.184 (0.035) | -0.140 (0.020) | 0.242 (0.059)
Table 1: Bias and MSEs (in parentheses) for $H^{-1}_{k_{n},n}$ over $1{,}000$
iterations of a MIRG model with $n=1{,}000{,}000$ and
$\alpha\in\\{1,1.2,1.4,1.6,1.8,2\\}$. For each network, $H^{-1}_{k_{n},n}$ is
computed for $k_{n}\in\\{100,200,500,10^{3},5\times 10^{3},10^{4},10^{5}\\}$.

##  5 Conclusion

In this paper we have introduced a multilayer inhomogeneous random graph
(MIRG) model and studied its theoretical properties. The model is inherently
flexible, allowing for layer-wise heterogeneity of edge counts and node
attractiveness as well as complete specification of the asymptotic extremal
dependence structure. When the latent weights are multivariate regularly
varying, we also confirm that tail index estimation is feasible in such a
model; with a restriction on the number of order statistics used in
estimation, the Hill estimator based on the norm of the layer-wise degrees is
consistent. Simulations indicate that, for practical sample sizes commonly
encountered in network science, the asymptotic degree distribution accurately
describes the behavior of extreme degrees in simulated networks. Additionally,
the simulations indicate that a restriction on the number of order statistics
is indeed necessary to achieve consistency of the Hill estimator in the MIRG
model.

The proposed model and results produce additional research questions to
consider in future work. First, we may consider the asymptotic behavior of the
degree distribution when $\alpha\in(0,1]$ in the MIRG model. Here, the ratio
of the maximum weight and sum of the weights converge in distribution, which
should produce interesting behavior. We also conjecture that the MIRG model
can be imbued with more complex weight distributions, such as independent and
non-identically distributed data, and still produce scale-free behavior. From
a statistical perspective, methods developed in [22] and [21] suggest that the
Hill estimator would still be consistent in this case. Additionally, given a
realization from the MIRG model, one would like to make inference on the
extremal dependence structure. This includes estimating the support of the
angular measure and distinguishing between different types of dependence.
Since the tail empirical measure of the degree radii may be approximated by
the tail empirical measure of independent data, methods developed in [44] may
apply. Such methods rely of asymptotic normality of the Hill estimator in the
MIRG model, an important result to evaluate in its own right.

##  6 Proofs

###  6.1 Weight approximation lemmas

In this section we present lemmas that help us establish the approximation
(6). The formal lemma is given in Lemma 6.3, though we first present some
helpful building block lemmas that will be used throughout the paper. Fix
$l\in[L]$. From condition (C1), we expect $g_{l}(W_{1l}W_{jl}/T_{l}(n))\approx
c_{l}W_{1l}W_{jl}/T_{l}(n)$ to hold for all $j\in[n]$ when
$W_{1l}W_{(1)l}/T_{l}(n)$ is small. Lemma 6.1 establishes the rate at which
$W_{1l}W_{(1)l}(n)/T_{l}(n)$ tends to zero under condition (C3). Naturally,
this requires that $W_{(1)l}(n)/T_{l}(n)$ tends to zero as
$n\rightarrow\infty$. It is well known that,

$\displaystyle W_{(1)l}(n)/T_{l}(n)\xrightarrow{p}0,\qquad\text{as
}n\rightarrow\infty$ (24)

if and only if $\mathbb{E}[W_{1l}]<\infty$ [33]. This necessitates the
requirement that $\alpha>1$ in condition (C2). A useful result that helps
characterize the rate of convergence of (24) is given in Theorem 9.1 of [19],
which states that if the distribution of $W_{1l}$ is regularly varying with
tail index $\alpha>1$, then

$\displaystyle\mathbb{E}\left[W_{(1)l}(n)/T_{l}(n)\right]\sim\frac{\mathbb{E}[W_{(1)l}(n)]}{nE[W_{1l}]},\qquad\text{as
}n\rightarrow\infty.$ (25)

We now present Lemma 6.1.

######  Lemma 6.1.

Let $\mathbf{W}_{[n]}$ be such that (C2) holds. Suppose $k_{n}$ is a sequence
satisfying $k_{n}\rightarrow\infty$ and $k_{n}/n\rightarrow 0$ as
$n\rightarrow\infty$. Further assume that (C3) holds. Fix $\epsilon>0$ and
suppose $\beta\in(0,\alpha)$. Then for each $l\in[L]$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W^{\beta}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}>\epsilon\right)\rightarrow
0,$

as $n\rightarrow\infty$.

###### Proof.

Note that

$\displaystyle W^{\beta}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}\leq$
$\displaystyle\frac{W^{1+\beta}_{1l}}{T_{l}(n)}+\frac{W^{\beta}_{1l}\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{T_{l}(n)}$
$\displaystyle\leq$
$\displaystyle\frac{W^{1+\beta}_{1l}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}+\frac{W^{\beta}_{1l}\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}.$

In addition it is easily seen that since the distribution of $W_{1l}$ is
continuous

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W_{1l}>\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right)=\frac{n}{k_{n}}\frac{1}{n}=\frac{1}{k_{n}}\rightarrow
0,\qquad\text{as }n\rightarrow\infty.$ (26)

Hence when $W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}$, we may
bound

$\displaystyle W^{\beta}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}\leq$
$\displaystyle\frac{2W^{\beta}_{1l}\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}.$
(27)

Applying (26) and (27) thus gives

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W^{\beta}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}>\epsilon\right)\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W_{1l}>\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right)+\frac{n}{k_{n}}\mathbb{P}\left(\frac{2W^{\beta}_{1l}\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}>\epsilon\right).$

The first term on the right-hand side of the inequality converges to $0$ by
(26). The convergence of the second term to $0$ is given Lemma 6.2. ∎

######  Lemma 6.2.

Let $\mathbf{W}_{[n]}$ be such that (C2) holds. Suppose $k_{n}$ is a sequence
satisfying $k_{n}\rightarrow\infty$ and $k_{n}/n\rightarrow 0$ as
$n\rightarrow\infty$. Further assume that (C3) holds. Fix $\epsilon>0$ and
suppose $\beta\in(0,\alpha)$. Then for each $l\in[L]$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W^{\beta}_{1l}\frac{\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}>\epsilon\right)\rightarrow
0,$ (28)

as $n\rightarrow\infty$.

###### Proof.

An application of Markov’s inequality and independence across the weights
gives that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W^{\beta}_{1l}\frac{\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}>\epsilon\right)\leq$
$\displaystyle\epsilon^{-1}\mathbb{E}[W^{\beta}_{1l}]\frac{n}{k_{n}}\mathbb{E}\left[\frac{\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}\right]$
and since $W_{1l}$ is regularly varying, we may apply Theorem 9.1 of [19] to
achieve that $\displaystyle\sim$
$\displaystyle\epsilon^{-1}\mathbb{E}[W^{\beta}_{1l}]\frac{n}{k_{n}}\frac{\mathbb{E}\left[\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right]}{(n-1)\mathbb{E}\left[W_{1l}\right]}$
and applying Proposition 2.1 of [36] $\displaystyle\sim$
$\displaystyle\epsilon^{-1}\frac{\mathbb{E}[W^{\beta}_{1l}]}{\mathbb{E}\left[W_{1l}\right]}\frac{1}{k_{n}}\Gamma\left(1-\frac{1}{\alpha}\right)b^{\star}(n),$

where $b^{\star}(t)$ is the $1-1/t$ quantile function of the distribution of
$W_{1l}$ for $t\geq 1$. Since $b^{\star}(t)$ is regularly varying with index
$1/\alpha$, we may apply (C3) to obtain (28). ∎

We now present the lemma that formalizes the approximation (6).

######  Lemma 6.3.

Assume (C1) and that $\mathbb{E}[\|\mathbf{W}_{1}\|_{1}]<\infty$. Then for
every $l\in[L]$,

$\displaystyle\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\xrightarrow{p}c_{l}W_{1l},\qquad\text{as
}n\rightarrow\infty.$ (29)

###### Proof.

Let $l\in[L]$. Recall that since $\mathbb{E}[\|\mathbf{W}_{1}\|_{1}]<\infty$,

$\displaystyle W_{(1)l}(n)/T_{l}(n)\xrightarrow{p}0,\qquad\text{as
}n\rightarrow\infty.$

In particular, since $W_{1l}$ is finite almost surely.

$\displaystyle W_{1l}W_{(1)l}(n)/T_{l}(n)\xrightarrow{p}0.$ (30)

Recall $M$ and $\delta$ from (5). From condition (C1), when
$W_{1l}W_{(1)l}(n)/T_{l}(n)<\delta$

$\displaystyle\left|\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))-c_{l}W_{1l}\right|\leq$
$\displaystyle MW^{1+\nu}_{1l}\sum_{j=1}^{n}W^{1+\nu}_{jl}/T^{1+\nu}_{l}(n)$
$\displaystyle\leq$ $\displaystyle
MW^{1+\nu}_{1l}W^{\nu}_{(1)l}(n)/T^{\nu}_{l}(n).$

Fix $\epsilon>0$. Then

$\displaystyle\mathbb{P}\left(\left|\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))-c_{l}W_{1l}\right|>\epsilon\right)\leq$
$\displaystyle\mathbb{P}\left(MW^{1+\nu}_{1l}W^{\nu}_{(1)l}(n)/T^{\nu}_{l}(n)>\epsilon\right)$
$\displaystyle+\mathbb{P}\left(W_{1l}W_{(1)l}(n)/T_{l}(n)\geq\delta\right).$

Again, since $W^{1+\nu}_{11}$ is finite almost surely, as $n\rightarrow\infty$

$\displaystyle
W^{1+\nu}_{1l}W^{\nu}_{(1)l}(n)/T^{\nu}_{l}(n)\xrightarrow{p}0.$ (31)

Hence applying (30) and (31) completes the proof. ∎

###  6.2 Coupling strategy

In this section we present a coupling that is used to translate results
regarding the degrees in the more theoretically amenable Poisson
$\mathcal{L}_{1}$ layers to the degrees in the Bernoulli $\mathcal{L}_{2}$
layers in the MIRG. Let $\mathcal{B}_{X\mid Y}$ denote the law of a random
variable $X$ conditioned on the random variable $Y$. Construct a coupled
version of $\mathbf{A}(n)$,
$\tilde{\mathbf{A}}(n)=\\{\tilde{A}_{ijl}\\}_{ijl}$, as follows. For
$l\in\mathcal{L}_{1}$, set $\tilde{A}_{ijl}=A_{ijl}$ for $1\leq i\leq j\leq
n$. For $l\in\mathcal{L}_{2}$, let

