Freezing Limits for Beta-Cauchy Ensembles

Freezing Limits for Beta-Cauchy Ensembles

Michael VOIT

M. Voit Fakultät Mathematik, Technische Universität Dortmund,
Vogelpothsweg 87, D-44221 Dortmund, Germany<EMAIL_ADDRESS>http://www.mathematik.tu-dortmund.de/lsiv/voit/voit.html

Received May 19, 2022, in final form September 15, 2022; Published online
September 28, 2022

Bessel processes associated with the root systems $A_{N-1}$ and $B_{N}$
describe interacting particle systems with $N$ particles on $\mathbb{R}$; they
form dynamic versions of the classical $\beta$-Hermite and Laguerre ensembles.
In this paper we study corresponding Cauchy processes constructed via some
subordination. This leads to $\beta$-Cauchy ensembles in both cases with
explicit distributions. For these distributions we derive central limit
theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend
to infinity. The results are closely related to corresponding known freezing
results for $\beta$-Hermite and Laguerre ensembles and for Bessel processes.

Cauchy processes; Bessel processes; $\beta$-Hermite ensembles;
$\beta$-Laguerre ensembles; freezing; zeros of classical orthogonal
polynomials; Calogero–Moser–Sutherland particle models

60F05; 60B20; 70F10; 82C22; 33C45

##  1 Introduction

By a classical result in probability (see, e.g., [12, 31]), a subordination of
Brownian motions on $\mathbb{R}^{N}$ by inverse Gaussian Lévy processes on
$[0,\infty[$ leads to classical Cauchy processes on $\mathbb{R}^{N}$. In the
one-dimensional case and for a start in the origin, these Cauchy processses
$(X_{t})_{t\geq 0}$ are Cauchy-distributed with the densities

$f_{t}(x)=\frac{1}{\pi}{\frac{t}{t^{2}+x^{2}}},\qquad x\in\mathbb{R},\quad
t>0.$

Motivated by the theory of Bessel processes associated with root systems on
Weyl chambers $C_{N}\subset\mathbb{R}^{N}$ and the distributions of the
classical $\beta$-Hermite and Laguerre ensembles, one can transfer this
subordination principle from Brownian motions to these Bessel processes and
obtain some kinds of Cauchy–Bessel processes on $C_{N}$. This construction in
particular leads to Lebesgue densities of the form

$C(k,N)\cdot\frac{1}{(1+\|y\|^{2})^{\gamma_{k}+(N+1)/2}}w_{k}(y)$ (1.1)

with some constant $\gamma_{k}\geq 0$, a norming constant $C(k,N)>0$, and some
weight functions $w_{k}$ where $k$ is some positive, possibly multivariate
multiplicity constant; see [28]. For the most relevant root systems of types
$A_{N-1}$ and $B_{N}$, these weights are given by

$w_{k}^{A}(x):=\prod_{i,j\colon i<j}(x_{i}-x_{j})^{2k},\qquad
w_{k}^{B}(x):=\prod_{i,j\colon
i<j}\big{(}x_{i}^{2}-x_{j}^{2}\big{)}^{2k_{2}}\prod_{i=1}^{N}x_{i}^{2k_{1}}$

with $k,k_{1},k_{2}\geq 0$ respectively. Due to the analogous construction and
shape to the classical setting, we call the distributions with the densities
(1.1) Cauchy–Bessel distributions of types A or B respectively.

We shall prove explicit central limit theorems (CLTs) for these distributions
for fixed dimensions $N$ when the parameters $k$ or $(k_{1},k_{2})$ tend to
infinity. The limit distributions here are non-Gaussian and live on certain
halfspaces in $\mathbb{R}^{N}$ where the limit distributions are composed in
some way of a $(N-1)$-dimensional normal distribution and some distribution on
$[0,\infty[$ which is related to inverse Gaussian distributions. For the
details for the types A or B we refer to Theorems 3.8 and 4.4 below. We point
out that the identification of the $(N-1)$-dimensional subspaces as well as of
the covariance matrices of the $(N-1)$-dimensional normal distributions are
expressed in terms of the ordered zeroes of the classical Hermite polynomal
$H_{N}$ and some Laguerre polynomial $L_{N}^{(\alpha)}$ respectively. We shall
present two different proofs for the central limit Theorems 3.8 and 4.4, where
both are closely related to the corresponding CLTs for the Bessel processes of
types A and B as well for $\beta$-Hermite and Laguerre ensembles in [2, 3, 4,
6, 7, 19, 20, 21, 35, 36]. The first approach, which is carried out for the
central limit Theorem 3.8, consists in some way of a copy of the corresponding
proof of the CLT for $\beta$-Hermite ensembles in [35] and will be based on
the explicit densities (1.1). The second approach, which is carried out for
the central limit Theorem 3.8, and which also works for $\beta$-Hermite
ensembles, uses the construction of the Cauchy–Bessel processes via
subordination and the known CLTs for Bessel processes from [35]. From a
structural point of view, this second proof seems to be more natural; however,
the complexity of both proofs is about the same.

The Bessel processes of types A and B describe Calogero–Moser–Sutherland
particle systems where the parameters $k$ or $(k_{1},k_{2})$ correspond to
inverse temperatures; see, e.g., [33]. Therefore our limits correspond to
freezing limits. Clearly, this interpretation is also available for the
Cauchy–Bessel processes and distributions in this paper.

This paper is organized as follows. Section 2 contains some background
information on Bessel and Cauchy–Bessel processes associated with root systems
from [8, 14, 26, 27, 28, 29, 33]. Sections 3 and 4 then are devoted to the
limit results for the root systems of types A and B respectively. We point out
that besides the central limit Theorems 3.8 and 4.4 we also present a further
asymptotic result in Theorem 3.3 where another norming of the given
Cauchy–Bessel distributions of type A is used and no weak convergence is
available. Furthermore, we briefly study the root systems of type D in Section
5; this will be applied to some singular case for the root systems of type B
there.

We finally point out that the Cauchy–Bessel ensembles in this paper are
different from the Hua–Pickrell ensembles, which are studied, e.g., in [9, 10,
11, 13, 22, 24, 25], and which are also called Cauchy ensembles in some
papers. However, we expect that these Hua–Pickrell ensembles can be partially
handled in a similar way as the Cauchy–Bessel ensembles in Section 3 of this
paper.

##  2 Cauchy–Bessel processes

Bessel processes associated with root systems can be used to describe several
integrable interacting particle systems of Calogero–Moser–Sutherland type on
the real line $\mathbb{R}$ or $[0,\infty[$ with $N$ particles; see for
instance [8, 14, 26, 27, 28, 29, 33] and references there for the background
in analysis, probability, and mathematical physics. We here mainly restrict
our attention to the two most relevant classes, namely the root systems
$A_{N-1}$ and $B_{N}$. The root systems $D_{N}$ will be discussed briefly in
Section 5.

In the cases $A_{N-1}$ and $B_{N}$, these processes are time-homogeneous
diffusion processes $(X_{t,k})_{t\geq 0}$ living on the closed Weyl chambers

$C_{N}^{A}:=\big{\\{}x\in\mathbb{R}^{N}\colon x_{1}\geq x_{2}\geq\dots\geq
x_{N}\big{\\}},\qquad C_{N}^{B}:=\big{\\{}x\in\mathbb{R}^{N}\colon x_{1}\geq
x_{2}\geq\dots\geq x_{N}\geq 0\big{\\}}$

of types A and B. Here, $k$ is a parameter with $k\subset[0,\infty[$ and
$k=(k_{1},k_{2})\subset[0,\infty[^{2}$ for the root systems of types A and B
respectively. The generators of the transition semigroups are given by

$\displaystyle L_{A}f:=\frac{1}{2}\Delta
f+k\sum_{i=1}^{N}\Bigg{(}\sum_{j\colon j\neq
i}\frac{1}{x_{i}-x_{j}}\Bigg{)}\frac{\partial}{\partial
x_{i}}f\qquad\text{and}$ (2.1) $\displaystyle L_{B}f:=\frac{1}{2}\Delta
f+k_{2}\sum_{i=1}^{N}\sum_{j\colon j\neq
i}\left(\frac{1}{x_{i}-x_{j}}+\frac{1}{x_{i}+x_{j}}\right)\frac{\partial}{\partial
x_{i}}f+k_{1}\sum_{i=1}^{N}\frac{1}{x_{i}}\frac{\partial}{\partial x_{i}}f,$

where in both cases reflecting boundaries are assumed, i.e., the generators
are applied to $C^{2}$-functions which are invariant under the corresponding
Weyl groups.

In both cases, the transition probabilities of the Bessel processes are given
as follows; see [26, 27, 28, 29]. For $t>0$, $x\in C_{N}$, $A\subset C_{N}$ a
Borel set,

$K_{t}(x,A)=c_{k}\int_{A}\frac{1}{t^{\gamma_{k}+N/2}}{\rm
e}^{-(\|x\|^{2}+\|y\|^{2})/(2t)}J_{k}\left(\frac{x}{\sqrt{t}},\frac{y}{\sqrt{t}}\right)w_{k}(y)\,{\rm
d}y$ (2.2)

with the weights

$w_{k}^{A}(x):=\prod_{i,j\colon i<j}(x_{i}-x_{j})^{2k},\qquad
w_{k}^{B}(x):=\prod_{i,j\colon
i<j}\big{(}x_{i}^{2}-x_{j}^{2}\big{)}^{2k_{2}}\prod_{i=1}^{N}x_{i}^{2k_{1}},$

the exponents

$\gamma_{k}^{A}(k)=kN(N-1)/2,\qquad\gamma_{k}^{B}(k_{1},k_{2})=k_{2}N(N-1)+k_{1}N,$
(2.3)

and the Selberg norming constants

$c_{k}^{A}:=\bigg{(}\int_{C_{N}^{A}}{\rm e}^{-\|y\|^{2}/2}w_{k}^{A}(y)\,{\rm
d}y\bigg{)}^{-1}=\frac{N!}{(2\pi)^{N/2}}\prod_{j=1}^{N}\frac{\Gamma(1+k)}{\Gamma(1+jk)}$
(2.4)

and

$\displaystyle c_{k}^{B}:={}$ $\displaystyle\bigg{(}\int_{C_{N}^{B}}{\rm
e}^{-\|y\|^{2}/2}w_{k}^{B}(y)\,{\rm d}y\bigg{)}^{-1}$ $\displaystyle={}$
$\displaystyle\frac{N!}{2^{N(k_{1}+(N-1)k_{2}-1/2)}}\prod_{j=1}^{N}\frac{\Gamma(1+k_{2})}{\Gamma(1+jk_{2})\Gamma\big{(}\frac{1}{2}+k_{1}+(j-1)k_{2}\big{)}},$
(2.5)

respectively. Notice that $w_{k}$ is homogeneous of degree $2\gamma_{k}$.
Furthermore, $J_{k}$ is a multivariate Bessel function of type $A_{N-1}$ or
$B_{N}$ with multiplicities $k$ or $(k_{1},k_{2})$ respectively; see, e.g.,
[26, 27, 28, 29]. We do not need much information about $J_{k}$. We only
notice that $J_{k}$ is analytic on $\mathbb{C}^{N}\times\mathbb{C}^{N}$ with
$J_{k}(x,y)>0$ for $x,y\in\mathbb{R}^{N}$. Moreover, $J_{k}(x,y)=J_{k}(y,x)$
and $J_{k}(0,y)=1$ for $x,y\in\mathbb{C}^{N}$. In particular, for the starting
point $x=0\in C_{N}$, (2.2) leads to the distributions

$c_{k}\frac{1}{t^{\gamma_{k}+N/2}}{\rm e}^{-\|y\|^{2}/(2t)}w_{k}(y)\,{\rm
d}y,$ (2.6)

which are just the distributions of the $\beta$-Hermite and Laguerre ensembles
from random matrix theory; see, e.g., [1, 16, 18]. In the last decade, several
freezing limit theorems were derived for the distributions in (2.6) and also
in (2.2) for general starting points for fixed $N$ and $k\to\infty$; see
Dumitriu and Edelman [18] for an approach via their tridiagonal random matrix
models for $x=0$ and [2, 3, 4, 6, 7, 21, 35, 36] for further limit results in
this context.

