# Analogues of Poisson type limit theorems in discrete bm-Fock spaces

Lahcen Oussi 1,2 1 Institute of Mathematics, Wroclaw University, Pl.
Grunwaldzki 2, 50-384 Wroclaw, Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Janusz Wysoczański 1,3

###### Abstract.

We present analogues of the Poisson limit distribution for the noncommutative
bm-independence, which is associated with several positive symmetric cones. We
construct related discrete Fock spaces with creation, annihilation and
conservation operators, and prove Poisson type limit theorems for them.
Properties of the positive cones, in particular the volume characteristic
property they enjoy, and the combinatorics of labelled noncrossing partitions,
play crucial role in these considerations.

###### Key words and phrases:

Noncommutative probability. Bm-independence. Positive symmetric cones. Fock
space. Poisson type limit distribution. Labelled noncrossing partitions

###### 2020 Mathematics Subject Classification:

46L53. 05A18. 30H20. 60F05

##  1\. Introduction

In this paper we study analogues of the classical Poisson limit theorem in the
noncommutative probability framework set by bm-independence. Our study can be
considered as parallel to the results of Crismale, Griseta and the second-
named author [1] and [2], where the weakly monotone case has been considered.
Here we develop theory extending the monotone case of Muraki [9].

In general, a noncommutative probability space is a unital $*-$algebra
$\mathcal{A}$ (or a $C^{*}-$algebra) with a normalized state $\varphi$ on it,
which plays the role of classical expectation.

There are several notions of independence in such framework, which are the
rules for computing mixed moments, i.e. expressions of the form
$\varphi(a_{i_{1}}a_{i_{2}}\ldots a_{i_{k}})$ for
$a_{i_{1}},a_{i_{2}},\ldots,a_{i_{k}}\in\mathcal{A}$. The universal ones are
the freeness of Voiculescu [16], the monotonic independence of Muraki [10] and
the Boolean independence [15]. There are other notions of noncommutative
independence, which are mixtures of these. For more information on this topic
we refer to the Introduction in [12].

Here we develop the theory of bm-independence introduced in [18, 19], which
combines Boolean and monotonic ones. A particular feature of this notion is
that it is defined for random variables indexed by a partially ordered set.
The definition of this notion is presented in the next section. In this study,
however, we consider only specific partially ordered sets, which are defined
by positive symmetric cones in Euclidean spaces, classification of which can
be found in [4]. These cones have an additional geometric property called the
volume characteristic, which allows to develop analogues of classical
properties in such general framework.

Noncommutative analogues of the classical Poisson limit theorem arise in two
ways. The first one is by considering the convolution powers of Bernoulli type
distributions, and this can be done for the free, monotone and boolean
convolutions. This can be generalized to arrays of independent random
variables (see [14] for free independence and [12] for bm-independence). The
second method, used by Muraki [9], comes from constructions of creation
$A_{i}^{+}$, annihilation $A_{i}^{-}$ and conservation $A_{i}^{\circ}$
operators ($i\in\mathbb{N}$) on appropriate Fock space associated, as the toy
model, with a given noncommutative independence. In particular, these
operators are independent (in the given noncommutative sense). For a parameter
$\lambda\geq 0$, one considers the sums

(1)
$S_{N}(\lambda):=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}(A_{i}^{+}+A_{i}^{-})+\lambda\sum_{i=1}^{N}A_{i}^{\circ}$

and one studies the limits of moments (for every $p\in\mathbb{N}$)

$m_{p}(\lambda):=\lim_{N\to+\infty}\varphi((S_{N}(\lambda))^{p}).$

Then these are moments of Poisson type measure (which is to be identified in
each case) associated with the given noncommutative independence. This type of
Poisson limit is motivated by the Fock space of the classical Poisson process
[7]. In particular, for free independence this is the free Poisson (i.e.
Marchenko-Pastur) measure [14], whereas for monotonic independence the measure
has been identified by Muraki [9]. Our constructions and results extend those
of Muraki, so we recall his description of moments of the limit measure in the
monotonic case (for details we refer to [9]):

(2)
$m_{p}(\lambda):=\lim_{N\rightarrow+\infty}\varphi\left((S_{N}(\lambda))^{p}\right)=\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}V(\pi)\lambda^{s(\pi)},$

where $\mathcal{NC}_{2}^{1,i}(p)$ denotes the set of all noncrossing
partitions of the set $[p]:=\\{1,\ldots,p\\}$, with pair blocks and inner
singleton blocks. In addition, $V(\pi)$ counts (approximately, as
$N\rightarrow+\infty$) the ratio of all blocks of
$\pi\in\mathcal{NC}_{2}^{1,i}(p)$ with monotonic labellings by integers
$\\{1,2,\ldots,N\\}$, divided by $N^{\frac{p-s(\pi)}{2}}$, and $s(\pi)$ is the
number of singleton blocks in $\pi$.