$\displaystyle\tilde{A}_{ijl}\mid\mathbf{W}_{[n]}\sim\text{Poisson}(g_{l}(W_{il}W_{jl}/T_{l}(n)))\qquad\text{independently,
}1\leq i\leq j\leq n,$

where

$\displaystyle
d_{\text{TV}}\left(\mathcal{B}_{\tilde{A}_{ijl}\mid\mathbf{W}_{[n]}},\mathcal{B}_{A_{ijl}\mid\mathbf{W}_{[n]}}\right)\leq$
$\displaystyle g_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right),\qquad
1\leq i\leq j\leq n.$ (32)

Such a coupling is given by a maximal coupling between Poisson and Bernoulli
random variables [see Section 2.2 of 40]. Conditional on $\mathbf{W}_{[n]}$,
we enforce independence of the pairs $(\tilde{A}_{ijl},A_{ijl})$ across edges
and layers. Under said coupling, one may easily compute

$\displaystyle\mathbb{E}\left[\left|\tilde{A}_{ijl}-A_{ijl}\right|\ \bigg{|}\
\mathbf{W}_{[n]}\right]\leq$ $\displaystyle
Kg_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)\qquad 1\leq i\leq j\leq
n,$ (33)

for some $K>0$. Additionally, Theorem 2.10 of [40] gives that

$\displaystyle
d_{\text{TV}}\left(\mathcal{B}_{\sum_{j=1}^{n}\tilde{A}_{ijl}\mid\mathbf{W}_{[n]}},\mathcal{B}_{\sum_{j=1}^{n}A_{ijl}\mid\mathbf{W}_{[n]}}\right)\leq$
$\displaystyle\sum_{j=1}^{n}g_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right),\qquad
i\in[n].$ (34)

Let the degree of node $i$ in the coupled graph be given by
$\tilde{\mathbf{D}}_{i}=(\tilde{D}_{i1},\dots,\tilde{D}_{iL})$ where
$\tilde{D}_{il}=\sum_{j=1}^{n}\tilde{A}_{ijl}$ for $l\in[L]$. Our first lemma
of this section provides a useful bound for the right-hand side of (34) that
we will repeatedly refer to.

######  Lemma 6.4.

Suppose (C1) holds. There exists a constant $C>0$ such that for any $i\in[n]$
and $l\in[L]$ with $W_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta$

$\displaystyle\sum_{j=1}^{n}g_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)\leq
CW^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}.$

###### Proof.

From (C1) and (5), we have that if $x<\delta$, then

$\displaystyle g^{2}_{l}(x)=$
$\displaystyle(g_{l}(x)-c_{l}x)^{2}+c^{2}_{l}x^{2}+2c_{l}x(g_{l}(x)-c_{l}x)$
$\displaystyle\leq$ $\displaystyle
M^{2}x^{2+2\nu}+c^{2}_{l}x^{2}+2c_{l}Mx^{2+\nu}.$

Hence

$\displaystyle\sum_{j=1}^{n}$ $\displaystyle
g_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)$ $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{n}\left(M^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)^{2+2\nu}+c_{l}^{2}\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)^{2}+2c_{l}M\left(\frac{W_{il}W_{jl}}{T_{l}(n)}\right)^{2+\nu}\right)$
$\displaystyle\leq$ $\displaystyle
M^{2}W^{2+2\nu}_{il}\left(\frac{W_{(1)l}(n)}{T_{l}(n)}\right)^{1+2\nu}+c_{l}^{2}W^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}+2c_{l}MW^{2+\nu}_{il}\left(\frac{W_{(1)l}(n)}{T_{l}(n)}\right)^{1+\nu}$
$\displaystyle\leq$ $\displaystyle
M^{2}\delta^{2\nu}W^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}+c_{l}^{2}W^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}+2c_{l}M\delta^{\nu}W^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)}$
$\displaystyle\leq$ $\displaystyle CW^{2}_{il}\frac{W_{(1)l}(n)}{T_{l}(n)},$

where we have chosen
$C=M^{2}\delta^{2\nu}+\max_{l\in[L]}c_{l}^{2}+2M\delta^{\nu}\max_{l\in[L]}c_{l}$.
∎

###  6.3 Properties of the degree distribution

In this section we present proofs of Lemma 1.1 and Theorem 2.4. The proofs
rely on tools developed in Sections 6.1 and 6.2

####  6.3.1 Proof of Lemma 1.1

###### Proof.

See that by conditioning on $\\{\mathbf{W}_{i}\\}_{i=1}^{n}$

$\displaystyle\mathbb{P}\left(\mathbf{D}_{1}(n)=\mathbf{m}\right)=$
$\displaystyle\mathbb{E}\left[\mathbb{P}\left(\mathbf{D}_{1}(n)=\mathbf{m}\mid\mathbf{W}_{[n]}\right)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\prod_{l=1}^{L}\mathbb{P}\left(D_{1l}(n)=m_{l}\mid\mathbf{W}_{[n]}\right)\right].$

For $l\in\mathcal{L}_{1}$, see that by applying Lemma 6.3

$\displaystyle\mathbb{P}\left(D_{1l}(n)=m_{l}\mid\mathbf{W}_{[n]}\right)=$
$\displaystyle\frac{\left(\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right)^{m_{l}}e^{-\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))}}{m_{l}!}$
$\displaystyle\xrightarrow{p}$
$\displaystyle\frac{\left(c_{l}W_{1l}\right)^{m_{l}}e^{-c_{l}W_{1l}}}{m_{l}!},\qquad\text{as
}n\rightarrow\infty.$

For $l\in\mathcal{L}_{2}$, we employ the coupled version of $\mathbf{A}(n)$,
$\tilde{\mathbf{A}}(n)$, developed in Section 6.2. From (34) and Lemma 6.4, we
have that

$\displaystyle
d_{\text{TV}}\left(\mathcal{B}_{\sum_{j=1}^{n}\tilde{A}_{1jl}\mid\mathbf{W}_{[n]}},\mathcal{B}_{\sum_{j=1}^{n}A_{1jl}\mid\mathbf{W}_{[n]}}\right)\leq$
$\displaystyle\sum_{j=1}^{n}g_{l}^{2}\left(\frac{W_{1l}W_{jl}}{T_{l}(n)}\right),$

and when $W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta$, by Lemma 6.4

$\displaystyle
d_{\text{TV}}\left(\mathcal{B}_{\sum_{j=1}^{n}\tilde{A}_{1jl}\mid\mathbf{W}_{[n]}},\mathcal{B}_{\sum_{j=1}^{n}A_{1jl}\mid\mathbf{W}_{[n]}}\right)\leq
CW^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}.$

Hence, for any $\epsilon>0$

$\displaystyle\mathbb{P}\left(d_{\text{TV}}\left(\mathcal{B}_{\sum_{j=1}^{n}\tilde{A}_{1jl}\mid\mathbf{W}_{[n]}},\mathcal{B}_{\sum_{j=1}^{n}A_{1jl}\mid\mathbf{W}_{[n]}}\right)>\epsilon\right)\leq$
$\displaystyle\mathbb{P}\left(CW^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}>\epsilon\right)$
$\displaystyle+\mathbb{P}\left(W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}\geq\delta\right).$

Recalling that $\mathbb{E}[\|\mathbf{W}_{1}\|_{1}]<\infty$,
$W^{j}_{1l}W_{(1)l}(n)/T_{l}(n)$ tends to $0$ in probability since
$W_{(1)l}(n)/T_{l}(n)$ tends to zero in probability and $W^{j}_{1l}$ is finite
almost surely for $j=1,2$. Hence for $l\in\mathcal{L}_{2}$,

$\displaystyle\mathbb{P}\left(D_{1l}(n)=m_{l}\mid\mathbf{W}_{[n]}\right)\xrightarrow{p}\frac{\left(c_{l}W_{1l}\right)^{m_{l}}e^{-c_{l}W_{1l}}}{m_{l}!},$

as $n\rightarrow\infty$. Thus, dominated convergence gives that as
$n\rightarrow\infty$

$\displaystyle\mathbb{P}\left(\mathbf{D}_{1}(n)=\mathbf{m}\right)=$
$\displaystyle\mathbb{E}\left[\prod_{l=1}^{L}\mathbb{P}\left(D_{1l}(n)=m_{l}\mid\mathbf{W}_{[n]}\right)\right]\rightarrow\mathbb{E}\left[\prod_{l=1}^{L}\frac{\left(c_{l}W_{1l}\right)^{m_{l}}e^{-c_{l}W_{1l}}}{m_{l}!}\right].$

∎

####  6.3.2 Proof of Theorem 2.4

In this section we prove Theorem 2.4. Given that
$\mathbb{P}\left(\boldsymbol{\mathcal{D}}_{1}\in\cdot\right)$ in Lemma 1.1
emits a mixed-Poisson representation, the proof of of Theorem 2.4 relies
heavily on the concentration of the Poisson distribution for large rates. Such
properties are also used in the single-layer case [see (A4) and Section 4 of
6]. We now recall concentration results for the Poisson distribution.

######  Lemma 6.5.

Suppose $X\sim\text{Poisson}(\lambda)$. For each $m\in\mathbb{N}$, there
exists nonnegative constants $a_{m}$ and $C_{m}$ depending only on $m$ such
that

$\displaystyle\mathbb{E}\left(\left|X-\lambda\right|^{m}\right)\leq
a_{m}\lambda^{m/2}+C_{m}.$

In order to form a multivariate extension of Lemma 6.5, we note that for
$X_{l}\sim\text{Poisson}(\lambda_{l})$ for $l=1,\dots,L$,
$\mathbf{X}=(X_{1},\dots,X_{L})$ and
$\boldsymbol{\lambda}=(\lambda_{1},\dots,\lambda_{L})$

$\displaystyle\mathbb{E}\left(\|\mathbf{X}-\boldsymbol{\lambda}\|^{m}_{m}\right)\leq
a_{m}\|\boldsymbol{\lambda}\|^{m/2}_{m/2}+LC_{m},$ (35)

which, when combined with the equivalence $\ell_{p}$ norms, gives the
following lemma.