In this paper we transfer some of these limit results to Cauchy-type
distributions. To motivate these distributions we recapitulate the
subordination procedure which leads from the Bessel processes
$(X_{t,k})_{t\geq 0}$ above and the classical convolution semigroup of inverse
Gaussian distributions to Cauchy-type processes on the Weyl chambers from
[28]. For this we consider the classical convolution semigroup
$(\mu_{t})_{t\geq 0}$ of inverse Gaussian measures on $(\mathbb{R},+)$ with
$\mu_{0}=\delta_{0}$ and, for $t>0$,

${\rm d}\mu_{t}(s)=\frac{{\bf
1}_{]0,\infty[}(s)}{\sqrt{4\pi}}ts^{-3/2}\exp\big{(}{-}t^{2}/(4s)\big{)}\,{\rm
d}s,$ (2.7)

see, e.g., [12, Section 9]. Moreover, let $(T_{t})_{t\geq 0}$ be an associated
Lévy process starting in $0$ with càdlàg paths. Then the process
$(Y_{t})_{t\geq 0}$ with $Y_{t}:=X_{T_{t}}$ for $t\geq 0$ is a Feller process
on $C_{N}$ whose transition probabilities are given by

$Q_{t}(x,A)=\int_{0}^{\infty}K_{s}(x,A)\,{\rm d}\mu_{t}(s),\qquad t\geq
0,\quad x\in C_{N},\quad A\subset C_{N}.$ (2.8)

As this construction is analogous to the classical construction of Cauchy
processes from Brownian motions, we call the processes $(Y_{t})_{t\geq 0}$
Cauchy–Bessel processes of type A or B, respectively. It seems to be difficult
to compute the densities of these distributions explicitly for general
starting points $x\in C_{N}$ like in (2.2) in terms of Bessel functions. On
the other hand, for $x=0$ and $t>0$ one obtains that the probability measures
$Q_{t}(0,\cdot)$ have the explicit Lebesgue densities

$\frac{c_{k}t\Gamma(\gamma_{k}+(N+1)/2)}{\sqrt{4\pi}}\left(\frac{4}{t^{2}+2\|y\|^{2}}\right)^{\gamma_{k}+(N+1)/2}w_{k}(y)$
(2.9)

with $c_{k}$, $\gamma_{k}$, $w_{k}$ as above depending on $k$ and the root
system by some elementary calculus; see also [28, Section 5] with a slightly
different $t$-scaling. In the next sections we study limits of these
distributions for $k\to\infty$. Due to the homogeneity property of these
Cauchy–Bessel distributions w.r.t. the scaling parameter $t$, we there
restrict our attention to the case $t=\sqrt{2}$ w.l.o.g.

We point out that for the root system $A_{N-1}$ and $k=1/2,1,2$, the densities
(2.6) admit the well-known interpretation as the distributions of the ordered
eigenvalues of Gaussian orthogonal, unitary, and symplectic ensembles (GOE,
GUE, GSE) respectively. Therefore, the subordination above leading to the
densities (2.9) corresponds to an analogous subordination of normal
distributions on the vector spaces associated with GOE, GUE, GSE, and the
corresponding time normalizations. Therefore, the densities (2.9) belong in
these cases to Cauchy–Bessel distributions on these vector spaces where the
entries of these matrices are no longer independent. A similar interpretation
exists for the root systems $B_{N}$ via subordinations of Laguerre ensembles.

##  3 Limit theorems for the root system $\boldsymbol{A_{N-1}}$

In this section we study the Cauchy–Bessel distributions with the densities
(2.9) of type $A$ with parameters $t=\sqrt{2}$ and $k\geq 0$. Taking the
constants in (2.3), (2.4) into account, we thus study the distributions with
the density

$f_{k}(y):=C(k,N)\frac{1}{(1+\|y\|^{2})^{kN(N-1)/2+(N+1)/2}}\prod_{i,j\colon
i<j}(y_{i}-y_{j})^{2k}$ (3.1)

on $C_{N}^{A}$ with the norming constant

$C(k,N)=\frac{2^{kN(N-1)/2}N!\Gamma(kN(N-1)/2+(N+1)/2)}{\pi^{(N+1)/2}}\prod_{j=1}^{N}\frac{\Gamma(1+k)}{\Gamma(1+jk)}.$
(3.2)

We first determine the maxima of $f_{k}$. In fact, as $f_{k}$ is equal to 0 on
the boundary $\partial C_{N}^{A}$ with $\lim_{\|y\|\to\infty}f_{k}(y)=0$,
$f_{k}$ has at least one maximum, and all maxima are in the interior of
$C_{N}^{A}$. To determine these maxima, we need the classical Hermite
polynomials $(H_{N})_{N\geq 0}$ which are orthogonal w.r.t. the density ${\rm
e}^{-x^{2}}$ on $\mathbb{R}$. We normalize the $H_{N}$ as usual as, e.g., in
[32] with the three-term-recurrence

$H_{0}=1,\qquad H_{1}(x)=x,\qquad H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x),\qquad
n\geq 1.$

Consider the vector

${\bf z}=(z_{1},\dots,z_{N})\in C_{N}^{A}$

whose entries are the ordered zeros of $H_{N}$. We need the following known
facts:

######  Lemma 3.1.

For $N\geq 2$ and $y\in C_{N}^{A}$, the following statements are equivalent:

  1. (1)

The function $\sum_{i,j\colon i<j}\ln(x_{i}-x_{j})-\|x\|^{2}/2$ is maximal at
$y\in C_{N}^{A}$;

  2. (2)

For $i=1,\dots,N$: $y_{i}=\sum_{j\colon j\neq i}\frac{1}{y_{i}-y_{j}}$;

  3. (3)

$y={\bf z}$.

Furthermore,

$\sum_{i=1}^{N}z_{i}^{2}=\frac{N(N-1)}{2}$ (3.3)

and

$2\sum_{i<j}\ln(z_{i}-z_{j})=-\frac{N(N-1)}{2}\ln 2+\sum_{j=1}^{N}j\ln j.$
(3.4)

###### Proof.

For the equivalence of (1)–(3) see [32, Section 6.7]; see also [3, 35]. For
(3.3) and (3.4) we refer to [3, Appendix D]; see in particular (D.22) and
(D.30) there. ∎

Now let $x\in C_{N}^{A}$ be a maximum of $f_{k}$. This implies that $\nabla
l(x)=0$ for

$l(y):=\ln\left(\frac{1}{(1+\|y\|^{2})^{kN(N-1)/2+(N+1)/2}}\prod_{i<j}(y_{i}-y_{j})^{2k}\right).$

Therefore, for $i=1,\dots,N$,

$\sum_{j\colon j\neq
i}\frac{1}{x_{i}-x_{j}}=\frac{kN(N-1)+N+1}{2k}\frac{1}{1+\|x\|^{2}}x_{i},$

i.e., $c_{1}x_{i}=\sum_{j\colon j\neq i}\frac{1}{x_{i}-x_{j}}$ for
$i=1,\dots,N$ with some constant $c_{1}>0$. A short computation now shows that
for some constant $c_{2}>0$, the vector $y:=c_{2}x$ satisfies the condition in
Lemma 3.1(2) and hence, by Lemma 3.1, ${\bf z}=c_{2}x$. In summary, with (3.3)
and a short computation, we obtain:

######  Lemma 3.2.

The density $f_{k}$ has a unique maximum on $C_{N}^{A}$. This maximum is
located in $\sqrt{\frac{2k}{N+1}}{\bf z}$.

This elementary observation together with the following known CLT for the
densities (2.6) of Bessel processes will be the motivation to study the
densities $f_{k}$ around these maxima for $k\to\infty$.

######  Theorem 3.3.

Let $X_{{\rm Bessel},k,N}$ be random variables with the densities (2.6) for
the root system $A_{N-1}$ for $N\geq 2$. Then the random variables $X_{{\rm
Bessel},k,N}-\sqrt{2k}{\bf z}$ converge in distribution for $k\to\infty$ to
the $N$-dimensional centered normal distribution ${\mathcal{N}}(0,\Sigma)$
where the covariance matrix $\Sigma$ is regular and has the following
properties:

  1. $(1)$

$\Sigma^{-1}=(s_{i,j})_{i,j=1,\dots,N}$ satisfies

$s_{i,j}:=\begin{cases}\displaystyle 1+\sum_{l\neq
i}(z_{i,N}-z_{l,N})^{-2}&\text{for}\ i=j,\\\
-(z_{i,N}-z_{j,N})^{-2}&\text{for}\ i\neq j.\end{cases}$ (3.5)

  2. $(2)$

$\Sigma^{-1}$ has the eigenvalues $1,2,3,4,\dots,N$, and consequently
$\det\Sigma^{-1}=N!$.

  3. $(3)$

$\Sigma=(\sigma_{i,j})_{i,j=1,\dots,N}$ satisfies

$\sigma_{i,j}=(-1)^{i+j}\frac{\sum_{k=0}^{N-1}\frac{H_{k}(z_{i,N})H_{k}(z_{j,N})}{2^{k}k!(N-k)}}{\sqrt{\sum_{k=0}^{N-1}\frac{(H_{k}(z_{i,N}))^{2}}{2^{k}k!}\sum_{l=0}^{N-1}\frac{(H_{l}(z_{j,N}))^{2}}{2^{l}l!}}}.$
(3.6)

Theorem 3.3 without parts (1) and (2) was obtained first by Dumitriu and
Edelman [19] via their tridiagonal random matrices in [18] with different
formulas for the entries of $\Sigma$. It was then reproved in a direct way in
[35] with the entries of $\Sigma^{-1}$ in (1). Furthermore, in [6] the
eigenvalues and eigenvectors of $\Sigma^{-1}$ were determined, and in [2] the
theory of dual orthogonal polynomials in the sense of De Boor and Saff (see
[15, 23, 34]) was used to obtain part (3). This CLT with part (3) was also
obtained in a different way by Gorin and Kleptsyn [21]. We notice that it
seems to be difficult to verify that the formulas in [19] and those in part
(3) are identical, as both formulas contain complicated expressions regarding
the zeros of $H_{N}$.

We now return to our Cauchy–Bessel distributions and try to copy the proof of
the central limit Theorem 3.3 from [35]. It turns out that here a centering
with the maxima from Lemma 3.2 does not lead to a full central limit theorem,
but to the following weaker asymptotic limit result only:

######  Theorem 3.4.

For $k>0$ and $N\geq 2$ let $X_{k}$ be a $C_{N}^{A}$-valued random variable
with density $f_{k}$. Moreover, let $\tilde{f}_{k}$ be the density of
$X_{k}-\sqrt{\frac{2k}{N+1}}{\bf z}$. Then there is a unique centered normal
distribution ${\mathcal{N}}(0,\Sigma_{\rm Cauchy})$ on $\mathbb{R}^{N}$ with
some regular covariance matrix $\Sigma_{\rm Cauchy}$ and density $f$ such that

$\lim_{k\to\infty}\frac{\tilde{f}_{k}(x)}{f(x)}k^{1/2}(N+1)^{N/2}\sqrt{N(N-1)}{\rm
e}^{(N+1)/2}=1$ (3.7)

holds locally uniformly for $x\in\mathbb{R}^{N}$. The matrix $\Sigma_{\rm
Cauchy}$ has the following properties:

  1. $(1)$

$\Sigma_{\rm Cauchy}^{-1}=(s_{i,j})_{i,j=1,\dots,N}$ satisfies

$s_{i,j}:=(N+1)\cdot\begin{cases}\displaystyle 1+\sum_{l\colon l\neq
i}(z_{i}-z_{l})^{-2}+\frac{4z_{i}^{2}}{N(N-1)}&\text{for}\ i=j,\\\
\displaystyle-(z_{i}-z_{j})^{-2}+\frac{4z_{i}z_{j}}{N(N-1)}&\text{for}\ i\neq
j.\end{cases}$ (3.8)

  2. $(2)$

$(N+1)^{-1}\Sigma_{\rm Cauchy}^{-1}$ has the eigenvalues $1,4,3,4,5,\dots,N$,
i.e., $\det\Sigma_{\rm Cauchy}^{-1}=2(N+1)^{N}N!$.

  3. $(3)$

$\Sigma_{\rm Cauchy}=(\sigma_{i,j})_{i,j=1,\dots,N}$ satisfies

$\sigma_{i,j}=\frac{(-1)^{i+j}}{N+1}\left(\frac{\sum_{k=0}^{N-1}\frac{H_{k}(z_{i,N})H_{k}(z_{j,N})}{2^{k}k!(N-k)}}{\sqrt{\sum_{k=0}^{N-1}\frac{(H_{k}(z_{i,N}))^{2}}{2^{k}k!}\sum_{l=0}^{N-1}\frac{(H_{l}(z_{j,N}))^{2}}{2^{l}l!}}}-\frac{z_{i}z_{j}}{2N(N-1)}\right).$

The proof of Theorem 3.4 is divided into two steps. In a first step we show
that (3.7) holds with $\Sigma_{\rm Cauchy}^{-1}$ as part (1) up to a positive
multiplicative constant in the limit. In the second step of the proof we then
use Theorem 3.3 and show that parts (2) and (3) hold, and that the constant in
(3.7) is the correct one.