The independence we study here can be defined for any partially ordered set,
however such generality does not allow for specific results. In particular one
needs additional structure of the poset. One of the most interesting ones is
guaranteed by partial orders defined by positive symmetric cones,
classification and properties of which can be found in [4].

More in detail we consider the following three classes of positive symmetric
cones $\Pi_{d}$ (with $d\in\mathbb{N}$) in particular vector spaces
$\mathtt{X}_{d}$:

  1. (1)

The cones
$\Pi_{d}={\mathbb{R}}_{+}^{d}:=\\{\xi=(\xi_{1},\xi_{2},\ldots,\xi_{d})\in{\mathbb{R}}^{d}:\xi_{1},\ldots,\xi_{d}\geq
0\\}$ in the vector spaces ${\mathtt{X}_{d}}={\mathbb{R}}^{d}$, where for
$\xi=(\xi_{1},\ldots,\xi_{d}),\rho=(\rho_{1},\ldots,\rho_{d})\in{\mathbb{R}}^{d}$,
the partial order $\preceq$ is given by coordinatewise comparison:

$\xi\preceq\rho\quad\text{if}\quad\xi_{1}\leq\rho_{1},\ldots,\xi_{d}\leq\rho_{d}.$

  2. (2)

The Lorentz light cones $\Lambda_{d}^{1}$ in $(d+1)$-Minkowski space time
${\mathtt{X}_{d}}={\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$ with the positive
cone
$\Pi_{d}=\Lambda_{d}^{1}:=\left\\{(t;x_{1},...,x_{d})\in{\mathbb{X}_{d}}:t\geq\left(\sum_{i=1}^{d}x_{i}^{2}\right)^{\frac{1}{2}}\right\\}$.
In this case, for $\xi=(t;x_{1},...,x_{d})$,
$\rho=(s;y_{1},...,y_{d})\in{\mathbb{R}}_{+}\times{\mathbb{R}}^{d}$, we have

$\rho\preceq\xi\quad\text{if}\quad
t-s\geq\left(\sum_{i=1}^{d}(x_{i}-y_{i})^{2}\right)^{\frac{1}{2}}.$

  3. (3)

The cones $\Pi_{d}={\mathtt{Symm}}_{d}^{+}(\mathbb{R})$ of positive
semidefinite real symmetric $(d\times d)$ matrices:

${\mathtt{Symm}}_{d}^{+}(\mathbb{R}):=\\{\xi\in\mathbb{M}_{d}({\mathbb{R}}):\xi=\xi^{T},\
\xi\geq 0\\}\subset\mathtt{X}_{d}=\mathbb{M}_{d}({\mathbb{R}}).$

For $\xi,\rho\in\mathbb{M}_{d}({\mathbb{R}})$, one defines

$\rho\preceq\xi\quad\text{ if
}\quad\xi-\rho\in{\mathtt{Symm}}_{d}^{+}(\mathbb{R}).$

These cones satisfy the property of _volume characteristic_ which plays
crucial role in our studies. This property, introduced in [6], reads as
follows:

######  Theorem 1.1 (volume characteristic [6]).

For each of the positive symmetric cones $\Pi_{d}$ we consider, there exists a
sequence $(\gamma_{m}(\Pi_{d}))_{m\geq 1}$ such that for any $\xi\in\Pi_{d}$
and any $m\in{\mathbb{N}}$

$\gamma_{m}(\Pi_{d})=\frac{1}{\mathtt{v}(\xi)^{m}}\int_{\rho\in[0,\xi]}\mathtt{v}(\rho)^{m-1}d(\rho),$

where $\mathtt{v}(\xi):=\mathtt{vol}[0,\xi]$ denotes the Euclidean volume of
the interval $[0,\xi]\subset\Pi_{d}\subset{\mathbb{R}}^{m}$ and $d(\rho)$ is
the Lebesgue measure on ${\mathbb{R}}^{m}$ (the dimension $m$ is minimal for
the embedding $\Pi_{d}\subset{\mathbb{R}}^{m}$).