######  Lemma 6.6.

Suppose $X_{l}\sim\text{Poisson}(\lambda_{l})$ for $l=1,\dots,L$. Let
$\mathbf{X}=(X_{1},\dots,X_{L})$ and
$\boldsymbol{\lambda}=(\lambda_{1},\dots,\lambda_{L})$. For each
$m,p\in\mathbb{N}$ there exists nonnegative constants $a_{m}$ and $C_{m}$
depending only on $m$ such that

$\displaystyle\mathbb{E}\left(\|\mathbf{X}-\boldsymbol{\lambda}\|_{p}^{m}\right)\leq
L^{\frac{m}{p}+1}\left(a_{m}\|\boldsymbol{\lambda}\|^{m/2}_{p}+C_{m}\right).$

Which such results in hand, we are now prepared to prove Theorem 2.4.

###### Proof of Theorem 2.4.

We only prove part (b) since part (a) is simpler and proven similarly. Let
$f\in\mathcal{C}(\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$ be uniformly
continuous. It suffices to show that

$\displaystyle
t\left|\mathbb{E}[f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))]-\mathbb{E}[f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))]\right|\rightarrow
0,\qquad\text{as }t\rightarrow\infty.$ (36)

Since $f\in\mathcal{C}(\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$, it has
support bounded away from $\mathbb{C}_{0}$. That is, there exists an
$\theta>0$ such that $f(\mathbf{x})=0$ for all
$\mathbf{x}\in\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}$ with
$d_{p}(\mathbf{x},\mathbb{C}_{0})\leq\theta$. Thus

$\displaystyle t$
$\displaystyle\left|\mathbb{E}[f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))]-\mathbb{E}[f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))]\right|$
$\displaystyle\leq$ $\displaystyle
t\mathbb{E}\left[\left|f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))-f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))\right|\right]$
$\displaystyle=$ $\displaystyle
t\mathbb{E}\left[\left|f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))-f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))\right|1_{\left\\{d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right\\}}\right]$
.
$\displaystyle+t\mathbb{E}\left[f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))1_{\left\\{d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\leq\theta\right\\}}\right]$
$\displaystyle+t\mathbb{E}\left[f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))1_{\left\\{d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})\leq\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right\\}}\right]$
$\displaystyle\equiv$ $\displaystyle A_{1}(t)+A_{2}(t)+A_{3}(t).$

We show that $A_{1}(t),A_{2}(t)$ and $A_{3}(t)$ all tend to zero as
$t\rightarrow\infty$. For $A_{1}(t)$, recall that, conditional on
$\boldsymbol{\mathcal{W}}_{1}$, the elements of $\boldsymbol{\mathcal{D}}_{1}$
are conditionally independent Poisson random variables. Hence we may apply
Lemma 6.6 to supply the existence of constants $a_{p}$ and $C_{p}$ such that,
almost surely

$\displaystyle\mathbb{E}\left[\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|_{p}^{p}\mid\boldsymbol{\mathcal{W}}_{1}\right]\leq
L^{2}\left(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|^{p/2}_{p}+C_{p}\right).$ (37)

Fix $\epsilon>0$. Define the random variable

$\displaystyle\delta_{p}=\delta_{p}(\boldsymbol{\mathcal{W}}_{1},t)=\frac{L^{2/p}(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|^{p/2}_{p}+C_{p})^{1/p}}{b_{0}(t)\epsilon^{1/p}},$

Note that if
$\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)$
are bounded away from $\mathbb{C}_{0}$ and

$\displaystyle\|\boldsymbol{\mathcal{D}}_{1}/b_{0}(t)-\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)\|_{p}\leq\delta_{p},$

then

$\displaystyle\left|f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))-f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))\right|\leq\Delta_{f}^{p}(\delta_{p}).$

Otherwise, since $f\in\mathcal{C}(\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0})$,
there exists an $B>0$ such that for all
$\mathbf{x}\in\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}$, $f(\mathbf{x})\leq
B$. Thus we may bound $A_{1}(t)$ by

$\displaystyle t\mathbb{E}$
$\displaystyle\left[\left|f(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t))-f(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t))\right|1_{\left\\{d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle
t\mathbb{E}\left[\Delta_{f}^{p}(\delta_{p})1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right\\}}\right]+2Bt\mathbb{E}\left[1_{\left\\{\|\boldsymbol{\mathcal{D}}_{1}/b_{0}(t)-\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)\|_{p}>\delta_{p}\right\\}}1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right\\}}\right]$
$\displaystyle=$ $\displaystyle
t\mathbb{E}\left[\Delta_{f}^{p}(\delta_{p})1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]+2Bt\mathbb{E}\left[1_{\left\\{\|\boldsymbol{\mathcal{D}}_{1}/b_{0}(t)-\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)\|_{p}>\delta_{p}\right\\}}1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle\equiv$ $\displaystyle
A_{11}(t)+A_{12}(t),$

where we have noted that

$\displaystyle\left\\{\mathbf{x}\in\mathbb{R}_{+}^{L}:\inf_{\mathbf{y}\in\mathbb{C}_{0}}\left\|\frac{\mathbf{x}}{b_{0}(t)}-\mathbf{y}\right\|_{p}>\eta\right\\}=$
$\displaystyle\left\\{\mathbf{x}\in\mathbb{R}_{+}^{L}:\inf_{\mathbf{y}\in\mathbb{C}_{0}}\left\|\mathbf{x}-b_{0}(t)\mathbf{y}\right\|_{p}>\eta
b_{0}(t)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\mathbf{x}\in\mathbb{R}_{+}^{L}:\inf_{\mathbf{z}\in\mathbb{C}_{0}}\left\|\mathbf{x}-\mathbf{z}\right\|_{p}>\eta
b_{0}(t)\right\\},$

since $\mathbb{C}_{0}$ is a cone. Hence in order to prove that $A_{1}(t)$
tends to $0$ as $t\rightarrow\infty$, it suffices to show that $A_{11}(t)$ and
$A_{12}(t)$ converge to $0$ as $t\rightarrow\infty$. For $A_{11}(t)$, see that
since $\Delta_{f}^{p}(\delta)\leq 2B$ for any $\delta>0$ we have that
$A_{11}(t)$ is equivalent to

$\displaystyle t\mathbb{E}$
$\displaystyle\left[\Delta_{f}^{p}(\delta_{p})1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle=$ $\displaystyle
t\mathbb{E}\left[\Delta_{f}^{p}(\delta_{p})1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}1_{\left\\{\delta_{p}\leq\epsilon\right\\}}\right]+t\mathbb{E}\left[\Delta_{f}^{p}(\delta_{p})1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}1_{\left\\{\delta_{p}>\epsilon\right\\}}\right]$
$\displaystyle\leq$
$\displaystyle\Delta_{f}^{p}(\epsilon)t\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right)+2Bt\mathbb{P}\left(\delta_{p}>\epsilon\right)$
$\displaystyle=$
$\displaystyle\Delta_{f}^{p}(\epsilon)t\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right)+2Bt\mathbb{P}\left(L^{2/p}\left(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|_{p}^{p/2}+C_{p}\right)^{1/p}>b_{0}(t)\epsilon^{(p+1)/p}\right),$

Due to regular variation of $\boldsymbol{\mathcal{W}}_{1}$ on
$\mathbb{R}_{+}^{L}\setminus\mathbb{C}_{0}$, we have that by the GPOLAR
transform as $t\rightarrow\infty$

$\displaystyle
t\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right)\rightarrow\theta^{-\alpha_{0}}.$ (38)

Additionally, since $\boldsymbol{\mathcal{W}}_{1}$ is multivariate regularly
varying on $\mathbb{R}_{+}^{L}\setminus\\{\mathbf{0}\\}$ with tail index
$\alpha$, $\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$ is univariate regularly
varying with tail index $\alpha$. Thus the quantity
$(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|^{p/2}_{p}+C_{p})^{1/p}$ is regularly
varying with tail index $2\alpha$ and

$\displaystyle
t\mathbb{P}\left(L^{2/p}\left(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|_{p}^{p/2}+C_{p}\right)^{1/p}>b_{0}(t)\epsilon^{(p+1)/p}\right)\rightarrow
0,\qquad\text{as }t\rightarrow\infty,$ (39)

since $b_{0}(t)\in RV_{1/\alpha_{0}}$ and $\alpha_{0}\in[\alpha,2\alpha)$.
Thus, from (38) and (39), we have that

$\displaystyle\limsup_{t\rightarrow\infty}A_{1}(t)\leq\Delta_{f}^{p}(\epsilon)\theta^{-\alpha_{0}}.$

For $A_{12}(t)$, see that by tower property and Markov’s inequality

$\displaystyle A_{12}(t)=$ $\displaystyle
2Bt\mathbb{E}\left[\mathbb{P}\left(\|\boldsymbol{\mathcal{D}}_{1}/b_{0}(t)-\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)\|_{p}>\delta_{p}\mid\boldsymbol{\mathcal{W}}_{1}\right)1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle\leq$ $\displaystyle
2Bt\mathbb{E}\left[\delta_{p}^{-p}\mathbb{E}\left[\|\boldsymbol{\mathcal{D}}_{1}/b_{0}(t)-\boldsymbol{\mathcal{W}}_{1}/b_{0}(t)\|^{p}_{p}\mid\boldsymbol{\mathcal{W}}_{1}\right]1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle=$ $\displaystyle
2Bt\mathbb{E}\left[(b_{0}(t)\delta_{p})^{-p}\mathbb{E}\left[\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|^{p}_{p}\mid\boldsymbol{\mathcal{W}}_{1}\right]1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle\leq$ $\displaystyle
2Bt\mathbb{E}\left[(b_{0}(t)\delta_{p})^{-p}L^{2}\left(a_{p}\|\boldsymbol{\mathcal{W}}_{1}\|_{p}^{p/2}+C_{p}\right)1_{\left\\{d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right\\}}\right]$ $\displaystyle=$ $\displaystyle 2B\epsilon
t\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\theta
b_{0}(t)\right),$

where we have applied (37) in the second to last line and the definition of
$\delta_{p}$ in the penultimate step. From (38), we thus have that

$\displaystyle\limsup_{t\rightarrow\infty}A_{12}(t)\leq
2B\epsilon\theta^{-\alpha_{0}}.$