###### Proof.

Equation (3.1) shows that the random variable $X_{k}-\sqrt{\frac{2k}{N+1}}{\bf
z}$ has a density which can be written as

$\tilde{f}_{k}(y)=f_{k}\left(y+\sqrt{\frac{2k}{N+1}}{\bf
z}\right)=\tilde{c}_{k}{\rm e}^{h_{k}(y)}$

with the exponent

$\displaystyle h_{k}(y):={}$ $\displaystyle 2k\sum_{i,j\colon
i<j}\ln\left(1+\frac{(y_{i}-y_{j})\sqrt{N+1}}{\sqrt{2k}(z_{i}-z_{j})}\right)$
$\displaystyle{}-\frac{kN(N-1)+N+1}{2}\ln\left(1+\frac{(1+\|y\|^{2})(N+1)}{2k\|{\bf
z}\|^{2}}+2\sqrt{\frac{N+1}{2k}}\frac{\langle y,{\bf z}\rangle}{\|{\bf
z}\|^{2}}\right)$ (3.9)

and the constant

$\displaystyle\tilde{c}_{k}:={}$ $\displaystyle
C(k,N)\exp\bigg{(}2k\sum_{i,j\colon
i<j}\ln\bigg{(}\sqrt{\frac{2k}{N+1}}(z_{i}-z_{j})\bigg{)}\bigg{)}$
$\displaystyle{}\times\exp\left({-}\frac{kN(N-1)+N+1}{2}\ln\left(\frac{2k}{N+1}\|{\bf
z}\|^{2}\right)\right)$ $\displaystyle={}$ $\displaystyle
C(k,N)\left(\frac{2k}{N+1}\right)^{kN(N-1)/2}\exp\bigg{(}2k\sum_{i,j\colon
i<j}\ln(z_{i}-z_{j})\bigg{)}$ $\displaystyle{}\times\left(\frac{2k}{N+1}\|{\bf
z}\|^{2}\right)^{-(kN(N-1)+N+1)/2}$ (3.10)

on the shifted cone $C_{N}^{A}-\sqrt{2k}{\bf z}$ with $\tilde{f}_{k}(y)=0$
otherwise on $\mathbb{R}^{N}$.

We now study the exponent $h_{k}(y)$. The power series of $\ln(1+x)$ shows
that for $k\to\infty$,

$\ln\bigg{(}1+\frac{\sqrt{N+1}(y_{i}-y_{j})}{\sqrt{2k}(z_{i}-z_{j})}\bigg{)}=\frac{\sqrt{N+1}(y_{i}-y_{j})}{\sqrt{2k}(z_{i}-z_{j})}-\frac{(N+1)(y_{i}-y_{j})^{2}}{4k(z_{i}-z_{j})^{2}}+O\big{(}k^{-3/2}\big{)}$
(3.11)

and

$\displaystyle\ln\bigg{(}1+\frac{(1+\|y\|^{2})(N+1)}{2k\|{\bf
z}\|^{2}}+2\sqrt{\frac{N+1}{2k}}\frac{\langle y,{\bf z}\rangle}{\|{\bf
z}\|^{2}}\bigg{)}$ $\displaystyle\qquad{}=2\sqrt{\frac{N+1}{2k}}\frac{\langle
y,{\bf z}\rangle}{\|{\bf z}\|^{2}}+\frac{(1+\|y\|^{2})(N+1)}{2k\|{\bf
z}\|^{2}}-\frac{N+1}{k}\frac{\langle y,{\bf z}\rangle^{2}}{\|{\bf
z}\|^{4}}+O\big{(}k^{-3/2}\big{)}.$ (3.12)

Moreover, by Lemma 3.1(2),

$\sum_{i,j\colon i<j}\frac{y_{i}-y_{j}}{z_{i}-z_{j}}-\langle y,{\bf
z}\rangle=\sum_{i=1}^{N}y_{i}\bigg{(}\sum_{j\colon j\neq
i}\frac{1}{z_{i}-z_{j}}-z_{i}\bigg{)}=0.$ (3.13)

Therefore, by (3.9), (3.11)–(3.13), and (3.3),

$h_{k}(y)=-\frac{N+1}{2}\biggl{(}\sum_{i,j\colon
i<j}\frac{(y_{i}-y_{j})^{2}}{(z_{i}-z_{j})^{2}}+\big{(}1+\|y\|^{2}\big{)}+\frac{4}{N(N-1)}\langle
y,{\bf z}\rangle^{2}\biggr{)}+O\big{(}k^{-1/2}\big{)}.$

Therefore,

${\rm e}^{h_{k}(y)}\sim{\rm e}^{-(N+1)/2}\exp\bigl{(}{-}y^{\rm T}\Sigma_{\rm
Cauchy}^{-1}y/2\bigr{)}$ (3.14)

with the matrix $\Sigma_{\rm Cauchy}^{-1}$ defined in (3.8).

We next turn to the constants $\tilde{c}_{k}$ in (3.10). Here (3.4) and (3.3)
imply that

$\displaystyle\tilde{c}_{k}={}$ $\displaystyle
C(k,N)\left(\frac{2k}{N+1}\right)^{kN(N-1)/2}\exp\bigg{(}k\bigg{(}{-}\frac{N(N-1)}{2}\ln
2+\sum_{j=1}^{N}j\ln j\bigg{)}\bigg{)}$
$\displaystyle{}\times\left(\frac{N+1}{kN(N-1)}\right)^{kN(N-1)/2+(N+1)/2}$
$\displaystyle={}$ $\displaystyle
C(k,N)\left(\frac{N+1}{k}\right)^{(N+1)/2}\left(\frac{1}{N(N-1)}\right)^{kN(N-1)/2+(N+1)/2}\prod_{j=1}^{N}j^{kj}.$

If we use (3.2) and Stirling’s formula $\Gamma(k+1)\sim\sqrt{2\pi k}(k/{\rm
e})^{k}$ for $k\to\infty$, an elementary, but tedious calculation leads to

$\tilde{c}_{k}\sim
k^{-1/2}\frac{\sqrt{2}\sqrt{N!}}{(2\pi)^{N/2}\sqrt{N(N-1)}},\qquad
k\to\infty,$ (3.15)

and thus with (3.14) to

$\tilde{f}_{k}(y)\sim
k^{-1/2}\frac{\sqrt{2}\sqrt{N!}}{(2\pi)^{N/2}\sqrt{N(N-1)}{\rm
e}^{(N+1)/2}}\exp\bigl{(}{-}y^{\rm T}\Sigma_{\rm Cauchy}^{-1}y/2\bigr{)}.$
(3.16)

An inspection of the preceding computations shows that (3.16) holds locally
uniformly for $y\in\mathbb{R}^{N}$. On the other hand,
${\mathcal{N}}(0,\Sigma_{\rm Cauchy})$ has the density

$f_{(}y)=\frac{1}{(2\pi)^{N/2}\sqrt{\det\Sigma_{\rm
Cauchy}}}\exp\bigl{(}{-}y^{\rm T}\Sigma_{\rm Cauchy}^{-1}y/2\bigr{)}.$ (3.17)

In order to determine $\det\Sigma_{\rm Cauchy}$, we compare (3.5) and (3.8)
and use (3.3). We obtain that

$\Sigma_{\rm Cauchy}^{-1}=(N+1)\left(\Sigma^{-1}+\frac{2}{\|{\bf z}\|^{2}}{\bf
z}{\bf z}^{\rm T}\right),$ (3.18)

where, by Theorem 3.3, $\Sigma^{-1}$ has the eigenvalues $1,2,3,\dots,N$.
Moreover, by [6], ${\bf z}$ is an eigenvector of $\Sigma^{-1}$ for the
eigenvalue 2. As the eigenvectors of a symmetric matrix are orthogonal, we
conclude that $(N+1)^{-1}\Sigma_{\rm Cauchy}^{-1}$ has the eigenvalues
$1,2+2=4,3,4,5,\dots,N$ where the eigenvectors are the same as for
$\Sigma^{-1}$. This proves part (2) of Theorem 3.4 and yields that

$\det\Sigma_{\rm Cauchy}=\frac{1}{2(N+1)^{N}N!}.$

This, (3.16), and (3.17) now lead to (3.7).

We finally turn to part (3) of Theorem 3.4. Using (3.18) and the fact that
$\Sigma_{\rm Cauchy}^{-1}$ and $\Sigma^{-1}$ have the same orthogonal
transformation matrices $T$, we write these matrices as

$\Sigma_{\rm Cauchy}^{-1}=(N+1)T^{\rm
T}\operatorname{diag}(1,4,3,4,\dots,N)T,\qquad\Sigma^{-1}=T^{\rm
T}\operatorname{diag}(1,2,3,4,\dots,N)T.$

Thus

$\displaystyle\Sigma_{\rm Cauchy}$ $\displaystyle=\frac{1}{N+1}T^{\rm
T}\operatorname{diag}(1,1/4,1/3,1/4,\dots,1/N)T$
$\displaystyle=\frac{1}{N+1}\big{(}\Sigma-T^{\rm
T}\operatorname{diag}(0,1/4,0,0,\dots,0)T\big{)}=\frac{1}{N+1}\left(\Sigma-\frac{1}{4\|{\bf
z}\|^{2}}{\bf z}{\bf z}^{\rm T}\right).$

This and (3.6) now lead to part (3). ∎

Theorem 3.4 shows that we need a stronger scaling of our Cauchy–Bessel
distributions (3.1) than in this theorem in order to obtain a weak limit
result with a probability measure as limit. We now study some suitable scaling
where we use different scales on two complementary subspaces of
$\mathbb{R}^{N}$. To understand the idea, we first consider the case $N=2$.

######  Example 3.5.

For $N=2$ we consider the densities $f_{k}(y)$ from (3.1) with the new
orthogonal coordinates

$x_{1}:=(y_{1}+y_{2})/\sqrt{2},\qquad x_{2}:=(y_{1}-y_{2})/\sqrt{2},$

i.e., $x_{1}\in\mathbb{R}$ describes the center of gravity and $x_{2}>0$ the
distance between the two particles up to the precise scaling. By a short
computation, in the new rotated coordinates, we then have the densities

$\frac{2^{2k+1}\Gamma(k+3/2)\Gamma(k+1)}{\pi^{3/2}\Gamma(2k+1)}\frac{x_{2}^{2k}}{\big{(}1+x_{1}^{2}+x_{2}^{2}\big{)}^{k+3/2}}$

for $x_{2}>0$ with the value $0$ otherwise. If we rescale the distance
coordinate by $1/\sqrt{k}$, i.e., if we define a new coordinate
$\tilde{x}_{2}:=x_{2}/\sqrt{k}$, we obtain a density which we write as

$\tilde{f}_{k}(x_{1},\tilde{x}_{2}):=\frac{2^{2k+1}\Gamma(k+3/2)\Gamma(k+1)}{k\pi^{3/2}\Gamma(2k+1)}\left(1-\frac{1}{k}\frac{1+x_{1}^{2}}{\tilde{x}_{2}^{2}+\frac{1+x_{1}^{2}}{k}}\right)^{k}\left(\tilde{x}_{2}^{2}+\frac{1+x_{1}^{2}}{k}\right)^{-3/2}$

for $x_{1}\in\mathbb{R}$ and $\tilde{x}_{2}>0$. By Stirling’s formula
$\Gamma(k+1)\sim\sqrt{2\pi k}(k/{\rm e})^{k}$, these densities tend to

$f(x_{1},\tilde{x}_{2}):=\frac{2}{\pi}{\rm
e}^{-(1+x_{1}^{2})/\tilde{x}_{2}^{2}}\frac{1}{\tilde{x}_{2}^{3}}$ (3.19)

for $x_{1}\in\mathbb{R}$ and $\tilde{x}_{2}>0$ for $k\to\infty$. It can be
easily checked that $f$ is in fact the density of a probability measure which
has in the coordinate $\tilde{x}_{2}>0$ the image of the inverse Gaussian
distribution $\mu_{2}$ from (2.7) with parameter $t=2$ under the mapping
$\tilde{x}_{2}\mapsto\tilde{x}_{2}^{2}$ on $[0,\infty[$ as marginal
distribution. If this is shown, it can be derived from (3.19) that in the
coordinate $x_{1}$ a classical one-dimensional Cauchy distribution appears as
marginal distribution.