The volume characteristic property generalizes the following simple fact for
the positive cone ${\mathbb{R}}_{+}$, where
$\mathtt{v}(x):=\mathtt{vol}[0,x]=x$ for $x>0$:

$\frac{1}{\mathtt{v}(x)^{m}}\int_{y\in[0,x]}\mathtt{v}(y)^{m-1}dy=\frac{1}{x^{m}}\int_{0}^{x}y^{m-1}dy=\frac{1}{m}=\gamma_{m}({\mathbb{R}}_{+}),$

In what follows we will write simply $\gamma_{m}$ if the cone is specified.

In our studies of Poisson type limit theorems we deal with bm-independent
random variables, indexed by elements of positive symmetric cones $\Pi_{d}$.
This creates some challenges regarding the formulation of the theorems. These
challenges are the same as those given in our study [12] of the law of small
numbers for bm-independent random variables. For the reader’s convenience we
recall briefly how these problems has been solved in [12].

  1. (1)

Instead of the index set of positive integers we use the following discrete
subsets ${\mathbf{I}}\subset\Pi_{d}$ depending on the positive cone $\Pi_{d}$:

    1. (a)

${\mathbf{I}}:={\mathbb{N}}^{d}$ if $\Pi_{d}={\mathbb{R}}_{+}^{d}$,

    2. (b)

${\mathbf{I}}:={\mathbb{N}}\times{\mathbb{Z}}^{d}$ if
$\Pi_{d}=\Lambda_{d}^{1}$ is the Lorentz light cone,

    3. (c)

${\mathbf{I}}:={\mathtt{Symm}}_{d}^{+}(\mathbb{Z})$ if
$\Pi_{d}={\mathtt{Symm}}_{d}^{+}(\mathbb{R})\subset{\mathbb{M}}_{d}({\mathbb{R}})$.

  2. (2)

The range of summation $j\in\\{1,2,\ldots,N\\}=[1,N]\cap\mathbb{N}$ in the
partial sum $\sum\limits_{j=0}^{N}A_{j}^{\varepsilon}$ (with
$\varepsilon\in\\{-,+,\circ\\}$) in the formula (1), is replaced by the finite
summation $\sum\limits_{\xi\in[0,\rho]_{{\mathbf{I}}}}A_{\xi}^{\varepsilon}$,
where $[0,\rho]_{{\mathbf{I}}}:=[0,\rho]\cap{\mathbf{I}}$ for $\rho\in\Pi_{d}$
and $[0,\rho]:=\\{\xi\in\Pi_{d}:0\preceq\xi\preceq\rho\\}$ is an interval in
the positive cone $\Pi_{d}$.

  3. (3)

The replacement of the convergence $N\rightarrow+\infty$ in the formula (2)
has to be formulated separately for each positive symmetric cone under
consideration. We use the notation $\rho\xrightarrow[]{\Pi}\infty$ in the
following cases:

     * •

For $\Pi_{d}={\mathbb{R}}_{+}^{d}$ and $\rho:=(a_{1},\cdots,a_{d})\in\Pi_{d}$,
then $\rho\xrightarrow[]{\Pi}\infty$ means that $a_{j}\rightarrow\infty$ for
all $1\leq j\leq d$,

     * •

For $\Pi_{d}=\Lambda_{d}^{1}$ and $\rho:=(t;x_{1},\ldots,x_{d})\in\Pi_{d}$,
then $\rho\xrightarrow[]{\Pi}\infty$ means that
$t-\left(\sum_{i=1}^{d}x_{i}^{2}\right)^{\frac{1}{2}}\rightarrow\infty$,

     * •

For $\Pi_{d}={\mathtt{Symm}}_{d}^{+}(\mathbb{R})$ and $\rho\in\Pi_{d}$, if
$0<\rho_{1}\leq\rho_{2}\leq\cdots\leq\rho_{d}$ are the eigenvalues of $\rho$,
then $\rho\xrightarrow[]{\Pi}\infty$ means that $\rho_{1}\rightarrow\infty$
(and consequently $\rho_{j}\rightarrow\infty$ for all $1\leq j\leq d$).

  4. (4)

For a function $f:\Pi_{d}\mapsto{\mathbb{R}}$, we define
$\lim\limits_{\rho\xrightarrow[]{\Pi}\infty}f(\rho)=\alpha$ if for each
$\epsilon>0$ there exists $\mu\in\Pi_{d}$ such that for every $\rho\in\Pi_{d}$
if $\mu\preceq\rho$ then $|f(\rho)-\alpha|<\epsilon$.