Thus, returning to our original goal, we achieve that

$\displaystyle\limsup_{t\rightarrow\infty}A_{1}(t)\leq$
$\displaystyle\limsup_{t\rightarrow\infty}A_{11}(t)+\limsup_{t\rightarrow\infty}A_{12}(t)$
$\displaystyle\leq$
$\displaystyle(\Delta_{f}^{p}(\epsilon)+2K\epsilon)\theta^{-\alpha_{0}}.$

Since $\epsilon>0$ was arbitrary, we have that $A_{1}(t)\rightarrow 0$ as
$t\rightarrow\infty$. Fix $\eta\in(0,\theta)$. For $A_{2}(t)$ and $A_{3}(t)$,
note that

$\displaystyle A_{2}(t)\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\leq\theta\right)$
$\displaystyle\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\in(\theta-\eta,\theta]\right)+Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})-d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\eta
b_{0}(t)\right)$ $\displaystyle\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\in(\theta-\eta,\theta]\right)+Bt\mathbb{P}\left(\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b_{0}(t)\right)$ $\displaystyle=$ $\displaystyle A_{21}(t)+\tilde{A}(t),$

and similarly

$\displaystyle A_{3}(t)\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{D}}_{1}/b_{0}(t),\mathbb{C}_{0})\leq\theta,d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})>\theta\right)$
$\displaystyle\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\in(\theta,\theta+\eta]\right)+Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{D}}_{1},\mathbb{C}_{0})-d_{p}(\boldsymbol{\mathcal{W}}_{1},\mathbb{C}_{0})>\eta
b_{0}(t)\right)$ $\displaystyle\leq$ $\displaystyle
Bt\mathbb{P}\left(d_{p}(\boldsymbol{\mathcal{W}}_{1}/b_{0}(t),\mathbb{C}_{0})\in(\theta,\theta+\eta]\right)+Bt\mathbb{P}\left(\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b_{0}(t)\right)$ $\displaystyle=$ $\displaystyle A_{31}(t)+\tilde{A}(t).$

Analyzing $\tilde{A}(t)$, let $s\in(\alpha_{0},2\alpha)$ and $m=\lceil
s\rceil$. Then by Markov and Jensen inequality,

$\displaystyle\tilde{A}(t)\leq$
$\displaystyle\frac{Bt}{\eta^{s}b^{s}_{0}(t)}\mathbb{E}\left[\mathbb{E}\left[\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|^{s}_{p}\mid\boldsymbol{\mathcal{W}}_{1}\right]\right]$
$\displaystyle\leq$
$\displaystyle\frac{Bt}{\eta^{s}b^{s}_{0}(t)}\mathbb{E}\left[\left(\mathbb{E}\left[\|\boldsymbol{\mathcal{D}}_{1}-\boldsymbol{\mathcal{W}}_{1}\|^{m}_{p}\mid\boldsymbol{\mathcal{W}}_{1}\right]\right)^{\frac{s}{m}}\right]$
$\displaystyle\leq$
$\displaystyle\frac{Bt}{\eta^{s}b^{s}_{0}(t)}\mathbb{E}\left[\left(L^{\frac{m}{p}+1}\left(a_{m}\|\boldsymbol{\mathcal{W}}_{1}\|_{p}^{m/2}+C_{m}\right)\right)^{\frac{s}{m}}\right].$

The random variable inside the expectation is reguarly varying with tail index
$2\alpha/s>1$ and hence the expectation is finite. In addition, $b^{s}_{0}(t)$
is regularly varying with index $s/\alpha_{0}>1$ and thus
$\tilde{A}(t)\rightarrow 0$ as $t\rightarrow\infty$. For $A_{21}(t)$ and
$A_{31}(t)$, the GPOLAR transform gives that as $t\rightarrow\infty$,

$\displaystyle
A_{21}(t)\rightarrow(\theta-\eta)^{-\alpha_{0}}-\theta^{-\alpha_{0}},$
$\displaystyle
A_{31}(t)\rightarrow\theta^{-\alpha_{0}}-(\theta+\eta)^{-\alpha_{0}}.$

Since $\eta>0$ is arbitrary, $A_{2}(t),A_{3}(t)\rightarrow 0$ as
$t\rightarrow\infty$. Thus the proof is complete. ∎

####  6.3.3 Proof of (10) in Example Example

In order to prove (10), it suffices to consider the vague convergence of
measures; see for instance [35, Chapter 3.3.5]. We include the definition
below. Let $\mathbb{E}$ be a locally compact topological space that has a
countable base with Borel sigma algebra $\mathcal{E}$. Let $M_{+}(\mathbb{E})$
be the space of Radon measures on $\mathcal{E}$. Then a sequence of measures
$\mu_{n}\in M_{+}(\mathbb{E})$ converges vaguely to $\mu\in
M_{+}(\mathbb{E})$, written $\mu_{n}\rightarrow\mu$, if for all nonnegative,
compactly supported continuous functions on $\mathcal{E}$

$\displaystyle\int fd\mu_{n}\rightarrow\int fd\mu,\qquad\text{as
}n\rightarrow\infty.$

With vague convergence in hand, we may now prove (10).

###### Proof of (10).

We first show that

$\displaystyle
t\mathbb{P}\left(\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2W_{11}}},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|},\frac{\sqrt{W_{11}}}{b_{0}(t)}\right)\in\cdot\right)\xrightarrow{v}(S\times\nu_{2\alpha})(\cdot),$
(40)

in $M_{+}([0,\infty]^{2}\times(0,\infty])$ where $S$ is the degenerate
probability measure given by

$\displaystyle S(\cdot)=\mathbb{P}\left((|Z|,1/|Z|)\in\cdot\right).$

Since $\sqrt{W_{11}}$ is marginally regularly varying, it suffices to show
that for any $x>0$ and bounded, continuous function $h$ on $[0,\infty]^{2}$

$\displaystyle
t\mathbb{E}\left[1_{\left\\{\sqrt{W_{11}}>b_{0}(t)x\right\\}}h\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2W_{11}}},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)\right]\rightarrow
x^{-2\alpha}E\left[h\left(|Z|,\frac{1}{|Z|}\right)\right],$ (41)

as $t\rightarrow\infty$. First note that for independent Poisson random
variables $X,Y$ with common rate $\lambda$, it is easily seen that as
$\lambda\rightarrow\infty$

$\displaystyle|X-Y|/\sqrt{2\lambda}\xrightarrow{d}|Z|,\qquad
X/\lambda\xrightarrow{p}1,$

where $Z$ is a standard normal random variable. Hence the joint weak
convergence of $(|X-Y|/\sqrt{2\lambda},X/\lambda)$ is obtained and the
continuous mapping theorem gives that as $\lambda\rightarrow\infty$

$\displaystyle\left(\frac{|X-Y|}{\sqrt{2\lambda}},\frac{\sqrt{2}X/\sqrt{\lambda}}{|X-Y|}\right)\xrightarrow{d}\left(|Z|,1/|Z|\right),$

which, in particular, implies that as $\lambda\rightarrow\infty$

$\displaystyle\mathbb{E}\left[h\left(\frac{|X-Y|}{\sqrt{2\lambda}},\frac{\sqrt{2}X/\sqrt{\lambda}}{|X-Y|}\right)\right]\rightarrow\mathbb{E}\left[h\left(|Z|,1/|Z|\right)\right].$
(42)

Fix $\epsilon>0$. From (42), we may extract the existence of a
$\lambda^{\star}>0$ such that

$\displaystyle\sup_{\lambda>\lambda^{\star}}\left|\mathbb{E}\left[h\left(\frac{|X-Y|}{\sqrt{2\lambda}},\frac{\sqrt{2}X/\sqrt{\lambda}}{|X-Y|}\right)-h\left(|Z|,\frac{1}{|Z|}\right)\right]\right|<\epsilon.$
(43)

Towards proving (41) then, see that we may write

$\displaystyle
t\left|\mathbb{E}\left[1_{\left\\{\sqrt{W_{11}}>b_{0}(t)x\right\\}}h\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2W_{11}}},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)\right]-x^{-2\alpha}E\left[h\left(|Z|,\frac{1}{|Z|}\right)\right]\right|$
$\displaystyle\leq
t\left|\mathbb{E}\left[1_{\left\\{\sqrt{W_{11}}>b_{0}(t)x\right\\}}\left\\{h\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2W_{11}}},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)-h\left(|Z|,\frac{1}{|Z|}\right)\right\\}\right]\right|$
$\displaystyle\ \ \
+\left|tE\left[1_{\left\\{\sqrt{W_{11}}>b_{0}(t)x\right\\}}h\left(|Z|,\frac{1}{|Z|}\right)\right]-x^{-2\alpha}E\left[h\left(|Z|,\frac{1}{|Z|}\right)\right]\right|$
$\displaystyle=C_{1}(t)+C_{2}(t).$

Using the tower property, see that

$\displaystyle C_{1}(t)\leq$ $\displaystyle
t\mathbb{E}\left[1_{\left\\{\sqrt{W_{11}}>b_{0}(t)x\right\\}}\left|\mathbb{E}\left[h\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2W_{11}}},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)-h\left(|Z|,\frac{1}{|Z|}\right)\Bigg{|}\
W_{11}\right]\right|\right]$ and choosing $t$ large enough so that
$b^{2}_{0}(t)x^{2}>\lambda^{\star}$, we have that $\displaystyle\leq$
$\displaystyle\epsilon
t\mathbb{P}\left(\sqrt{W_{11}}>b_{0}(t)x\right)\rightarrow\epsilon
x^{-2\alpha},$

as $t\rightarrow\infty$. Hence, $C_{1}(t)\rightarrow 0$ as
$t\rightarrow\infty$. Since $h$ is bounded, we may apply regular variation of
$\sqrt{W_{11}}$ to conclude that $C_{2}(t)\rightarrow 0$ as
$t\rightarrow\infty$. Hence (41) is proven, as well as (40). We next define
the mapping
$\chi:[0,\infty]^{2}\times(0,\infty]\mapsto[0,\infty]^{2}\times(0,\infty]$

$\displaystyle\chi\left((x,y),a\right)=\left((ax,y),a\right).$

This mapping satisfies the conditions in Proposition 5.5 of [35] and thus we
may apply it to the convergence in (40) to obtain that