In particular, the classical Cauchy distribution as marginal distribution for
the center-of-gravity-part is no accident, and appears for all $N\geq 2$. To
explain this, we consider a diffusion process
$(X_{t}:=(X_{t,1},\dots,X_{t,N}))_{t\geq 0}$ on $C_{N}^{A}$ associated with
the generator (2.1) with start in $0$ for $k>0$ and $N\geq 2$. Moreover, for
the vector ${\bf 1}:=(1,\dots,1)\in\mathbb{R}^{N}$ we denote the orthogonal
projections from $\mathbb{R}^{N}$ onto $\mathbb{R}\cdot{\bf 1}$ and its
orthogonal complement ${\bf 1}^{\perp}$ by $p_{{\bf 1}}$ and $p_{{\bf
1}^{\perp}}$ respectively. We now consider the center-of-gravity-process

$\big{(}X_{t}^{\rm cg}:=(X_{t,1}+\dots+X_{t,N})/N\big{)}_{t\geq 0},$

which may be regarded as $(p_{{\bf 1}}(X_{t}))_{t\geq 0}$ by identifying
$x\in\mathbb{R}$ with $x\cdot{\bf 1}$. It can be easily seen from (2.1) (see
for instance [30, Lemma 3.2] or [6, Section 2]) that this process is a usual
one-dimensional Brownian motion (up to some scaling factor) which is
stochastically independent from the orthogonal projection

$\big{(}X_{t}^{\rm diff}:=p_{{\bf 1}^{\perp}}(X_{t})=X_{t}-X_{t}^{\rm
cg}\cdot{\bf 1}\big{)}_{t\geq 0}$

onto ${\bf 1}^{\perp}$. Notice that the $N-1$ coordinates of the diffusion
$\big{(}X_{t}^{\rm diff}\big{)}_{t\geq 0}$ describe the sucessive distances of
the neighbored particles, and that the center-of-gravity-part
$\big{(}X_{t}^{\rm cg}\big{)}_{t\geq 0}$ is independent from $k$. Using our
subordination procedure in (2.7), (2.8), and (2.9) we thus obtain readily that
the center-of-gravity marginal distributions of the Cauchy–Bessel
distributions with densities $f_{k}$ on $\mathbb{R}^{N}$ are a classical
standard Cauchy distribution on $\mathbb{R}$ independent from $k$. In summary
we obtain in this way:

######  Lemma 3.6.

Let $(X_{1},\dots,X_{N})$ be a $C_{N}^{A}$-valued random variable with the
Lebesgue density $f_{k}$ from (3.1) with $N\geq 2$ and $k>0$. Then
$(X_{1}+\dots+X_{N})/\sqrt{N}$ is standard Cauchy distributed on $\mathbb{R}$
with the density $\frac{1}{\pi}\frac{1}{1+x^{2}}$.

Therefore, also in the limit $k\to\infty$, a standard Cauchy distribution on
$\mathbb{R}$ appears for the center-of-gravity part.

Motivated by this result and Example 3.5 for $N=2$, we now turn to some weak
limit theorem for $N\geq 2$. We consider $C_{N}^{A}$-valued random variables
$X_{k}$ with the densities $f_{k}$ as above. Motivated by Example 3.5, we now
use different scalings on two complementary subspaces of $\mathbb{R}^{N}$,
namely the one-dimensional subspace $\mathbb{R}\cdot{\bf z}$ and its
orthogonal complement ${\bf z}^{\perp}$. Let $p_{{\bf
z}}\colon\mathbb{R}^{N}\to\mathbb{R}\cdot{\bf z}$ be the orthogonal projection
onto $\mathbb{R}\cdot{\bf z}$ which satisfies

$p_{{\bf z}}(y)=\frac{\langle y,{\bf z}\rangle}{\|{\bf z}\|^{2}}{\bf
z}=\frac{2\langle y,{\bf z}\rangle}{N(N-1)}{\bf z}.$

Moreover, the orthogonal projection onto ${\bf z}^{\perp}$ is given by

$p_{{\bf z}^{\perp}}(y)=y-\frac{\langle y,{\bf z}\rangle}{\|{\bf z}\|^{2}}{\bf
z}=y-\frac{2\langle y,{\bf z}\rangle}{N(N-1)}{\bf z}.$

We now define the rescaled random variables

$\tilde{X}_{k}:=\phi_{k}(X_{k})$ (3.20)

with the linear mappings

$\phi_{k}\colon\
\mathbb{R}^{N}\to\mathbb{R}^{N},\qquad\phi_{k}(y):=\frac{1}{\sqrt{k}}p_{{\bf
z}}(y)+p_{{\bf z}^{\perp}}(y)=y+\left(\frac{1}{\sqrt{k}}-1\right)p_{{\bf
z}}(y)$

for $k\geq 1$. The random variables $\tilde{X}_{k}$ then have values in the
sets $C_{N,k}^{A}:=\phi_{k}\big{(}C_{N}^{A}\big{)}$. These sets have the
following properties:

######  Lemma 3.7.

The closure of $\bigcup_{k\geq 1}C_{N,k}^{A}$ is the closed half space

$B_{N}:=\big{\\{}y\in\mathbb{R}^{N}\colon\langle y,{\bf z}\rangle\geq
0\big{\\}}.$

Moreover, for $1\leq k_{1}\leq k_{2}$, $C_{N,k_{1}}^{A}\subset
C_{N,k_{2}}^{A}$.

###### Proof.

For $y\in C_{N,k}^{A}$ we have

$\langle\phi_{k}(y),{\bf z}\rangle=\langle y,{\bf
z}\rangle+\left(\frac{1}{\sqrt{k}}-1\right)\langle y,{\bf
z}\rangle=\frac{1}{\sqrt{k}}\langle y,{\bf z}\rangle\geq 0$

and thus $\bigcup_{k\geq 1}C_{N,k}^{A}\subset B_{N}$. For the converse
statement we first consider some $w$ in the interior of $B_{N}$, i.e., with
$\langle w,{\bf z}\rangle>0$. We now choose some $t>0$ sufficiently large with
$y:=w+tz\in C_{N}^{A}$. Notice that this is possible for any vector $w$, as
${\bf z}$ is in the interior of $C_{N}^{A}$. Then we obtain for all $k\geq 1$
that

$\phi_{k}(y)=w+\left(\frac{t}{\sqrt{k}}+\left(\frac{1}{\sqrt{k}}-1\right)\langle
w,{\bf z}\rangle\right){\bf z}.$

Therefore, if we take the unique $k=k(t)\geq 1$ with
$t=\big{(}\sqrt{k}-1\big{)}\frac{\langle w,{\bf z}\rangle}{\|{\bf
z}\|^{2}}>0$, we obtain $\phi_{k}(y)=w$. We thus conclude that the interior of
$B_{N}$ is contained in $\bigcup_{k\geq 1}C_{N,k}^{A}$. This completes the
proof of the first statement of the lemma. The second statement can be checked
in a similar way. ∎

With the first statement of Lemma 3.7 on the ranges of the random variables
$X_{k}$ in mind, we now turn to the following limit theorem.

######  Theorem 3.8.

For $k>0$ and $N\geq 2$ let $X_{k}$ be a $C_{N}^{A}$-valued random variable
with density $f_{k}$. Then the $\mathbb{R}^{N}$-valued rescaled random
variables $\tilde{X}_{k}$ from (3.20) converge in distribution for
$k\to\infty$ to some probability measure $\mu\in
M^{1}\big{(}\mathbb{R}^{N}\big{)}$ with $B_{N}$ as support. $\mu$ has the
Lebesgue density

$\displaystyle f(y):={}$ $\displaystyle\frac{\sqrt{N!}(N(N-1))^{N/2}{\rm
e}^{N(N-1)}}{\pi^{N/2}2^{(N-1)/2}}\exp\Biggl{(}{-}\frac{\|{\bf
z}\|^{2}}{\|p_{{\bf z}}(y)\|^{2}}\Biggl{(}\sum_{i,j\colon
i<j}\frac{(y_{i}-y_{j})^{2}}{(z_{i}-z_{j})^{2}}+\|y\|^{2}\Biggr{)}\Biggr{)}$
$\displaystyle{}\times\exp\left(\frac{-N(N-1)}{2\|p_{{\bf
z}}(y)\|^{2}}\right)\frac{1}{\|p_{{\bf z}}(y)\|^{N+1}}$ (3.21)

for $y$ in the interior of the half space $B_{N}$.

A short calculation shows that for $N=2$, the measure $\mu$ with density
(3.21) is in fact equal to the limit in Example 3.5 where one has to take the
rotation in the coordinates there into account.

The proof of Theorem 3.8 will be decomposed into two parts. In the first part
we show that distributions of the random variables $X_{k}$ converge vaguely to
the measure $\mu$ on $\mathbb{R}^{N}$ with the density $f$ from (3.21) on the
interior of $B_{N}$. In a second step we then check that the density $f$ from
(3.21) is in fact the density of a probability measure, which then implies
weak convergence.

###### First part of the proof of Theorem 3.8.

Let $k\geq 1$. By using our rescaling on the one-dimensional subspace
$\mathbb{R}\cdot{\bf z}$ together with the transformation formula for the
densities of transformed random variables, the random variable $\tilde{X}_{k}$
has the density

$\tilde{f}_{k}(y):=\sqrt{k}f_{k}(\tilde{y})\quad\quad\text{with}\quad\tilde{y}:={\sqrt{k}}p_{{\bf
z}}(y)+p_{{\bf z}^{\perp}}(y)=y+\big{(}\sqrt{k}-1\big{)}p_{{\bf z}}(y)$

on the interior of $C_{N,k}$. Using (3.1) we write this density as

$\displaystyle\tilde{f}_{k}(y)=\sqrt{k}C(k,N)\Biggl{(}\prod_{i,j\colon
i<j}\\!\frac{(\tilde{y}_{i}-\tilde{y}_{j})^{2}}{\|\tilde{y}\|^{2}}\Biggr{)}^{k}\\!\left(\frac{\|\tilde{y}\|^{2}}{1+\|\tilde{y}\|^{2}}\right)^{kN(N-1)/2}\\!\left(\frac{1}{1+\|\tilde{y}\|^{2}}\right)^{(N+1)/2}.\\!\\!\\!$
(3.22)

We next notice that

$\|\tilde{y}\|^{2}=k\|p_{{\bf z}}(y)\|^{2}+\|p_{{\bf
z}^{\perp}}(y)\|^{2}=\|y\|^{2}+(k-1)\|p_{{\bf z}}(y)\|^{2}.$

Hence, for $k\to\infty$,

$\displaystyle\left(\frac{\|\tilde{y}\|^{2}}{1+\|\tilde{y}\|^{2}}\right)^{kN(N-1)/2}$
$\displaystyle=\left(1-\frac{1}{k(\|p_{{\bf z}}(y)\|^{2}+(1+\|p_{{\bf
z}^{\perp}}(y)\|^{2})/k)}\right)^{kN(N-1)/2}$
$\displaystyle\to\exp\left(\frac{-\|{\bf z}\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}\right)$ (3.23)

and

$\left(\frac{1}{1+\|\tilde{y}\|^{2}}\right)^{(N+1)/2}\sim\frac{1}{k^{(N+1)/2}\|p_{{\bf
z}}(y)\|^{N+1}}.$ (3.24)

Furthermore, we write the remaining term in (3.22) as

$\Biggl{(}\prod_{i,j\colon
i<j}\frac{(\tilde{y}_{i}-\tilde{y}_{j})^{2}}{\|\tilde{y}\|^{2}}\Biggr{)}^{k}={\rm
e}^{h_{k}(y)}$ (3.25)

with

$\displaystyle h_{k}(y):={}$ $\displaystyle 2k\sum_{i,j\colon
i<j}\ln(\tilde{y}_{i}-\tilde{y}_{j})-\frac{kN(N-1)}{2}\ln\big{(}\|\tilde{y}\|^{2}\big{)}$
$\displaystyle={}$ $\displaystyle 2k\sum_{i,j\colon
i<j}\ln\left(\frac{(\sqrt{k}-1)\langle y,{\bf z}\rangle}{\|{\bf
z}\|^{2}}(z_{i}-z_{j})+y_{i}-y_{j}\right)$
$\displaystyle{}-\frac{kN(N-1)}{2}\ln\big{(}\|y\|^{2}+(k-1)\|p_{{\bf
z}}(y)\|^{2}\big{)}.$

We now use $\langle y,{\bf z}\rangle>0$ and write $h_{k}(y)$ as

$\displaystyle h_{k}(y)={}$ $\displaystyle 2k\sum_{i,j\colon
i<j}\ln\left(1+\frac{\|{\bf z}\|^{2}(y_{i}-y_{j})}{\sqrt{k}\langle y,{\bf
z}\rangle(z_{i}-z_{j})}-\frac{1}{\sqrt{k}}\right)$
$\displaystyle{}-\frac{kN(N-1)}{2}\ln\left(1+\frac{1}{k}\left(\frac{\|y\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}-1\right)\right)+R_{k}$ (3.26)

with

$\displaystyle R_{k}:={}$ $\displaystyle 2k\sum_{i,j\colon
i<j}\ln\left(\frac{\sqrt{k}\langle y,{\bf z}\rangle(z_{i}-z_{j})}{\|{\bf
z}\|^{2}}\right)-\frac{kN(N-1)}{2}\ln\left(\frac{k\langle y,{\bf
z}\rangle^{2}}{\|{\bf z}\|^{2}}\right)$ $\displaystyle={}$ $\displaystyle
2k\sum_{i,j\colon i<j}\ln\left(\frac{\sqrt{k}(z_{i}-z_{j})}{\|{\bf
z}\|^{2}}\right)-\frac{kN(N-1)}{2}\ln\left(\frac{k}{\|{\bf z}\|^{2}}\right).$