  5. (5)

Furthermore, the normalization factor $\sqrt{N}$ in (1) is replaced by
$\sqrt{\\#[0,\rho]_{\mathbf{I}}}$, where $\\#[0,\rho]_{\mathbf{I}}$ is the
number of elements in $[0,\rho]_{\mathbf{I}}:=[0,\rho]\cap\mathbf{I}$, which
is asymptotically the same as the Euclidean volume
$\mathtt{v}(\rho):=\mathtt{vol}[0,\rho]$ of an interval
$[0,\rho]\subset\Pi_{d}$.

For the reader’s convenience we recall the following formulas for the
Euclidean volume of intervals $[0,\rho]$ (cf. [4]):

    1. (a)

$\displaystyle\mathtt{v}(\rho)=\prod_{j=1}^{d}a_{j}$ if
$\rho:=(a_{1},\ldots,a_{d})\in\Pi_{d}=\mathbb{R}_{+}^{d}$,

    2. (b)

$\displaystyle\mathtt{v}(\rho)=\alpha_{d}(t^{2}-\|x\|^{2})^{\frac{d+1}{2}}$
for some constant $\alpha_{d}$, if $\rho:=(t;x)\in\Pi_{d}=\Lambda_{d}^{1}$,

    3. (c)

$\displaystyle\mathtt{v}(\rho)=\beta_{d}\left(\prod_{j=1}^{d}\lambda_{j}\right)^{\frac{d+1}{2}}=\beta_{d}\Big{(}\det(\rho)\Big{)}^{\frac{d+1}{2}}$
for some constant $\beta_{d}$, if $(\lambda_{1},\ldots,\lambda_{d})$ are the
eigenvalues of
$\rho\in\Pi_{d}={{\mathtt{Symm}}_{d}^{+}(\mathbb{R})}\subset{\mathbb{M}}_{d}(\mathbb{R})$.

With these changes our bm-analogue of the formula (1) can be written as

(3)
$S_{\rho}(\lambda):=\frac{1}{\sqrt{\mathtt{v}(\rho)}}\sum_{\xi\in[0,\rho]_{\mathbf{I}}}(A_{\xi}^{+}+A_{\xi}^{-})+\lambda\sum_{\xi\in[0,\rho]_{\mathbf{I}}}A_{\xi}^{\circ}.$

The main goal of our study is the limit

$\lim_{\rho\xrightarrow[]{\Pi}\infty}\varphi((S_{\rho}(\lambda))^{p}),\quad
p\in{\mathbb{N}},$

which we shall describe in a combinatorial manner.

Our starting point is the construction of _discrete bm-Fock spaces_ (for each
positive symmetric cone) and the creation $A_{\xi}^{+}$, annihilation
$A_{\xi}^{-}$ and conservation $A_{\xi}^{\circ}$ operators on it. The
construction is related to the bm-product Fock space [17] and generalizes the
ideas of Muraki.

In [20] the second-named author provided a model for bm-independence, where
bm-Fock space has been constructed and the algebras
$\\{\mathcal{A}_{\xi}:\xi\in\mathbb{X}\\}$, generated by creation
$A^{+}_{\xi}:=A^{+}(e_{\xi})$ and annihilation $A^{-}_{\xi}:=A^{-}(e_{\xi})$
operators, are bm-independent if the vectors $\\{e_{\xi}:\xi\in\mathbb{X}\\}$
are mutually orthogonal. This construction, however, is not suitable for the
present study, which requires a discrete version of it. Therefore, in the next
section, we present new constructions of discrete bm-Fock spaces, each of
which is related with the given positive symmetric cone.

##  2\. Preliminaries

In this section we present the framework for our study.

A _noncommutative probability space_ $(\mathcal{A},\varphi)$ consists of a
unital $C^{*}$-algebra $\mathcal{A}$ and a state $\varphi$ on it, that is, a
linear functional $\varphi:\mathcal{A}\rightarrow{\mathbb{C}}$ which is
positive $(\varphi(a^{*}a)\geq 0$ for all $a\in\mathcal{A})$ and unital
$(\varphi(1_{\mathcal{A}})=1$ for the unit $1_{\mathcal{A}}\in\mathcal{A})$.
Self-adjoint elements $a=a^{*}\in\mathcal{A}$ are called _noncommutative
random variables_ and the distribution of a random variable
$a=a^{*}\in\mathcal{A}$ is a probability measure $\mu$ with the moments given
by the sequence $\displaystyle m_{n}(\mu):=\varphi(a^{n})$. It always exists,
since the sequence of moments is positive definite.