$\displaystyle
t\mathbb{P}\left(\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|},\frac{\sqrt{W_{11}}}{b_{0}(t)}\right)\in\cdot\right)\xrightarrow{v}(S\times\nu_{2\alpha})\circ\chi^{-1}(\cdot),$
(44)

in $M_{+}([0,\infty]^{2}\times(0,\infty])$. From (44), we aim to prove that

$\displaystyle
t\mathbb{P}\left(\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)\in\cdot\right)\xrightarrow{v}(S\times\nu_{2\alpha})\circ\chi^{-1}(\cdot\times(0,\infty]),$
(45)

in $M_{+}((0,\infty]\times[0,\infty])$. Let $f$ be a positive, bounded and
continuous function with compact support in $(0,\infty]\times[0,\infty]$.
Since the support is compact, we may posit the existence of a $\theta>0$ such
that the compact support of $f$ lies outside of $[0,\theta]\times[0,\infty]$.
From (44), we then have for any $\tau>0$

$\displaystyle
t\mathbb{E}\left[f\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W}_{11}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)1_{\left\\{\sqrt{W_{11}}>b_{0}(t)\tau\right\\}}\right]$
$\displaystyle\rightarrow\int_{\tau}^{\infty}$
$\displaystyle\mathbb{E}\left(f\left(x|Z|,\frac{1}{|Z|}\right)\right)\nu_{2\alpha}(dx).$

Hence, in order to show (45), it suffices to show that

$\displaystyle\lim_{\tau\rightarrow
0}\limsup_{t\rightarrow\infty}t\mathbb{E}\left[f\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)1_{\left\\{\sqrt{W_{11}}\leq
b_{0}(t)\tau\right\\}}\right]=0.$

Since $f$ is bounded and supported on the complement of
$[0,\theta]\times[0,\infty]$, we have that

$\displaystyle t\mathbb{E}$
$\displaystyle\left[f\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)},\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W}_{11}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}\right)1_{\left\\{\sqrt{W_{11}}\leq
b_{0}(t)\tau\right\\}}\right]$ $\displaystyle\leq$
$\displaystyle\sup_{\mathbf{x}\in\mathbb{R}_{+}^{2}}|f(\mathbf{x})|t\mathbb{P}\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)}>\theta,\sqrt{W_{11}}\leq
b_{0}(t)\tau\right)$ $\displaystyle\equiv$
$\displaystyle\sup_{\mathbf{x}\in\mathbb{R}_{+}^{2}}|f(\mathbf{x})|B(t,\tau).$

Hence, it suffices to show that $\lim_{\tau\rightarrow
0}\limsup_{t\rightarrow\infty}B(t,\tau)=0$. For $B(t,\tau)$, we employ Lemma
6.5 to achieve that for any $m\in\mathbb{N}$

$\displaystyle B(t,\tau)=$ $\displaystyle
t\mathbb{E}\left[\mathbb{P}\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)}>\theta\mid
W_{11}\right)1_{\left\\{\sqrt{W_{11}}\leq b_{0}(t)\tau\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle
2^{m/2}\theta^{-m}\frac{t}{b^{m}_{0}(t)}\mathbb{E}\left[(a_{m}W^{m/2}_{11}+C_{m})1_{\left\\{\sqrt{W_{11}}\leq
b_{0}(t)\tau\right\\}}\right]$ $\displaystyle\leq$ $\displaystyle
2^{m/2}\theta^{-m}a_{m}\frac{t}{b^{m}_{0}(t)}\mathbb{E}\left[W^{m/2}_{11}1_{\left\\{\sqrt{W_{11}}\leq
b_{0}(t)\tau\right\\}}\right]+2^{m/2}\theta^{-m}C_{m}\frac{t}{b^{m}_{0}(t)}.$

Choosing $m>2\alpha$ makes the second hand term in the previous display tend
to zero. By Karamata’s Theorem, we also achieve that

$\displaystyle\frac{t}{b^{m}_{0}(t)}\mathbb{E}\left[W^{m/2}_{11}1_{\left\\{\sqrt{W_{11}}\leq
b_{0}(t)\tau\right\\}}\right]\leq$
$\displaystyle\frac{t}{b^{m}_{0}(t)}\int_{0}^{b_{0}^{m}(t)\tau^{m}}\mathbb{P}\left(W_{11}^{m/2}>x\right)dx$
$\displaystyle\sim$
$\displaystyle\left(1-\frac{2\alpha}{m}\right)\tau^{m}t\mathbb{P}\left(W_{11}^{m/2}>b_{0}^{m}(t)\tau^{m}\right)$
$\displaystyle\sim$
$\displaystyle\left(1-\frac{2\alpha}{m}\right)\tau^{m-2\alpha}.$

Hence $\lim_{\tau\rightarrow 0}\limsup_{t\rightarrow\infty}B(t,\tau)=0$ and we
have proven (45). One may evaluate the convergence (45) on sets of the form
$(u,\infty]\times(v,\infty]$ $u,v\geq 0$ to obtain that as
$t\rightarrow\infty$

$\displaystyle
t\mathbb{P}\left(\frac{|\mathcal{D}_{11}-\mathcal{D}_{12}|}{\sqrt{2}b_{0}(t)}>u,\frac{\sqrt{2}\mathcal{D}_{11}/\sqrt{W_{11}}}{|\mathcal{D}_{11}-\mathcal{D}_{12}|}>v\right)\rightarrow
u^{-2\alpha}\cdot 2\int_{0}^{1/v}z^{2\alpha}\phi(z)dz,$

where $\phi$ is the standard normal density. ∎

###  6.4 Tail empirical measure approximation

In this section we present results that allow us to approximate the tail
empirical measure of the degrees (15) by the tail empirical measure of the
weights (13). From (3), we expect that the large degrees in layers
$\mathcal{L}_{1}$ will concentrate around their conditional mean due to the
strong concentration of Poisson distributions with large rates. Additionally,
we expect the conditional means to concentrate around the scaled weights, as
indicated by Lemma 6.3. We also expect results derived from the layers in
$\mathcal{L}_{1}$ to translate to layers in $\mathcal{L}_{2}$ using the
coupling developed in Section 6.2. Thus, our proof strategy proceeds as
follows.

  * 1.

In Section 6.4.1, we prove concentration of the degrees around the conditional
means for two subsets of models:

    * (i)

$|\mathcal{L}_{1}|=L$ and

    * (ii)

$|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$, $\alpha\in(1,2]$.

We are not able to use the coupling in Section 6.2 to handle case (ii) since
it is seemingly inapplicable in the infinite-variance setting. Such
limitations also appear in [6] (see Section 4.2).

  * 2.

In Section 6.4.2, we prove concentration of the conditional mean degree around
the scaled weights. Note that we need not distinguish between models (i) and
(ii) in this case.

  * 3.

In Section 6.4.3, we connect results developed in Sections 6.4.1 and 6.4.2 to
produce the tail empirical approximation for submodels (i) and (ii). We then
use the coupling developed in Section 6.2 to prove the tail empirical measure
approximation for the submodel $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$,
$\alpha>2$ by relating it to (i).

####  6.4.1 Concentration of the degrees

Lemma 6.7 considers concentration of the degrees around their conditional
means when $|\mathcal{L}_{1}|=L$. Define the conditional mean
$\boldsymbol{g}_{1}(n)=\mathbb{E}[\mathbf{D}_{1}(n)\mid\mathbf{W}_{[n]}]$ and
recall $\boldsymbol{\mathcal{W}}_{1}=(c_{1}W_{11},\dots,c_{L}W_{1L})$.

######  Lemma 6.7.

Suppose $|\mathcal{L}_{1}|=L$. Assume (C1)-(C3) hold. Let $\eta$ and $u$ be
strictly positive real numbers. As $n\rightarrow\infty$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n})\right)\rightarrow 0,$ (46)
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n})\right)\rightarrow 0.$ (47)

###### Proof.

Fix $\epsilon\in(0,\delta)$, where $\delta$ is as in (5). Define the event

$\displaystyle\mathcal{A}_{n}(\epsilon)=\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\epsilon\right\\}.$

Note that due to Lemma 6.1, in suffices to prove that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)\rightarrow 0,$ (48)
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)\rightarrow 0.$ (49)

We start by proving (48). Let $m$ be a positive integer such that $m>2\alpha$.
We may apply Markov’s inequality and Lemma 6.6 to achieve that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)$
$\displaystyle\leq\left(\frac{1}{\eta
b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{m}_{p}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right]$
$\displaystyle\leq\left(\frac{L^{\frac{1}{p}+\frac{1}{m}}}{\eta
b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\mathbb{E}\left[\left(a_{m}\|\boldsymbol{g}_{1}(n)\|^{m/2}_{p}+C_{m}\right)1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right].$

Since $m>\alpha$, we may determine that as $n\rightarrow\infty$

$\displaystyle\left(\frac{1}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)\leq\left(\frac{1}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\rightarrow
0.$

Hence in order to prove (48), it suffices to show that

$\displaystyle
H_{1}(n)\equiv\left(\frac{1}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\mathbb{E}\left[\|\boldsymbol{g}_{1}(n)\|_{p}^{m/2}1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right]\rightarrow 0,$

as $n\rightarrow\infty$. Note that when
$\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\epsilon<\delta$,

$\displaystyle\sum_{l=1}^{L}\left(\sum_{j=1}^{n}g_{l}\left(W_{1l}W_{jl}/T_{l}(n)\right)\right)^{p}\leq$
$\displaystyle\sum_{l=1}^{L}\left(MW^{1+\nu}_{1l}\sum_{j=1}^{n}\frac{W^{1+\nu}_{jl}}{T^{1+\nu}_{l}(n)}+c_{l}W_{1l}\right)^{p}$
$\displaystyle\leq$
$\displaystyle\sum_{l=1}^{L}\left(MW^{1+\nu}_{1l}\frac{W^{\nu}_{(1)l}(n)}{T^{\nu}_{l}(n)}+c_{l}W_{1l}\right)^{p}$
$\displaystyle\leq$
$\displaystyle\sum_{l=1}^{L}\left(M\epsilon^{\nu}W_{1l}+c_{l}W_{1l}\right)^{p}$
$\displaystyle\leq$ $\displaystyle
2^{p}\|\boldsymbol{\mathcal{W}}_{1}\|^{p}_{p},$

where, if necessary, we have taken $\epsilon$ small enough so that
$M\epsilon^{\nu}<\min_{l\in[L]}c_{l}$. Hence
$\|\boldsymbol{g}_{1}(n)\|_{p}^{m/2}\leq
2^{m/2}\|\boldsymbol{\mathcal{W}}_{1}\|^{m/2}_{p}$ and we may write