This, (3.4), (3.3), and elementary calculus now lead to

${\rm e}^{R_{k}}=(N(N-1))^{-kN(N-1)/2}\prod_{j=1}^{N}j^{kj}.$ (3.27)

Moreover, the power series of $\ln(1+x)$ for the logarithms in (3.26) shows
for $k\to\infty$ that

$\displaystyle\ln\bigg{(}1+\frac{\|{\bf z}\|^{2}(y_{i}-y_{j})}{\sqrt{k}\langle
y,{\bf z}\rangle(z_{i}-z_{j})}-\frac{1}{\sqrt{k}}\bigg{)}$
$\displaystyle\qquad{}=\frac{1}{\sqrt{k}}\left(\frac{\|{\bf
z}\|^{2}(y_{i}-y_{j})}{\langle y,{\bf
z}\rangle(z_{i}-z_{j})}-1\right)-\frac{1}{2k}\left(\frac{\|{\bf
z}\|^{2}(y_{i}-y_{j})}{\langle y,{\bf
z}\rangle(z_{i}-z_{j})}-1\right)^{2}+O\big{(}k^{-3/2}\big{)}$ (3.28)

and

$\ln\left(1+\frac{1}{k}\left(\frac{\|y\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}-1\right)\right)=\frac{1}{k}\left(\frac{\|y\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}-1\right)+O\big{(}k^{-2}\big{)}.$ (3.29)

We next use (3.13) and (3.3) and observe that

$\sum_{i,j\colon i<j}\left(\frac{\|{\bf z}\|^{2}(y_{i}-y_{j})}{\langle y,{\bf
z}\rangle(z_{i}-z_{j})}-1\right)=0.$ (3.30)

In summary, we conclude from (3.26), (3.28), (3.29), and (3.30) that

$\displaystyle h_{k}(y)={}$ $\displaystyle-\sum_{i,j\colon
i<j}\left(\frac{\|{\bf z}\|^{2}(y_{i}-y_{j})}{\langle y,{\bf
z}\rangle(z_{i}-z_{j})}-1\right)^{2}-\frac{N(N-1)}{2}\left(\frac{\|y\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}-1\right)+R_{k}+O\big{(}k^{-1/2}\big{)}$ $\displaystyle={}$
$\displaystyle-\sum_{i,j\colon i<j}\frac{\|{\bf
z}\|^{4}(y_{i}-y_{j})^{2}}{\langle y,{\bf
z}\rangle^{2}(z_{i}-z_{j})^{2}}-\frac{N(N-1)}{2}+2\sum_{i,j\colon
i<j}\frac{\|{\bf z}\|^{2}(y_{i}-y_{j})}{\langle y,{\bf
z}\rangle(z_{i}-z_{j})}$ $\displaystyle{}-\|{\bf
z}\|^{2}\left(\frac{\|y\|^{2}\|{\bf z}\|^{2}}{\|\langle y,{\bf
z}\rangle^{2}}-1\right)+R_{k}+O\big{(}k^{-1/2}\big{)}$ $\displaystyle={}$
$\displaystyle-\sum_{i,j\colon i<j}\\!\\!\frac{\|{\bf
z}\|^{4}(y_{i}-y_{j})^{2}}{\langle y,{\bf
z}\rangle^{2}(z_{i}-z_{j})^{2}}+\frac{N(N-1)}{2}-\frac{\|{\bf
z}\|^{4}\|y\|^{2}}{\langle y,{\bf z}\rangle^{2}}+\|{\bf
z}\|^{2}+R_{k}+O\big{(}k^{-1/2}\big{)}.\\!\\!\\!\\!\\!\\!\\!$ (3.31)

We next consider the constant $C(k,N)$ from (3.2). Stirling’s formula
$\Gamma(k+1)\sim\sqrt{2\pi k}(k/{\rm e})^{k}$ for $k\to\infty$, and an
elementary, but tedious calculation as in (3.15) leads to

$C(k,N)\sim\frac{\sqrt{N!}k^{N/2}(N(N-1))^{kN(N-1)/2+N/2}}{\pi^{N/2}2^{(N-1)/2}\prod_{j=1}^{N}j^{kj}},\qquad
k\to\infty.$

This, (3.22), (3.23), (3.24), (3.25), (3.27), and (3.31) now show that

$\lim_{k\to\infty}\tilde{f}_{k}(y)=f(y)$

for the density $f$ from (3.21) and for $y$ with $\langle y,{\bf z}\rangle>0$,
i.e., for $y$ in the interior of the half space $B_{N}$. We also observe by
inspection of the preceding arguments that the convergence above holds locally
uniformly in $y$ in the interior of the half space $B_{N}$. We thus conclude
that the distributions of the random variables $\tilde{X}_{k}$ tend vaguely to
the measure $\mu$ with density $f$ in the interior of $B_{N}$. ∎

###### Second part of the proof of Theorem 3.8.

In order to complete the proof of Theorem 3.8 we now check that the density
$f$ with $f(y):=0$ for $y$ on the boundary of $B_{N}$, i.e., for $y$ with
$\langle y,{\bf z}\rangle=0$ is in fact the density of a probability measure.
If this is shown, a standard argument in probability applied to the interior
of $B_{N}$ then implies weak convergence as claimed.

In order to compute $\int_{B_{N}}f(y)\,{\rm d}y$, we first observe that

$\exp\bigg{(}{-}\frac{\|{\bf z}\|^{2}}{\|p_{{\bf
z}}(y)\|^{2}}\bigg{(}\sum_{i,j\colon
i<j}\frac{(y_{i}-y_{j})^{2}}{(z_{i}-z_{j})^{2}}+\|y\|^{2}\bigg{)}\bigg{)}=\exp\bigg{(}{-}\frac{\|{\bf
z}\|^{2}}{\|p_{{\bf z}}(y)\|^{2}}y^{\rm T}\Sigma^{-1}y\bigg{)}$

with the matrix $\Sigma^{-1}=(s_{i,j})_{i,j=1,\dots,N}$ defined in (3.5) where
$\Sigma^{-1}$ has the eigenvalues $1,2,3,4,\allowbreak\dots,N$ by Theorem 3.3.
We also recapitulate that the vectors ${\bf 1}$, ${\bf z}$ are eigenvectors of
$\Sigma^{-1}$ for the eigenvalues 1, 2 respectively. We now choose an
orthonormal basis of $\mathbb{R}^{N}$ consisting of eigenvectors of
$\Sigma^{-1}$ associated with the eigenvalues $1,3,4,5,\dots,N$ and $2$
respectively where we choose the vector ${\bf z}/\|{\bf z}\|$ as “the”
eigenvector associated with the eigenvalue 2. Moreover, for
$y=(y_{1},\dots,y_{N-1})\in\mathbb{R}^{N-1}$ and $t\in\mathbb{R}$ let
$(y,t):=(y_{1},\dots,y_{N-1},t)\in\mathbb{R}^{N}$. With this notation and the
constant

$D(N):=\frac{\sqrt{N!}(N(N-1))^{N/2}{\rm e}^{N(N-1)}}{\pi^{N/2}2^{(N-1)/2}}$

from the density in (3.21), we obtain by elementary calculations with a
orthogonal transformation and with the norming of the inverse Gaussian density
in (2.7) that

$\displaystyle\int_{B_{N}}f(y)\,{\rm d}y={}$ $\displaystyle
D(N)\int_{0}^{\infty}\Biggl{(}\int_{\mathbb{R}^{N-1}}\exp\left(-\frac{\|{\bf
z}\|^{2}}{t^{2}}(y,t)^{\rm
T}\operatorname{diag}(1,3,4,5,\dots,N,2)(y,t)\right){\rm d}y\Biggr{)}$
$\displaystyle{}\times\exp\left(\frac{-N(N-1)}{2t^{2}}\right)\frac{1}{t^{N+1}}\,{\rm
d}t$ $\displaystyle={}$ $\displaystyle D(N){\rm
e}^{-N(N-1)}\int_{0}^{\infty}\Biggl{(}\int_{\mathbb{R}^{N-1}}\exp\left(-\frac{\|{\bf
z}\|^{2}}{t^{2}}y^{\rm T}\operatorname{diag}(1,3,4,5,\dots,N)y\right){\rm
d}y\Biggr{)}$
$\displaystyle{}\times\exp\left(\frac{-N(N-1)}{2t^{2}}\right)\frac{1}{t^{N+1}}\,{\rm
d}t$ $\displaystyle={}$ $\displaystyle D(N){\rm
e}^{-N(N-1)}\frac{(2\pi)^{N-1)/2}\sqrt{2}}{(N(N-1))^{(N-1)/2}\sqrt{N!}}\int_{0}^{\infty}\exp\left(\frac{-N(N-1)}{2t^{2}}\right)\frac{1}{t^{2}}\,{\rm
d}t=1$

as claimed. This completes the proof. ∎

######  Remark 3.9.

  1. (1)

Let $X$ be a random variable with values in the half space $B_{N}$ and density
(3.21) as in Theorem 3.8. Then $(X_{1}+\dots+X_{N})/\sqrt{N}$ is standard
Cauchy distributed on $\mathbb{R}$. This follows immediately from Lemma 3.6
and the fact that the maps $\phi_{k}$ leave $x_{1}+\dots+x_{N}$ invariant for
each $x\in\mathbb{R}^{N}$ because of $\langle{\bf 1},{\bf z}\rangle=0$.

  2. (2)

There is a second, more structural proof of Theorem 3.8 which explains the
limit density (3.21) in terms of subordination; see Remark 4.5(1) below.

##  4 Limit theorems for the root system $\boldsymbol{B_{N}}$

We now study the Cauchy–Bessel distributions with the densities (2.9) of type
$B$ with the parameters $t=\sqrt{2}$ and $k=(k_{1},k_{2})$ with
$k_{1},k_{2}\geq 0$. Following [4, 6, 7, 35] we write $k$ as
$(k_{1},k_{2})=(\nu\beta,\beta)$ where we fix $\nu>0$ and investigate limits
for $\beta\to\infty$. Taking these new parameters and the constants in (2.3)
and (2.5) into account, we thus study the distributions $\tau_{\nu,\beta}$
with the density

$f_{\nu,\beta}(y):=C_{B}(\nu,\beta,N)\frac{1}{\big{(}1+\|y\|^{2}\big{)}^{\beta
N(N+\nu-1)+(N+1)/2}}\prod_{i,j\colon
i<j}\big{(}y_{i}^{2}-y_{j}^{2}\big{)}^{2\beta}\prod_{i=1}^{N}y_{i}^{2\nu\beta}$

on the Weyl chambers $C_{N}^{B}$ with the norming constants

$\displaystyle C_{B}(\nu,\beta,N)=\frac{N!2^{N}\Gamma(\beta
N(N+\nu-1)+(N+1)/2)}{\sqrt{\pi}}$
$\displaystyle\hphantom{C_{B}(\nu,\beta,N)=}{}\times\prod_{j=1}^{N}\frac{\Gamma(1+\beta)}{\Gamma(1+j\beta)\Gamma(\frac{1}{2}+\beta(j+\nu-1))}.$
(4.1)

We now proceed as in Section 3 and use the Laguerre polynomials
$L_{N}^{(\nu-1)}$ instead of the Hermite polynomials. Recapitulate that the
$L_{N}^{(\nu-1)}$ are orthogonal w.r.t. the density ${\rm e}^{-x}x^{\nu-1}$ on
$]0,\infty[$ for $\nu>0$ as defined in [32]. We recapitulate the following
facts about the zeros of $L_{N}^{(\nu-1)}$.