$\displaystyle H_{1}(n)\leq$
$\displaystyle\left(\frac{2^{1/2}}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\mathbb{E}\left[\|\boldsymbol{\mathcal{W}}_{1}\|_{p}^{m/2}1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
ub(n/k_{n})\right\\}}\right]$ $\displaystyle\leq$
$\displaystyle\left(\frac{2^{1/2}}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\int_{0}^{(ub(n/k_{n}))^{m/2}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|^{m/2}_{p}>t\right)dt,$
and recalling that $m>2\alpha$ we may apply Karamata’s Theorem [35, Theorem
2.1] to achieve that $\displaystyle\sim$
$\displaystyle\left(\frac{2^{1/2}}{b(n/k_{n})}\right)^{m}\frac{n}{k_{n}}\left(1-\frac{2\alpha}{m}\right)^{-1}\left(ub(n/k_{n})\right)^{m/2}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>ub(n/k_{n})\right)$
$\displaystyle\sim$
$\displaystyle\left(\frac{2}{b(n/k_{n})}\right)^{m/2}\left(1-\frac{2\alpha}{m}\right)^{-1}u^{\frac{m}{2}-\alpha}\rightarrow
0,$

as $n\rightarrow\infty$. Hence (48) and is proven and thus so is (46). We now
prove (47), for which it suffices to prove (49). By similar calculations as
before

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)$
$\displaystyle\leq\left(\frac{1}{\eta
b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{2}_{p}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right]$
$\displaystyle\leq\left(\frac{L^{\frac{1}{p}+\frac{1}{2}}}{\eta
b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{E}\left[\left(a_{2}\|\boldsymbol{g}_{1}(n)\|_{p}+C_{2}\right)1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right].$

For the second summand, note that

$\displaystyle\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right)$
$\displaystyle\leq\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n})\right)\rightarrow 0\cdot u^{-\alpha}=0,$

as $n\rightarrow\infty$, by regular variation of
$\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$. Hence in order to prove (49), it
suffices to show that

$\displaystyle
H_{2}(n)\equiv\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{E}\left[\|\boldsymbol{g}_{1}(n)\|_{p}1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n}),\mathcal{A}_{n}(\epsilon)\right\\}}\right]\rightarrow 0,$

as $n\rightarrow\infty$. Recall that
$\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\epsilon<\delta$ implies
that $\|\boldsymbol{g}_{1}(n)\|_{p}\leq 2\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$
for $\epsilon$ small enough and thus

$\displaystyle H_{2}(n)\leq$ $\displaystyle
2\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\mathbb{E}\left[\|\boldsymbol{\mathcal{W}}_{1}\|_{p}1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n})\right\\}}\right],$ $\displaystyle=$ $\displaystyle
2\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\int_{0}^{\infty}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}1_{\left\\{\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n})\right\\}}>t\right)dt$ $\displaystyle=$ $\displaystyle
2\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}ub(n/k_{n})\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq
ub(n/k_{n})\right)$
$\displaystyle+2\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}\int_{ub(n/k_{n})}^{\infty}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>t\right)dt$
$\displaystyle\sim$ $\displaystyle
2\frac{1}{b(n/k_{n})}u^{1-\alpha}+2\left(\frac{1}{b(n/k_{n})}\right)^{2}\frac{n}{k_{n}}(\alpha-1)^{-1}ub(n/k_{n})\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>ub(n/k_{n})\right)$
$\displaystyle\sim$ $\displaystyle
2\frac{1}{b(n/k_{n})}u^{1-\alpha}+2\frac{1}{b(n/k_{n})}(\alpha-1)^{-1}u^{1-\alpha},$

where we have again applied Karamata’s Theorem since
$\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$ has a tail index of $\alpha>1$. Hence
$H_{2}(n)\rightarrow 0$ as $n\rightarrow\infty$ and (49) is proven. Thus we
have proved (47). ∎

Lemma 6.8 considers concentration of the degrees around their conditional
means when $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ and $\alpha\in(1,2]$.

######  Lemma 6.8.

Suppose $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ and that (C1), (C3)
hold. Assume (C2) holds with the restriction that $\alpha\in(1,2]$. Then

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})\right)\rightarrow 0,$

as $n\rightarrow\infty$.

###### Proof.

Let $\delta$ be as in (5). in From Lemma 6.1, it suffies to show that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)\rightarrow
0,$

as $n\rightarrow\infty$. Suppose $\alpha\in(1,2)$. By equivalence of
$\ell_{p}$ norms, we have that

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{2}\geq\eta
b(n/k_{n})/L^{1/p},\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)$
$\displaystyle=$
$\displaystyle\frac{n}{k_{n}}\mathbb{E}\left[\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{2}\geq\eta
b(n/k_{n})/L^{1/p}\ \bigg{|}\
\boldsymbol{\mathcal{W}}_{[n]}\right)1_{\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{2}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{2}_{2}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right\\}}\right].$

Note that for $l\in\mathcal{L}_{1}$

$\displaystyle\mathbb{E}\left[\left(D_{1l}(n)-\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right)^{2}\
\Bigg{|}\
\boldsymbol{\mathcal{W}}_{[n]}\right]=\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n)),$

and for $l\in\mathcal{L}_{2}$

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left(D_{1l}(n)-\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right)^{2}\
\Bigg{|}\ \boldsymbol{\mathcal{W}}_{[n]}\right]$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))(1-g_{l}(W_{1l}W_{jl}/T_{l}(n)))$
$\displaystyle\leq$ $\displaystyle\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n)).$

Additionally, when $\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta$,

$\displaystyle\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\leq$ $\displaystyle
c_{l}W_{1l}+M\sum_{j=1}^{n}\left(\frac{W_{1l}W_{jl}}{T_{l}(n)}\right)^{1+\nu}$
$\displaystyle\leq$ $\displaystyle
c_{l}W_{1l}+MW^{1+\nu}_{1l}\left(\frac{W_{(1)l}(n)}{T_{l}(n)}\right)^{\nu}$
$\displaystyle\leq$ $\displaystyle\left(c_{l}+M\delta^{\nu}\right)W_{1l}.$

Hence

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{2}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{2}_{2}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{2}\sum_{l=1}^{L}\mathbb{E}\left[\left(c_{l}+M\delta^{\nu}\right)W_{1l}\right].$

Since $-\alpha<-1$, $\mathbb{E}[W_{1l}]<\infty$ and since $b(t)$ is regularly
varying with index $1/\alpha$, we have that

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)\rightarrow
0\qquad\text{as }n\rightarrow\infty.$

Now suppose $\alpha=2$. Using similar steps we find that

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{3}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{3}_{3}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right\\}}\right]$

Note that for $l\in\mathcal{L}_{1}$, by Lemma 6.5

$\displaystyle\mathbb{E}\left[\left|D_{1l}(n)-\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right|^{3}\
\Bigg{|}\ \boldsymbol{\mathcal{W}}_{[n]}\right]\leq
a_{3}\left(\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right)^{3/2}+C_{3}.$

Applying a bound for the absolute third central moment of Poisson binomial
random variables developed in Lemma 6.9, we further find that for
$l\in\mathcal{L}_{2}$

$\displaystyle\mathbb{E}\left[\left|D_{1l}(n)-\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\right|^{3}\
\Bigg{|}\ \boldsymbol{\mathcal{W}}_{[n]}\right]\leq$ $\displaystyle
2\sum_{j=1}^{n}g(W_{1l}W_{jl}/T_{l}(n))$
$\displaystyle+2\left(\sum_{j=1}^{n}g(W_{1l}W_{jl}/T_{l}(n))\right)^{3/2}.$

When $\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta$,

$\displaystyle\sum_{j=1}^{n}g_{l}(W_{1l}W_{jl}/T_{l}(n))\leq\left(c_{l}+M\delta^{\nu}\right)W_{1l},$

and hence

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{3}\mathbb{E}\left[\mathbb{E}\left[\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|^{3}_{3}\mid\boldsymbol{\mathcal{W}}_{[n]}\right]1_{\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right\\}}\right]$
$\displaystyle\leq$ $\displaystyle\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{3}\sum_{l\in\mathcal{L}_{1}}\left(a_{3}\left(c_{l}+M\delta^{\nu}\right)^{3/2}\mathbb{E}\left[W^{3/2}_{1l}\right]+C_{3}\right)$
$\displaystyle+2\frac{n}{k_{n}}\left(\frac{L^{1/p}}{\eta
b(n/k_{n})}\right)^{3}\sum_{l\in\mathcal{L}_{2}}\left(\left(c_{l}+M\delta^{\nu}\right)\mathbb{E}\left[W_{1l}\right]+\left(c_{l}+M\delta^{\nu}\right)^{3/2}\mathbb{E}\left[W^{3/2}_{1l}\right]\right).$

Since $\alpha=2$,
$\mathbb{E}\left[W_{1l}\right],\mathbb{E}\left[W^{3/2}_{1l}\right]<\infty$.
Additionally, since $b(t)$ is regularly varying with index $1/2$, we have that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta\right)\rightarrow
0,\qquad\text{as }n\rightarrow\infty.$

∎

######  Lemma 6.9.

Fix $n\in\mathbb{N}$. Suppose $X_{1},X_{2},\dots,X_{n}$ are independent
Bernoulli random variables with success probabilities
$p_{1},p_{2},\dots,p_{n}$, respectively. Then

$\displaystyle\mathbb{E}\left[\left|\sum_{i=1}^{n}(X_{i}-p_{i})\right|^{3}\right]\leq
2\sum_{i=1}^{n}p_{i}+2\left(\sum_{i=1}^{n}p_{i}\right)^{3/2}.$ (50)

###### Proof.

A tedious calculation gives that

$\displaystyle\mathbb{E}\left[\left(\sum_{i=1}^{n}(X_{i}-p_{i})\right)^{4}\right]=$
$\displaystyle\sum_{i=1}^{n}p_{i}(1-p_{i})\left((1-2p_{i})^{2}-2p_{i}(1-p_{i})\right)+3\left(\sum_{i=1}^{n}p_{i}(1-p_{i})\right)^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{n}p_{i}+3\left(\sum_{i=1}^{n}p_{i}\right)^{2}.$