######  Lemma 4.1.

Let $\nu>0$. For $r=(r_{1},\dots,r_{N})\in C_{N}^{B}$, the following
statements are equivalent:

  1. $(1)$

The function

$W_{B}(y):=2\sum_{i<j}\ln\big{(}y_{i}^{2}-y_{j}^{2}\big{)}+2\nu\sum_{i}\ln
y_{i}-\|y\|^{2}/2$

is maximal at $r\in C_{N}^{B}$;

  2. $(2)$

For $i=1,\dots,N$, $\frac{1}{2}r_{i}=\sum_{j\colon j\neq
i}\frac{2r_{i}}{r_{i}^{2}-r_{j}^{2}}+\frac{\nu}{r_{i}}=\sum_{j:j\neq
i}\big{(}\frac{1}{r_{i}-r_{j}}+\frac{1}{r_{i}+r_{j}}\big{)}+\frac{\nu}{r_{i}}$;

  3. $(3)$

If $z_{1}^{(\nu-1)}\geq\dots\geq z_{N}^{(\nu-1)}$ are the ordered zeros of
$L_{N}^{(\nu-1)}$, then

$2\big{(}z_{1}^{(\nu-1)},\dots,z_{N}^{(\nu-1)}\big{)}=\big{(}r_{1}^{2},\dots,r_{N}^{2}\big{)}.$

The vector $r$ of $(1)$–$(3)$ satisfies

$\|r\|^{2}=N(N+\nu-1)$ (4.2)

and

$\displaystyle-\frac{1}{2}\|r\|^{2}+\nu\sum_{j=1}^{N}\ln
r_{j}^{2}+2\sum_{i<j}\ln\big{(}r_{i}^{2}-r_{j}^{2}\big{)}$
$\displaystyle\qquad{}=N(N+\nu-1)(-1+\ln 2)+\sum_{j=1}^{N}j\ln
j+\sum_{j=1}^{N}(\nu+j-1)\ln(\nu+j-1).$ (4.3)

###### Proof.

See [4] or [35]. Parts are also in [32, Section 6.3].∎

This result leads to the following CLT in the Bessel case; see [35, Theorem
3.3].

######  Theorem 4.2.

Let $\nu>0$, $N\geq 1$ an integer, and $(X_{t,\beta})_{t\geq 0}$ a Bessel
process of type $B_{N}$ on $C_{N}^{B}$ starting in $0\in C_{N}^{B}$ with
parameter $k=(\nu\beta,\beta)$. Then, for the vector $r\in C_{N}^{B}$ from
Lemma 4.1,

$\frac{X_{t,\beta}}{\sqrt{t}}-\sqrt{\beta}r$

converges for $\beta\to\infty$ to the centered $N$-dimensional distribution
${\mathcal{N}}(0,\Sigma)$ with the regular covariance matrix $\Sigma$ with
$\Sigma^{-1}=(s_{i,j})_{i,j=1,\dots,N}$ with

$s_{i,j}:=\begin{cases}\displaystyle 1+\frac{2\nu}{r_{i}^{2}}+2\sum_{l\neq
i}(r_{i}-r_{l})^{-2}+2\sum_{l\neq i}(r_{i}+r_{l})^{-2}&\text{for}\ i=j,\\\
2(r_{i}+r_{j})^{-2}-2(r_{i}-r_{j})^{-2}&\text{for}\ i\neq j.\end{cases}$

The matrix $\Sigma^{-1}$ has the eigenvalues $2,4,\dots,2N$.

We now derive an associated weak limit law for the Cauchy–Bessel distributions
$\tau_{\nu,\beta}$ analogous to Theorem 3.8. We here again use different
scalings on two complementary subspaces of $\mathbb{R}^{N}$, namely on
$\mathbb{R}\cdot r$ and its orthogonal complement $r^{\perp}$. Let
$p_{r}\colon\mathbb{R}^{N}\to\mathbb{R}\cdot r$ be the orthogonal projection
onto $\mathbb{R}\cdot r$, and $p_{{r}^{\perp}}$ the orthogonal projection onto
${r}^{\perp}$. Now let $X_{\beta}$ be a $\tau_{\nu,\beta}$-distributed random
variable. We again define the rescaled random variables
$\tilde{X}_{\beta}:=\phi_{\beta}(X_{\beta})$ with the linear mappings

$\phi_{\beta}\colon\
\mathbb{R}^{N}\to\mathbb{R}^{N},\qquad\phi_{\beta}(y):=\frac{1}{\sqrt{\beta}}p_{r}(y)+p_{{r}^{\perp}}(y)=y+\left(\frac{1}{\sqrt{\beta}}-1\right)p_{{r}}(y).$
(4.4)

The random variables $\tilde{X}_{\beta}$ then have values in the sets
$C_{N,\beta}^{B}:=\phi_{\beta}\big{(}C_{N}^{B}\big{)}$. These sets have the
following property analogous to Lemma 3.7.

######  Lemma 4.3.

The closure of $\bigcup_{\beta\geq 1}C_{N,\beta}^{B}$ is the closed half space

$B_{N}:=\big{\\{}y\in\mathbb{R}^{N}\colon\langle y,{r}\rangle\geq 0\big{\\}}.$

Moreover, for $1\leq\beta_{1}\leq\beta_{2}$, $C_{N,\beta_{1}}^{B}\subset
C_{N,\beta_{2}}^{B}$.

The following limit theorem is analogous to Theorem 3.8.

######  Theorem 4.4.

For $\beta>0$ and $N\geq 1$ let $X_{\beta}$ be a $C_{N}^{A}$-valued,
$\tau_{\nu,\beta}$-distributed random variable. Then the rescaled random
variables $\tilde{X}_{\beta}$ converge in distribution for $\beta\to\infty$ to
some probability measure $\mu$ on $\mathbb{R}^{N}$ with the half space $B_{N}$
as support. This measure $\mu$ is given by

$\mu:=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}{\mathcal{N}}\big{(}\sqrt{s}r,sA\Sigma
A\big{)}s^{-3/2}\exp(-1/(2s))\,{\rm d}s$ (4.5)

with the matrix $\Sigma$ from Theorem 4.2 where $A$ is the matrix belonging to
the orthogonal projection $p_{{r}^{\perp}}$ in the standard coordinates on
$\mathbb{R}^{N}$. Moreover, $\mu$ has the Lebesgue density

$\displaystyle
f(y):=D(N)\exp\left(-\frac{N(N+\nu-1)}{2\|p_{{r}}(y)\|^{2}}y^{\rm
T}\Sigma^{-1}y\right)\exp\left(\frac{-N(N+\nu-1)}{2\|p_{{r}}(y)\|^{2}}\right)\frac{1}{\|p_{{r}}(y)\|^{N+1}}$
(4.6)

with

$D(N):=\frac{\sqrt{2}\sqrt{N!}(N(N+\nu-1))^{N/2}{\rm
e}^{N(N+\nu-1)}}{\pi^{N/2}}$

for $y$ in the interior of $B_{N}$.

Please notice that the normal distributions in the mixing formula (4.5) are
singular, and that the existence of the density (4.6) on the half space
$B_{N}$ is a consequence of the integration w.r.t. the mean vectors of the
normal distributions which compensate the singular direction.

Theorem 4.4 with the limit with density (4.6) can be derived in the same way
as Theorem 3.8 by using Lemmas 4.1 and 4.3. We skip this direct approach and
present a second, Fourier-analytic proof which is based on the central limits
Theorem 4.2 and the very construction of the Cauchy–Bessel distributions
$\tau_{\nu,\beta}$ via the subordination (2.8). We point out that this
approach also works for Theorem 3.8.

###### Proof.

Fix $\nu>0$, and consider the inverse Gaussian convolution semigroup
$(\mu_{t})_{t\geq 0}$ on $(\mathbb{R},+)$ as in (2.7). For $s\geq 0$ and
$\beta>0$ let $\rho_{s,\beta}\in M^{1}\big{(}C_{N}^{B}\big{)}$ be the
distributions of a Bessel process $(X_{s,\beta})_{s\geq 0}$ of type $B_{N}$
starting in $0$ with parameter $k=(\nu\beta,\beta)$ as in Theorem 4.2. Hence,
by Theorem 4.2, $X_{1,\beta}-\sqrt{\beta}r$ tends in distribution to
${\mathcal{N}}(0,\Sigma)$ with the covariance matrix $\Sigma$ from Theorem
4.2. Therefore, in terms of the classical convolution $*$ of measures on
$(\mathbb{R}^{n},+)$,

$\rho_{1,\beta}*\delta_{-\sqrt{\beta}r}\longrightarrow{\mathcal{N}}(0,\Sigma),\qquad\beta\to\infty,$

weakly. Hence, using the classical Fourier transform
$\hat{\mu}(w):=\int_{\mathbb{R}^{N}}{\rm e}^{-{\rm i}\langle w,x\rangle}\,{\rm
d}\mu(x)$ of measures $\mu$ on $\mathbb{R}^{N}$ and Levy’s continuity theorem,
we get

${\rm e}^{{\rm i}\sqrt{\beta}\langle
w,r\rangle}\hat{\rho}_{1,\beta}(w)\longrightarrow{\rm e}^{-w^{\rm T}\Sigma
w/2},\qquad\beta\to\infty,$ (4.7)

locally uniformly for $w\in\mathbb{R}^{N}$. Moreover, by the scaling
properties of the $\rho_{s,\beta}$ we have
$\hat{\rho}_{s,\beta}(w)=\hat{\rho}_{1,\beta}\big{(}\sqrt{s}w\big{)}$ for
$s\geq 0$ and $w\in\mathbb{R}^{N}$.

We now consider the Cauchy–Bessel distributions $\tau_{\nu,\beta}$ which are
related to the $\rho_{s,\beta}$ via the subordination (2.8) by

$\tau_{\nu,\beta}=\int_{0}^{\infty}\rho_{s,\beta}\,{\rm d}\mu_{\sqrt{2}}(s)$

in the sense of concatenation of a Markov kernel with a measure. Using the
definition (2.7) of $\mu_{\sqrt{2}}$, we obtain

$\displaystyle\hat{\tau}_{\nu,\beta}(w)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\hat{\rho}_{s,\beta}(w)s^{-3/2}\exp(-1/(2s))\,{\rm
d}s$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\hat{\rho}_{1,\beta}\big{(}\sqrt{s}w\big{)}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s.$

Now consider the linear mappings (4.4) which transform the given random
variables $X_{\beta}$ with the distributions $\tau_{\nu,\beta}$ into the
rescaled random variables $\tilde{X}_{\beta}:=\phi_{\beta}(X_{\beta})$. As the
$\phi_{\beta}$ are symmetric linear operators, we conclude that the Fourier
transforms $\hat{\tau}_{\nu,\beta,\phi}$ of the distributions
$\tau_{\nu,\beta,\phi}$ of the $\tilde{X}_{\beta}$ satisfy

$\hat{\tau}_{\nu,\beta,\phi}(w)=\hat{\tau}_{\nu,\beta}(\phi_{\beta}(w)).$

Using (4.7) and dominated convergence, we hence obtain that for
$w\in\mathbb{R}^{N}$,

$\displaystyle\lim_{\beta\to\infty}\hat{\tau}_{\nu,\beta,\phi}(w)$
$\displaystyle=\lim_{\beta\to\infty}\hat{\tau}_{\nu,\beta}(\phi_{\beta}(w))$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\lim_{\beta\to\infty}\hat{\rho}_{1,\beta}\big{(}\sqrt{s}\phi_{\beta}(w)\big{)}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\lim_{\beta\to\infty}\big{(}{\rm
e}^{-s\phi_{\beta}(w)^{\rm T}\Sigma\phi_{\beta}(w)/2}{\rm e}^{-{\rm
i}\sqrt{s\beta}\langle\phi_{\beta}(w),r\rangle}\big{)}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s.$

We next observe from the definition of $\phi_{\beta}$ that

$\sqrt{\beta}\langle\phi_{\beta}(w),r\rangle=\langle p_{r}(w),r\rangle=\langle
w,r\rangle$

and

$\lim_{\beta\to\infty}\phi_{\beta}(w)^{\rm
T}\Sigma\phi_{\beta}(w)=p_{r^{\perp}}(w)^{\rm T}\Sigma p_{r^{\perp}}(w).$

Therefore,

$\lim_{\beta\to\infty}\hat{\tau}_{\nu,\beta,\phi}(w)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}{\rm
e}^{-sp_{r^{\perp}}(w)^{\rm T}\Sigma p_{r^{\perp}}(w)/2}{\rm e}^{-{\rm
i}\sqrt{s}\langle w,r\rangle}s^{-3/2}\exp(-1/(2s))\,{\rm d}s.$

Clearly, the r.h.s. is just the Fourier transform of the probability measure

$\mu:=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}{\mathcal{N}}(0,sA\Sigma
A)*\delta_{\sqrt{s}r}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s=\int_{0}^{\infty}{\mathcal{N}}\big{(}\sqrt{s}r,sA\Sigma A\big{)}\,{\rm
d}\mu_{\sqrt{2}}(s)$

from (4.5). Hence, by Levy’s continuity theorem, we have weak convergence to
this $\mu$.

We finally check that $\mu$ has the density (4.6). Clearly we may restrict our
attention to the interior of the half space $B_{N}$. Moreover, by standard
arguments from measure theory it suffices to check this by comparing $\mu$
with the measure with density (4.6) for sets of the form
$T([c_{1},d_{1}]\times R)\subset B_{N}$ for $0\leq c_{1}\leq d_{1}$, a Borel
set $R\subset\mathbb{R}^{n-1}$, and $T$ the map belonging to the change of
coordinates from the given standard coordinates $e_{1},\dots,e_{N}$ into the
orthogonal coordinates belonging to the normalized eigenvectors
$v_{1},\dots,v_{N}$ of $\Sigma^{-1}$ associated with the eigenvalues
$2,4,\dots,N$. We recapitulate that by [6, Theorem 4.3] and by (4.2),
$v_{1}=r/\sqrt{N(N+\nu-1)}$ holds. With these notations and the substitution
$t=\sqrt{s}\|r\|$ we obtain for the probability measure $\mu$ from (4.5) that

$\displaystyle\mu(T([c_{1},d_{1}]\times R))$
$\displaystyle\qquad{}=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}{\mathcal{N}}\left(\sqrt{s}\|r\|e_{1},s\operatorname{diag}\left(0,\frac{1}{4},\frac{1}{6},\dots,\frac{1}{2N}\right)\right)([c_{1},d_{1}]\times
R)\frac{\exp(-1/(2s))}{s^{3/2}}\,{\rm d}s$
$\displaystyle\qquad{}=\frac{2\|r\|}{\sqrt{2\pi}}\int_{0}^{\infty}{\mathcal{N}}\left(te_{1},\frac{t^{2}}{\|r\|^{2}}\operatorname{diag}\left(0,\frac{1}{4},\frac{1}{6},\dots,\frac{1}{2N}\right)\right)([c_{1},d_{1}]\times
R)\frac{{\rm e}^{-\|r\|^{2}/(2t^{2})}}{t^{2}}\,{\rm d}t$
$\displaystyle\qquad{}=\frac{2\|r\|}{\sqrt{2\pi}}\int_{c_{1}}^{d_{1}}{\mathcal{N}}_{N-1}\left(0,\frac{t^{2}}{\|r\|^{2}}\operatorname{diag}\left(\frac{1}{4},\frac{1}{6},\dots,\frac{1}{2N}\right)\right)(R)\frac{{\rm
e}^{-\|r\|^{2}/(2t^{2})}}{t^{2}}\,{\rm d}t,$ (4.8)

where ${\mathcal{N}}_{N-1}$ is an $(N-1)$-dimensional normal distribution. On
the other hand, with the same change of coordinates, (4.6) and (4.2) lead to

$\displaystyle\int_{T([c_{1},d_{1}]\times R)}f(y)\,{\rm d}y$
$\displaystyle\qquad{}=D(N)\int_{c_{1}}^{d_{1}}\Biggl{(}\int_{R}\exp\left(-\frac{\|{r}\|^{2}}{2t^{2}}(t,y)^{\rm
T}\operatorname{diag}(2,4,6,\dots,2N)(t,y)\right){\rm d}y\Biggr{)}$
$\displaystyle\qquad\quad{}\times\exp\left(\frac{-\|{r}\|^{2}}{2t^{2}}\right)\frac{1}{t^{N+1}}\,{\rm
d}t$ $\displaystyle\qquad{}D(N){\rm
e}^{-\|r\|^{2}}\int_{c_{1}}^{d_{1}}\Biggl{(}\int_{R}\exp\left({-}\frac{\|{r}\|^{2}}{2t^{2}}y^{\rm
T}\operatorname{diag}(4,6,\dots,2N)y\right){\rm d}y\Biggr{)}$
$\displaystyle\qquad\quad{}\times\exp\left(\frac{-\|r\|^{2}}{2t^{2}}\right)\frac{1}{t^{N+1}}\,{\rm
d}t.$ (4.9)

Using the definition of $D(N)$ and the constants of multivariate normal
distributions, we see that the expressions in the end of (4.8) and (4.9) are
equal. This completes the proof. ∎

######  Remark 4.5.

  1. (1)

Clearly, the central limits Theorem 3.8 can be also proved in the same way as
Theorem 4.4. Moreover the limit measure with density (3.21) there can be also
expressed in a form which corresponds to (4.5).

On the other hand, the methods of the proof of Theorem 3.8 can be also applied
in the situations of Theorems 3.8 and 4.4 in order to derive corresponding
limits for distributions of the form

$c(k,t)\left(\frac{4}{t^{2}+2\|y\|^{2}}\right)^{r_{k}}w_{k}(y)$

with more general exponents $r_{k}$ than in (2.9) and suitable norming
constants $c(k,t)>0$.

  2. (2)

The asymptotic Theorem 3.4 in the Hermite case can be also transfered to the
Laguerre case. We skip the details.

  3. (3)

The assertion of Theorem 4.4 remains valid for all Cauchy–Bessel distributions
of type $B_{N}$ as defined in (2.8) via subordination for all fixed starting
points $x\in C_{N}^{B}$ and not only for $x=0$.

This can be seen as follows. Lemma 5 of [4] implies that for all $x,y\in
C_{N}^{B}$ and $\nu>0$, the corresponding Bessel functions satisfy

$\lim_{\beta\to\infty}J_{(\nu\beta,\beta)}^{B}\big{(}\sqrt{\beta}x,y\big{)}=\exp\left(\frac{\|x\|^{2}\|y\|^{2}}{4N(\nu+N-1)}\right).$
(4.10)

This implies that Theorem 4.2 is available also for arbitrary starting points
$x\in C_{N}^{B}$; see [35, Theorem 3.3]. This shows that the proof of Theorem
4.4 also works for arbitrary starting points $x\in C_{N}^{B}$.

  4. (4)

The preceding result can be also stated for the root systems $A_{N-1}$ and
arbitrary starting points $x\in C_{N}^{A}$. However, the details of the proof
and of the result are slightly more complicated, as the root system is not
longer reduced, and as the center-of-gravity-part of the limit has a slightly
different behavior. In fact, the analogue of (4.10) for the Bessel functions
of type A is more complicated; see of [5, Corollary 8] as well as [6, Lemma
2.4 and Theorem 2.5]. This limit for the Bessel functions implies that the
limit distribution of the CLT for Bessel processes in [6, Theorem 2.3]
contains an additional drift in the center-of-gravity-direction. Having this
in mind, one can also restate central limits Theorem 3.8 in this way for
arbitrary starting points $x\in C_{N}^{A}$ by taking this drift into account.

  5. (5)

In [36], freezing limits are studied for Bessel processes with parameter
$k\to\infty$ where the starting points have the form $\sqrt{k}x$ with points
$x$ in the interior of the Weyl chamber. We do not know whether the CLTs there
can be transfered to Cauchy–Bessel processes.

  6. (6)

We expect that the methods of the proof of Theorem 4.4 can be used to study
freezing limits for further classes of distributions which appear form the
Bessel processes by different subordinations like general analogues of stable
distributions.

  7. (7)

In the singular case $\nu=0$ there exists an analogue of Theorem 4.4 where the
details are slightly different. We discuss this singular case in the next
section as a consequence of the corresponding results for the root systems
$D_{N}$.

##  5 Freezing limits for the root system $\boldsymbol{D_{N}}$
and an extremal $\boldsymbol{B_{N}}$-case

We here briefly study Bessel processes and related Cauchy–Bessel distributions
for the root system $D_{N}$ and an extremal $B_{N}$-case. We recapitulate that
the root system $D_{N}$ is given by

$D_{N}=\\{\pm e_{i}\pm e_{j}\colon 1\leq i<j\leq N\\}$

with the Weyl chamber

$C_{N}^{D}=\big{\\{}x\in\mathbb{R}^{N}\colon x_{1}\geq\dots\geq
x_{N-1}\geq|x_{N}|\big{\\}},$

which may be seen as a doubling of $C_{N}^{B}$ w.r.t. the last coordinate. We
have a multiplicity $k\in{}]0,\infty[$. The generator of the transition
semigroup of the Bessel process $(X_{t,k})_{t\geq 0}$ of type D is

$\displaystyle Lf:=\frac{1}{2}\Delta f+k\sum_{i=1}^{N}\sum_{j\neq
i}\left(\frac{1}{x_{i}-x_{j}}+\frac{1}{x_{i}+x_{j}}\right)\frac{\partial}{\partial
x_{i}}f.$

The transition probabilities are

$K_{t,k}(x,A)=c_{k}^{D}\int_{A}\frac{1}{t^{\gamma_{D}+N/2}}{\rm
e}^{-(\|x\|^{2}+\|y\|^{2})/(2t)}J_{k}^{D}\left(\frac{x}{\sqrt{t}},\frac{y}{\sqrt{t}}\right)w_{k}^{D}(y)\,{\rm
d}y$

with

$w_{k}^{D}(x):=\prod_{i<j}\big{(}x_{i}^{2}-x_{j}^{2}\big{)}^{2k},\qquad\gamma_{D}:=kN(N-1)$
(5.1)

and

$c_{k}^{D}=\frac{N!}{2^{N(N-1)k-N/2+1}}\prod_{j=1}^{N}\frac{\Gamma(1+k)}{\Gamma(1+jk)\Gamma(\frac{1}{2}+(j-1)k)};$
(5.2)

see Demni [17] and [7, 35] for the details.

We next recapitulate some fact on Laguerre polynomials. Using the
representation

$L_{N}^{(\alpha)}(x):=\sum_{k=0}^{N}{N+\alpha\choose N-k}\frac{(-x)^{k}}{k!}$

(see [32, equation (5.1.6)]), we can form the polynomial $L_{N}^{(-1)}$ of
order $N\geq 1$ where, by [32, equation (5.2.1)],

$L_{N}^{(-1)}(x)=-\frac{x}{N}L_{N-1}^{(1)}(x).$ (5.3)

Using the $N-1$ ordered zeros $z_{1}^{(1)}>\dots>z_{N-1}^{(1)}>0$ of
$L_{N-1}^{(1)}$, we define the vector $r=(r_{1},\dots,r_{N})\in[0,\infty[^{N}$
with

$2\big{(}z_{1}^{(1)},\dots,z_{N-1}^{(1)},0\big{)}=\big{(}r_{1}^{2},\dots,r_{N}^{2}\big{)}$
(5.4)

similar to Section 4. Notice that $r$ is in the interior of $C_{N}^{D}$, and
that (4.2) and (5.3) imply

$\|r\|^{2}=N(N-1).$

Most parts of the following CLT for the Bessel processes $(X_{t,k})_{t\geq 0}$
of type $D_{N}$ on $C_{N}^{D}$ with multiplicity $k>0$ with start in $0$ were
proved in [35]:

######  Theorem 5.1.

For each $t>0$, the random variables $\frac{X_{t,k}}{\sqrt{t}}-\sqrt{k}r$
converge for $k\to\infty$ to the centered $N$-dimensional distribution
${\mathcal{N}}(0,\Sigma_{D})$ with the regular covariance matrix $\Sigma_{D}$
with $\Sigma_{D}^{-1}=(s_{i,j})_{i,j=1,\dots,N}$ with

$s_{i,j}:=\begin{cases}\displaystyle 1+2\sum_{l\neq
i}(r_{i}-r_{l})^{-2}+2\sum_{l\neq i}(r_{i}+r_{l})^{-2}&\text{for}\ i=j,\\\
2(r_{i}+r_{j})^{-2}-2(r_{i}-r_{j})^{-2}&\text{for}\ i\neq j.\end{cases}$ (5.5)

The entries $s_{i,j}$ satisfy $s_{i,N}=s_{N,i}=0$ for $i=1,\dots,N-1$ and
$s_{N,N}=N$. The block $(s_{i,j})_{i,j=1,\dots,N-1}$ is the inverse covariance
matrix in Theorem 4.4 for the dimension $N-1$ with $\nu=2$.