Hence by Cauchy-Bunyakovsky-Schwarz,

$\displaystyle\mathbb{E}\left[\left|\sum_{i=1}^{n}(X_{i}-p_{i})\right|^{3}\right]\leq$
$\displaystyle\left(\mathbb{E}\left[\left(\sum_{i=1}^{n}(X_{i}-p_{i})\right)^{4}\right]\mathbb{E}\left[\left(\sum_{i=1}^{n}(X_{i}-p_{i})\right)^{2}\right]\right)^{1/2},$
and using the fact that
$\mathbb{E}\left[\left(\sum_{i=1}^{n}(X_{i}-p_{i})\right)^{2}\right]=\sum_{i=1}^{n}p_{i}(1-p_{i})\leq\sum_{i=1}^{n}p_{i}$,
$\displaystyle\leq$
$\displaystyle\left(\left(\sum_{i=1}^{n}p_{i}+3\left(\sum_{i=1}^{n}p_{i}\right)^{2}\right)\sum_{i=1}^{n}p_{i}\right)^{1/2}$
$\displaystyle=$
$\displaystyle\left(\left(\sum_{i=1}^{n}p_{i}\right)^{2}+3\left(\sum_{i=1}^{n}p_{i}\right)^{3}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
2\max\left\\{\sum_{i=1}^{n}p_{i},\left(\sum_{i=1}^{n}p_{i}\right)^{3/2}\right\\}$
$\displaystyle\leq$ $\displaystyle
2\sum_{i=1}^{n}p_{i}+2\left(\sum_{i=1}^{n}p_{i}\right)^{3/2}.$

∎

####  6.4.2 Concentration of the conditional mean

Lemma 6.10 considers concentration of the conditional mean around the scaled
weights.

######  Lemma 6.10.

Suppose (C1)-(C3) hold. For any $\eta>0,$ and $p\in\mathbb{N}$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b(n/k_{n})\right)\rightarrow 0,$ (51)

as $n\rightarrow\infty$.

###### Proof.

Fix $\epsilon\in(0,\delta)$, where $\delta$ is as in (5). If
$\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\epsilon<\delta$, we find
that

$\displaystyle\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq$
$\displaystyle\left(M^{p}\sum_{l=1}^{L}\left(W^{1+\nu}_{1l}\sum_{j=1}^{n}\frac{W^{1+\nu}_{jl}}{T^{1+\nu}_{l}(n)}\right)^{p}\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle
M\left(\sum_{l=1}^{L}\left(W^{1+\nu}_{1l}\frac{W^{\nu}_{(1)l}(n)}{T^{\nu}_{l}(n)}\right)^{p}\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\epsilon^{\nu}M\left(\sum_{l=1}^{L}W^{p}_{1l}\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon^{\nu}M}{c_{\text{min}}^{p}}\left(\sum_{l=1}^{L}c_{l}^{p}W^{p}_{1l}\right)^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\frac{\epsilon^{\nu}M}{c_{\text{min}}^{p}}\|\boldsymbol{\mathcal{W}}_{1}\|_{p},$

where we define $c_{\text{min}}=\min\\{c_{1},\dots,c_{L}\\}$. Hence

$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b(n/k_{n})\right)$ $\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b(n/k_{n}),\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\epsilon\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}>\epsilon\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>\frac{\eta
c_{\text{min}}^{p}}{\epsilon^{\nu}M}b(n/k_{n})\right)+\frac{n}{k_{n}}\sum_{l=1}^{L}\mathbb{P}\left(W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}>\epsilon\right)$
$\displaystyle\equiv$ $\displaystyle C_{1}(n)+C_{2}(n).$

From Lemma 6.1 we have that $C_{2}(n)\rightarrow 0$ as $n\rightarrow\infty$.
Note that due to regular variation of $\boldsymbol{\mathcal{W}}_{1}$ (and
hence $\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$),

$\displaystyle
C_{1}(n)=\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>\frac{\eta
c^{p}_{\text{min}}}{\epsilon^{\nu}M}b(n/k_{n})\right)\rightarrow\left(\frac{\epsilon^{\nu}M}{\eta
c^{p}_{\text{min}}}\right)^{\alpha},\quad\text{as }n\rightarrow\infty.$

Since $\epsilon>0$ was arbitrary, we may conclude that $C_{1}(n)\rightarrow 0$
as $n\rightarrow\infty$. ∎

####  6.4.3 Proof of tail empirical measure approximation

This subsection is devoted to the proof of Theorem 3.1. In order to prove
Theorem 3.1, we require the following lemma.

######  Lemma 6.11.

Suppose (C1)-(C3) hold. Then as $n\rightarrow\infty$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}>yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
yb(n/k_{n})\right)\rightarrow 0,$ (52)
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n}),\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n})\right)\rightarrow 0.$ (53)

We divide the proof of Lemma 6.11 into two parts. The first, presented in
Lemma 6.12, employs concentration results developed in Sections 6.4.1 and
6.4.2 to prove a tail empirical measure approximation for the submodels where
either $|\mathcal{L}_{1}|=L$ or $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$
and $\alpha\in(1,2]$. The second, presented in Lemma 6.14, employs the
coupling presented in Section 6.2 to approximate the tail empirical measure
when $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ and $\alpha>2$.

######  Lemma 6.12.

Suppose (C1)-(C3) hold. Additionally, suppose either $|\mathcal{L}_{1}|=L$ or
$|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ with the restriction that
$\alpha\in(1,2]$. Then as $n\rightarrow\infty$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}>yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
yb(n/k_{n})\right)\rightarrow 0,$ (54)
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n}),\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n})\right)\rightarrow 0.$ (55)

###### Proof.

Fix $\eta>0$. See that

$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\mathbf{D}_{1}(n)\|_{p}>yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
yb(n/k_{n})\right)$ $\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\in(y-\eta,y]b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}-\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$
$\displaystyle\equiv$ $\displaystyle A_{11}(n)+A_{12}(n).$

Similarly,

$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n}),\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n})\right)$ $\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\in(y,y+\eta]b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}-\|\mathbf{D}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle\equiv$ $\displaystyle A_{21}(n)+A_{22}(n).$

By regular variation of $\boldsymbol{\mathcal{W}}_{1}$, we have that as
$n\rightarrow\infty$

$\displaystyle A_{11}(n)=$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\in(y-\eta,y]b(n/k_{n})\right)\rightarrow(y-\eta)^{-\alpha}-y^{-\alpha},$
$\displaystyle A_{21}(n)=$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\in(y,y+\eta]b(n/k_{n})\right)\rightarrow
y^{-\alpha}-(y+\eta)^{-\alpha}.$

We now analye the behavior of $A_{12}(n)$. Using the fact that
$\|\mathbf{D}_{1}(n)\|_{p}-\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq\left|\|\mathbf{D}_{1}(n)\|_{p}-\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\right|\leq\|\mathbf{D}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}$,

$\displaystyle A_{12}(n)$
$\displaystyle=\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}-\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$
$\displaystyle\leq\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$

Define
$\boldsymbol{g}_{1}(n)=\mathbb{E}[\mathbf{D}_{1}(n)\mid\boldsymbol{\mathcal{W}}_{[n]}]$.
Then

$\displaystyle A_{12}(n)\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq(y-\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n})/2\right)$ $\displaystyle\equiv$ $\displaystyle
A_{121}(n)+A_{122}(n).$

From Lemmas 6.7 and 6.8, $A_{121}(n)\rightarrow 0$ as $n\rightarrow\infty$.
Additionally, Lemma 6.10 gives that $A_{122}(n)\rightarrow 0$ as
$n\rightarrow\infty$. Hence $A_{12}(n)\rightarrow 0$ as $n\rightarrow\infty$.
Similarly, for $A_{22}(n)$

$\displaystyle A_{22}(n)=$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}-\|\mathbf{D}_{1}(n)\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{g}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})/2,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq(y+\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{g}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}\geq\eta
b(n/k_{n})/2\right)$ $\displaystyle\equiv$ $\displaystyle
A_{221}(n)+A_{222}(n).$

From Lemmas 6.7, 6.8 and 6.10, $A_{221}(n)\rightarrow 0$ and
$A_{222}(n)\rightarrow 0$ as $n\rightarrow\infty$. Hence $A_{22}(n)\rightarrow
0$ as $n\rightarrow\infty$. In summary,

$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\mathbf{D}_{1}(n)\|_{p}>yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
yb(n/k_{n})\right)\leq
A_{11}(n)+A_{12}(n)\rightarrow(y-\eta)^{-\alpha}-y^{-\alpha},$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n}),\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n})\right)\leq A_{21}(n)+A_{22}(n)\rightarrow
y^{-\alpha}-(y+\eta)^{-\alpha},$

as $n\rightarrow\infty$. Since $\eta>0$ was arbitrary, we have proven (54) and
(55). ∎

In Lemma 6.12, we proved an approximation of the tail empirical measure for
the MIRG model with $|\mathcal{L}_{1}|=L$. We use this result, along with the
coupling presented in Section 6.2 to approximate the tail empirical measure
when $|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ and $\alpha>2$. In order to
do so, we first compare $\mathbf{D}_{1}(n)$ with $\tilde{\mathbf{D}}_{1}(n)$
presented in Section 6.2.

######  Lemma 6.13.

Suppose (C1) and(C3) hold. Suppose (C2) holds with $\alpha>2$. Let
$|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$. Then

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{p}\geq\eta
b(n/k_{n})\right)\rightarrow 0,$

as $n\rightarrow\infty$.