###### Proof.

By the proof of Theorem 5.2 in [35], the densities of the
$\frac{X_{t,k}}{\sqrt{t}}-\sqrt{k}r$ may be written as

$\displaystyle f_{k}^{D}(y)={}$ $\displaystyle
c_{k}^{D}\exp\Bigg{(}2k\sum_{i<j}\ln\bigg{(}1+\frac{y_{i}-y_{j}}{\sqrt{k}(r_{i}-r_{j})}\bigg{)}+2k\sum_{i<j}\ln\bigg{(}1+\frac{y_{i}+y_{j}}{\sqrt{k}(r_{i}+r_{j})}\bigg{)}\Bigg{)}$
$\displaystyle{}\times{\rm e}^{-\|y\|^{2}/2}{\rm e}^{-k\|r\|^{2}/2}{\rm
e}^{-\sqrt{k}\langle
y,r\rangle}\exp\Bigg{(}2k\sum_{i<j}\bigl{(}\ln\big{(}\sqrt{k}(r_{i}-r_{j})\big{)}+\ln\big{(}\sqrt{k}(r_{i}+r_{j})\big{)}\bigr{)}\Bigg{)}$

on the shifted cone $C_{N}^{D}-\sqrt{k}r$, with $f_{k}^{D}(y)=0$ elsewhere on
$\mathbb{R}^{N}$. We write this as

$f_{k}^{D}(y)=\tilde{c}_{k}^{D}h_{k}(y)$

with

$\displaystyle h_{k}(y):={}$
$\displaystyle\exp\Bigg{(}{-}\|y\|^{2}/2-\sqrt{k}\langle
y,r\rangle+2k\sum_{i<j}\bigg{(}\ln\bigg{(}1+\frac{y_{i}-y_{j}}{\sqrt{k}(r_{i}-r_{j})}\bigg{)}$
$\displaystyle{}+\ln\bigg{(}1+\frac{y_{i}+y_{j}}{\sqrt{k}(r_{i}+r_{j})}\bigg{)}\bigg{)}\Bigg{)}$

and

$\tilde{c}_{k}^{D}:=c_{k}^{D}{\rm
e}^{-k\|r\|^{2}/2}\exp\Bigg{(}2k\sum_{i<j}\big{(}\ln\big{(}\sqrt{k}(r_{i}-r_{j})\big{)}+\ln\big{(}\sqrt{k}(r_{i}+r_{j})\big{)}\big{)}\Bigg{)},$

where, by [35, equation (5.7)],

$\lim_{k\to\infty}h_{k}(y)=\exp\Bigg{(}{-}\frac{\|y\|^{2}}{2}-\sum_{i<j}\frac{(y_{i}-y_{j})^{2}}{(r_{i}-r_{j})^{2}}-\sum_{i<j}\frac{(y_{i}+y_{j})^{2}}{(r_{i}+r_{j})^{2}}\Bigg{)}.$

This implies by the arguments in the proofs of Theorem 5.2 in [35] (more
precisely, by the arguments in the proofs of Theorems 2.2 and 3.3 there) that
the probability measures with the densities $f_{k}^{D}$ tend weakly to
${\mathcal{N}}(0,\Sigma_{D})$ with $\Sigma_{D}^{-1}$ as in the theorem above.
Moreover, except for the statement $s_{N,N}=N$, all additional facts about the
entries of $\Sigma_{D}^{-1}$ in the theorem are clear by (5.3).

In order to prove $s_{N,N}=N$, we use (5.3) and (4.3) for $\nu=2$ (i.e.,
$\alpha=1$) and $N-1$ (instead of $N$), and we observe that in our situation
$r_{N}=0$ holds. These facts lead readily to

$2\ln\Bigg{(}\prod_{i<j}\big{(}r_{i}^{2}-r_{j}^{2}\big{)}\Bigg{)}=N(N-1)(-1/2+\ln
2)+\sum_{j=1}^{N}j\ln j+\sum_{j=1}^{N-1}j\ln j.$

This, (5.2), and Stirling’s formula applied to the Gamma functions in (5.2)
now imply that

$\lim_{k\to\infty}\tilde{c}_{k}^{D}=\frac{2^{(N-1)/2}\sqrt{N!}}{(2\pi)^{N/2}}.$

If we compare this with the normalization constants of
${\mathcal{N}}(0,\Sigma_{D})$ and use

$\det\big{(}\Sigma_{D}^{-1}\big{)}=\det((s_{i,j})_{i,j=1,\dots,N-1})s_{N,N}=2^{N-1}(N-1)!s_{N,N},$

we obtain $s_{N,N}=N$ as claimed. ∎

######  Remark 5.2.

If we combine $s_{N,N}=N$ with the $(N,N)$-entry in (5.5), we obtain that the
zeros $z_{1}^{(1)}>\dots>z_{N-1}^{(1)}>0$ of $L_{N-1}^{(1)}$ satisfy
$\sum_{l=1}^{N-1}\frac{1}{z_{l}^{(1)}}=\frac{N-1}{2}$. It was pointed out by
one of the referees that such sums over the inverses of the zeros can be
computed easily for all classical orthogonal polynomials. For this use the
elementary symmetric polynomials $e_{0},\dots,e_{N}$ in $N$ variables, and
write such a polynomial $P_{N}$ of order $N$ with zeros $z_{1},\dots,z_{N}$ as

$P_{N}(z)=c_{N}\prod_{j=1}^{N}(z-z_{j})=c_{N}\sum_{j=0}^{N}(-1)^{N-j}e_{N-j}(z_{1},\dots,z_{N})z^{j}.$
(5.6)

As

$\sum_{l=1}^{N}\frac{1}{z_{l}}=\frac{e_{N-1}(z_{1},\dots,z_{N})}{e_{N}(z_{1},\dots,z_{N})},$

we can derive this sum from (5.6) and the well-known formulas for the
coefficients of the classical orthognal polynomials in [32]. For instance,
equation (5.1.6) of [32] yields for $L_{N}^{(\alpha)}$ with the zeros
$z_{1}^{(\alpha)},\dots,z_{N}^{(\alpha)}$ that

$\sum_{l=1}^{N}\frac{1}{z_{l}^{(\alpha)}}=\frac{N}{\alpha+1}.$

This in particular leads to an alternative proof of the statement $s_{N,N}=N$
in the preceding theorem.

We next turn to Cauchy–Bessel distributions of type $D_{N}$ which are
constructed from the associated Bessel processes via subordination. More
precisely, we use the inverse Gaussian distribution $\mu_{t}$ with density
(2.7) for $t=\sqrt{2}$ as in the preceding sections, and obtain from the
densities (2.6) together with (2.9), (5.1), and (5.2) that the associated
Cauchy–Bessel ensembles have the densities

$f_{k,D}(y):=C_{D}(k,N)\frac{1}{(1+\|y\|^{2})^{kN(N-1)+(N+1)/2}}\prod_{i,j\colon
i<j}\big{(}y_{i}^{2}-y_{j}^{2}\big{)}^{2k}$ (5.7)

on the Weyl chambers $C_{N}^{D}$ with the norming constants

$C_{D}(k,N)=\frac{2^{N-1}N!\Gamma(kN(N-1)+(N+1)/2)}{\sqrt{\pi}}\prod_{j=1}^{N}\frac{\Gamma(1+k)}{\Gamma(1+jk)\Gamma(\frac{1}{2}+k(j-1))}.$

The Fourier-analytic proof of Theorem 4.4 leads to the following CLT where,
similar to Section 4, we use the normalization mappings

$\phi_{k}\colon\
\mathbb{R}^{N}\to\mathbb{R}^{N},\qquad\phi_{k}(y):=\frac{1}{\sqrt{k}}p_{r}(y)+p_{{r}^{\perp}}(y)=y+\left(\frac{1}{\sqrt{k}}-1\right)p_{{r}}(y).$
(5.8)

######  Theorem 5.3.

For $k>0$ and $N\geq 2$ let $X_{k}$ be $C_{N}^{D}$-valued random variables
with the Lebesgue densities (5.7). Then the rescaled random variables
$\tilde{X}_{k}:=\phi_{k}(X_{k})$ converge in distribution for $k\to\infty$ to
some $\mu\in M^{1}\big{(}\mathbb{R}^{N}\big{)}$ with the half space
$B_{N}:=\big{\\{}y\in\mathbb{R}^{N}\colon\langle y,r\rangle\geq 0\big{\\}}$ as
support. $\mu$ is given by

$\mu:=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}{\mathcal{N}}\big{(}\sqrt{s}r,sA\Sigma_{D}A\big{)}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s,$

where $A$ is the matrix of the orthogonal projection $p_{{r}^{\perp}}$. The
measure $\mu$ has the Lebesgue density

$\displaystyle f(y):=D(N)\exp\left(-\frac{N(N-1)}{2\|p_{{r}}(y)\|^{2}}y^{\rm
T}\Sigma_{D}^{-1}y\right)\exp\left(\frac{-N(N-1)}{\|p_{{r}}(y)\|^{2}}\right)\frac{1}{\|p_{{r}}(y)\|^{N+1}}$

for $y$ in the interior of $B_{N}$ with

$D(N):=\frac{\sqrt{N!}(N(N-1))^{N/2}{\rm e}^{N(N-1)}}{\pi^{N/2}}.$

The central limit Theorems 5.1 and 5.3 for Bessel and Cauchy–Bessel processes
of type D lead immediately to CLTs for the Bessel and Cauchy–Bessel processes
of type B with the multiplicities $(k_{1},k_{2}):=(0,\beta)$ for
$\beta\to\infty$, i.e., the case $\nu=0$ in Section 4.

For this we recapitulate the following fact from [7]. If
$\big{(}X_{t,k}^{D}\big{)}_{t\geq 0}$ is a Bessel process of type D with
multiplicity $k\geq 0$ on the chamber $C_{N}^{D}$ starting in $0$, then the
process $\big{(}X_{t,k}^{B}\big{)}_{t\geq 0}$ with

$X_{t,k}^{B,i}:=X_{t,k}^{D,i},\quad i=1,\dots,N-1,\qquad
X_{t,k}^{B,N}:=\big{|}X_{t,k}^{D,N}\big{|}$

is a Bessel process of type B with $(k_{1},k_{2}):=(0,k)$. This follows easily
from a comparison of the corresponding generators. The central limit Theorem
5.1 for $\big{(}X_{t,k}^{D}\big{)}_{t\geq 0}$ thus leads to the following
central limit Theorem 5.4 for Bessel processes of type B with the
multiplicities $(0,k)$ for $k\to\infty$ with one-sided normal distribution as
limit; see [35, Corollary 5.3]. By [35, Theorem 6.2], this CLT also holds for
the multiplicities $(k_{1},k_{2})$ form any fixed $k_{1}\geq 0$ and
$k_{2}\to\infty$.

To state the result we denote the image of a $N$-dimensional normal
distribution ${\mathcal{N}}(0,\Sigma)$ with covariance matrix $\Sigma$ under
the map

$\mathbb{R}^{N}\longrightarrow H_{N}:=\big{\\{}x\in\mathbb{R}^{N}\colon
x_{N}\geq
0\big{\\}},\qquad(x_{1},\dots,x_{N})\mapsto(x_{1},\dots,x_{N-1},|x_{N}|)$

by $|{\mathcal{N}}(0,\Sigma)|$, i.e., the support of
$|{\mathcal{N}}(0,\Sigma)|$ is contained in the half space $H_{N}$.

######  Theorem 5.4.

Consider the Bessel processes $(X_{t,(k_{1},k_{2})})_{t\geq 0}$ of type
$B_{N}$ on $C_{N}^{B}$ with multiplicities $(k_{1},k_{2})$ with start in $0$
and $k_{1}\geq 0$. Then, for the vector $r$ from (5.4) on the boundary of
$C_{N}^{B}$,

$\frac{X_{t,(k_{1},k_{2})}}{\sqrt{t}}-\sqrt{k_{2}}r$

converges for $k_{2}\to\infty$ in distribution to
$|{\mathcal{N}}(0,\Sigma_{D})|$ with $\Sigma_{D}$ as in Theorem 5.1.

This one-sided CLT leads to the following corresponding result for
Cauchy–Bessel distributions.

######  Corollary 5.5.

For $k_{1},k_{2}\geq 0$ and integer $N\geq 2$ let $X_{k_{1},k_{2}}$ be
$C_{N}^{B}$-valued random variables with the Cauchy–Bessel densities

$f_{k_{1},k_{2}}(y):=C_{B}(k_{1},k_{2},N)\frac{1}{\big{(}1+\|y\|^{2}\big{)}^{k_{2}N(N-1)+k_{1}N+(N+1)/2}}\prod_{i,j\colon
i<j}\big{(}y_{i}^{2}-y_{j}^{2}\big{)}^{2k_{2}}\prod_{i=1}^{N}y_{i}^{2k_{1}}$

with the norming constants as in (4.1). Then the rescaled random variables
$\tilde{X}_{k_{1},k_{2}}:=\phi_{k}(X_{k_{1},k_{2}})$ with $\phi_{k}$ as in
(5.8) converge in distribution for $k_{2}\to\infty$ to some $\mu\in
M^{1}\big{(}\mathbb{R}^{N}\big{)}$ with the quarter space

$B_{N,0}:=\big{\\{}y\in\mathbb{R}^{N}\colon y_{N}\geq 0,\langle y,r\rangle\geq
0\big{\\}}$

as support. $\mu$ is given by

$\mu:=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\big{|}{\mathcal{N}}\big{(}\sqrt{s}r,sA\Sigma_{D}A\big{)}\big{|}s^{-3/2}\exp(-1/(2s))\,{\rm
d}s,$

where $A$ is the matrix of the orthogonal projection $p_{{r}^{\perp}}$. The
measure $\mu$ has the Lebesgue density

$\displaystyle f(y):=2D(N)\exp\left(-\frac{N(N-1)}{2\|p_{{r}}(y)\|^{2}}y^{\rm
T}\Sigma_{D}^{-1}y\right)\exp\left(\frac{-N(N-1)}{\|p_{{r}}(y)\|^{2}}\right)\frac{1}{\|p_{{r}}(y)\|^{N+1}}$

for $y$ in the interior of $B_{N,0}$ with $D(N)$ as in Theorem 5.3.

### Acknowledgements

The author would like to thank the anonymous referees for their numerous
comments, which improved the paper considerably.

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