###### Proof.

It suffices to prove the case where $|\mathcal{L}_{2}|=L$ since the distance
$\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{p}$ is increasing with the
number of Bernoulli-layers. By equivalence of $\ell_{p}$ norms, we may instead
show that as $n\rightarrow\infty$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{1}\geq\eta
b(n/k_{n})/L^{1/p}\right)\rightarrow 0.$

Note that for every $l\in[L]$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(W_{1l}>\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right)=\frac{n}{k_{n}}\frac{1}{n}=\frac{1}{k_{n}}\rightarrow
0,\qquad\text{as }n\rightarrow\infty.$ (56)

Define the event

$\displaystyle\mathcal{H}_{n}\equiv\left\\{\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta,W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\quad\forall
l\in[L]\right\\}.$

By Lemma 6.2 and (56), it suffices to show that as $n\rightarrow\infty$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{1}\geq\eta
b(n/k_{n})/L^{1/p},\mathcal{H}_{n}\right)\rightarrow 0,$ (57)

where $\delta>0$ is as in (5). Note that by Markov’s inequality and (33),

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{1}\geq\eta
b(n/k_{n})/L^{1/p},\mathcal{H}_{n}\right)$
$\displaystyle=\frac{n}{k_{n}}\mathbb{E}\left[\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{1}\geq\eta
b(n/k_{n})/L^{1/p}\bigg{|}\ \mathbf{W}_{[n]}\right)1_{\mathcal{H}_{n}}\right]$
$\displaystyle\leq\frac{L^{1/p}}{\eta
b(n/k_{n})}K\frac{n}{k_{n}}\mathbb{E}\left[\sum_{l=1}^{L}\sum_{j=1}^{n}g_{l}^{2}\left(\frac{W_{1l}W_{jl}}{T_{l}(n)}\right)1_{\mathcal{H}_{n}}\right].$

Note that if $\max_{l\in[L]}W_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}<\delta$, by
Lemma 6.4

$\displaystyle\sum_{l=1}^{L}\sum_{j=1}^{n}g_{l}^{2}\left(\frac{W_{1l}W_{jl}}{T_{l}(n)}\right)\leq$
$\displaystyle\sum_{l=1}^{L}CW^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)},$

for some constant $C>0$. Hence

$\displaystyle\frac{n}{k_{n}}$
$\displaystyle\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{1}\geq\eta
b(n/k_{n})/L^{1/p},\mathcal{H}_{n}\right)$
$\displaystyle\leq\frac{L^{1/p}}{\eta
b(n/k_{n})}KC\frac{n}{k_{n}}\mathbb{E}\left[\sum_{l=1}^{L}W^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}1_{\left\\{\cap_{l\in[L]}\left\\{W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right\\}\right\\}}\right],$

and hence it suffices to show that for any $l\in[L]$

$\displaystyle\frac{1}{b(n/k_{n})}\frac{n}{k_{n}}\mathbb{E}\left[W^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}1_{\left\\{\bigcap_{l\in[L]}\left\\{W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right\\}\right\\}}\right]\rightarrow
0,\qquad\text{as }n\rightarrow\infty.$

See that if $W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}$, by
similar calculations in the proofs of Lemmas 6.1 and 6.2

$\displaystyle W^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}\leq$ $\displaystyle
2\frac{W^{2}_{1l}\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}.$

and

$\displaystyle\frac{1}{b(n/k_{n})}\frac{n}{k_{n}}$
$\displaystyle\mathbb{E}\left[W^{2}_{1l}\frac{W_{(1)l}(n)}{T_{l}(n)}1_{\left\\{\bigcap_{l\in[L]}\left\\{W_{1l}\leq\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right\\}\right\\}}\right]$
$\displaystyle\leq$
$\displaystyle\frac{2}{b(n/k_{n})}\frac{n}{k_{n}}\mathbb{E}[W^{2}_{1l}]\mathbb{E}\left[\frac{\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}}{\sum_{i\in[n]\setminus\\{1\\}}W_{il}}\right]$
$\displaystyle\sim$
$\displaystyle\frac{2}{b(n/k_{n})}\frac{n}{k_{n}}\frac{\mathbb{E}[W^{2}_{1l}]}{\mathbb{E}[W_{1l}]}\frac{\mathbb{E}\left[\underset{i\in[n]\setminus\\{1\\}}{\max}W_{il}\right]}{n-1}$
$\displaystyle\sim$
$\displaystyle\frac{2}{b(n/k_{n})}\frac{1}{k_{n}}\frac{\mathbb{E}[W^{2}_{1l}]}{\mathbb{E}[W_{1l}]}\Gamma\left(1-\frac{1}{\alpha}\right)b^{\star}(n),$

where we have applied Theorem 9.1 of [19] and Proposition 2.1 of [36]. Here,
$b^{\star}(t)$ is the $1-1/t$ quantile function of the distribution of
$W_{1l}$ for $t\geq 1$. Since $b^{\star}(t)$ is regularly varying with index
$1/\alpha$, we may apply (C3) to obtain that bound tends to zero and the proof
is complete. ∎

We now prove the tail empirical measure approximation when
$|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$ and $\alpha>2$.

######  Lemma 6.14.

Suppose (C1) and(C3) hold. Suppose (C2) holds with $\alpha>2$. Let
$|\mathcal{L}_{1}|<L,|\mathcal{L}_{2}|\leq L$. Then for $y>0$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}>yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\leq
yb(n/k_{n})\right)\rightarrow 0,$ (58)
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n})\right)\rightarrow
0,$ (59)

as $n\rightarrow\infty$.

###### Proof.

We only prove (59) as the proof for (58) is very similar. See that for
$\eta>0$

$\displaystyle\frac{n}{k_{n}}\mathbb{P}$
$\displaystyle\left(\|\mathbf{D}_{1}(n)\|_{p}\leq
yb(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>yb(n/k_{n})\right)$
$\displaystyle\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b(n/k_{n}),\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>(y+\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\boldsymbol{\mathcal{W}}_{1}\|_{p}\in(y,y+\eta]b(n/k_{n})\right)$
$\displaystyle=$ $\displaystyle B_{1}(n)+B_{2}(n).$

We prove that $B_{1}(n)$ and $B_{2}(n)$ converge to $0$ as
$n\rightarrow\infty$. For $B_{1}(n)$, note that

$\displaystyle B_{1}(n)\leq$
$\displaystyle\frac{n}{k_{n}}\mathbb{P}\left(\|\mathbf{D}_{1}(n)-\tilde{\mathbf{D}}_{1}(n)\|_{p}>\eta
b(n/k_{n})/3\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\tilde{\mathbf{D}}_{1}(n)-\tilde{\boldsymbol{g}}_{1}(n)\|_{p}>\eta
b(n/k_{n})/3,\|\boldsymbol{\mathcal{W}}_{1}\|_{p}>(y+\eta)b(n/k_{n})\right)$
$\displaystyle+\frac{n}{k_{n}}\mathbb{P}\left(\|\tilde{\boldsymbol{g}}_{1}(n)-\boldsymbol{\mathcal{W}}_{1}\|_{p}>\eta
b(n/k_{n})/3\right)$ $\displaystyle=$ $\displaystyle
B_{11}(n)+B_{12}(n)+B_{13}(n),$

where we have defined
$\tilde{\boldsymbol{g}}_{1}(n)=\mathbb{E}[\tilde{\mathbf{D}}_{1}(n)\mid\mathbf{W}_{[n]}]$.
Note that as $n\rightarrow\infty$, $B_{11}(n)\rightarrow 0$ by Lemma 6.13,
$B_{12}(n)\rightarrow 0$ by Lemma 6.7 and $B_{13}(n)\rightarrow 0$ by Lemma
6.10. Hence $B_{1}(n)\rightarrow 0$ as $n\rightarrow\infty$. From regular
variation of $\|\boldsymbol{\mathcal{W}}_{1}\|_{p}$, we additionally have that
as $n\rightarrow\infty$

$\displaystyle B_{2}(n)\rightarrow y^{-\alpha}-(y+\eta)^{-\alpha}.$

Since $\eta>0$ is arbitrary, the proof is complete ∎

With Lemma 6.11 in hand, we are now able to prove 3.1. The proof of Theorem
3.1 closely follows that of Proposition 3.5 of [6].

###### Proof of Theorem 3.1.

From Theorem 1.1 of [25], it suffices to show that for any $d\in\mathbb{N}$

$\displaystyle\left(\nu_{n}(I_{1}),\dots\nu_{n}(I_{d})\right)\Rightarrow\left(\nu_{\alpha}(I_{1}),\dots\nu_{\alpha}(I_{d})\right),$
(60)

in $\mathbb{R}^{d}$ as $n\rightarrow\infty$ for intervals of the form
$I_{k}=(a_{k},b_{k}]$, $0<a_{k}<b_{k}\leq\infty$, $k\in[d]$. Let $d$ denote
the Euclidean metric. By Slutsky’s Theorem, the convergence in (60) is
achieved by combining

$\displaystyle\left(\nu^{\star}_{n}(I_{1}),\dots\nu^{\star}_{n}(I_{d})\right)\Rightarrow\left(\nu_{\alpha}(I_{1}),\dots\nu_{\alpha}(I_{d})\right),\qquad\text{in
}\mathbb{R}^{d},$ (61) $\displaystyle
d\left(\left(\nu_{n}(I_{1}),\dots\nu_{n}(I_{d})\right),\left(\nu^{\star}_{n}(I_{1}),\dots\nu^{\star}_{n}(I_{d})\right)\right)\xrightarrow{p}0,$
(62)

$n\rightarrow\infty$, where (61) is given by (14). In order to prove (62), it
suffices to show that for any $0<a<b\leq\infty$

$\displaystyle\nu_{n}\left((a,b]\right)-\nu^{\star}_{n}\left((a,b]\right)\xrightarrow{p}0.$

Towards this end, see that

$\displaystyle\mathbb{E}\left|\nu_{n}\left((a,b]\right)-\nu^{\star}_{n}\left((a,b]\right)\right|=$
$\displaystyle\mathbb{E}\left|\nu_{n}\left((a,\infty]\right)-\nu_{n}\left((b,\infty]\right)-\left(\nu^{\star}_{n}\left((a,\infty]\right)-\nu^{\star}_{n}\left((b,\infty]\right)\right)\right|$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left|\nu_{n}\left((a,\infty]\right)-\nu^{\star}_{n}\left((a,\infty]\right)\right|+\mathbb{E}\left|\nu_{n}\left((b,\infty]\right)-\nu^{\star}_{n}\left((b,\infty]\right)\right|.$

Additionally note that for any $y\in(0,\infty]$

$\displaystyle\mathbb{E}\left|\nu_{n}\left((y,\infty]\right)-\nu^{\star}_{n}\left((y,\infty]\right)\right|\leq$
$\displaystyle\frac{1}{k_{n}}\sum_{i=1}^{n}\mathbb{E}\left[1_{\left\\{R_{i}(n)>yb(n/k_{n}),R^{\star}_{i}(n)\leq
yb(n/k_{n})\right\\}}\right]$
$\displaystyle+\frac{1}{k_{n}}\sum_{i=1}^{n}\mathbb{E}\left[1_{\left\\{R^{\star}_{i}(n)>yb(n/k_{n}),R_{i}(n)\leq
yb(n/k_{n})\right\\}}\right]$ $\displaystyle=$