General references on noncommutative probability are [5, 7, 13] and references
therein.

###  2.1. bm-independence

The general formulation of _bm-independence_ was given by the second-named
author in [20] for families of algebras indexed by partially ordered sets. If
$(\mathbb{X},\preceq)$ is a poset, we shall use the following notation: $x\sim
y$ if $x,y\in\mathbb{X}$ are comparable, $x\nsim y$ if $x,y\in\mathbb{X}$ are
incomparable and $x\prec y$ if $x\preceq y$ and $x\neq y$.

######  Definition 2.1 (bm-independence).

Let $(\mathcal{A},\varphi)$ be a noncommutative probability space. We say that
a family $\\{\mathcal{A}_{\xi}:\xi\in{\mathbb{X}}\\}$ of subalgebras of
$\mathcal{A}$, indexed by a partially ordered set $({\mathbb{X}},\preceq)$, is
bm-independent in $(\mathcal{A},\varphi)$ if the following conditions hold:

  * •

${\mathtt{BM1}}:$ If $\xi,\rho,\eta\in{\mathbb{X}}$ satisfy:
$\xi\prec\rho\succ\eta$ or $\xi\nsim\rho\succ\eta$ or $\xi\prec\rho\nsim\eta$,
then for any
$a_{1}\in\mathcal{A}_{\xi},a_{2}\in\mathcal{A}_{\rho},a_{3}\in\mathcal{A}_{\eta}$
we have

(4) $a_{1}a_{2}a_{3}=\varphi(a_{2})a_{1}a_{3}.$

  * •

${\mathtt{BM2}}:$ If
$\xi_{1}\succ\cdots\succ\xi_{m}\nsim\cdots\nsim\xi_{k}\prec\xi_{k+1}\prec\cdots\prec\xi_{n}$
for some $1\leq m\leq k\leq n$ and $\xi_{1},\cdots,\xi_{n}\in{\mathbb{X}}$,
with $a_{j}\in\mathcal{A}_{\xi_{j}}$ for $1\leq j\leq n$, then

(5) $\varphi(a_{1}\cdots a_{n})=\prod_{j=1}^{n}\varphi(a_{j}).$

Noncommutative random variables
$\\{a_{\xi}\in\mathcal{A}:\xi\in{\mathbb{X}}\\}$ are called _bm-independent_
if the subalgebras $\mathcal{A}_{\xi}$ they generate are bm-independent. For
more information about properties of bm-independence we refer to [20, 6].

######  Remark 2.2.

In the above definition, if ${\mathbb{X}}$ is totally ordered (i.e. every two
elements in ${\mathbb{X}}$ are comparable), then we obtain the monotone
independence. On the other hand, if $\mathbb{X}$ is totally disordered (i.e.
none of the elements of ${\mathbb{X}}$ are comparable), then we obtain the
Boolean independence.
The two conditions ${\mathtt{BM1}}$ and ${\mathtt{BM2}}$ allow to compute all
joint moments $\varphi(a_{1}\cdots a_{n})$ for bm-independent random variables
$a_{1},\ldots,a_{n}$ via marginals $\varphi_{j}:=\varphi|\mathcal{A}_{j}$,
i.e. by the restriction of $\varphi$ to the subalgebras they generate (c.f.
[20], Lemmas 2.3, 2.4). The idea is that first one applies ${\mathtt{BM1}}$ as
many times as possible, and then what remains is subject to ${\mathtt{BM2}}$.
We also refer to [12] for an algorithm which allows to evaluate explicitly
joint moments using the conditions ${\mathtt{BM1}}$ and ${\mathtt{BM2}}$.

###  2.2. Noncrossing partitions with pair or singleton blocks and bm-orders

For $p\in{\mathbb{N}}$, a _partition_ $\pi$ of the finite set
$[p]:=\\{1,\ldots,p\\}$ is a collection of disjoint nonempty subsets of $[p]$,
called _blocks_ , whose union is $[p]$. For a partition $\pi$ with $k$ blocks
$B_{1},\ldots,B_{k}$ we will write $\pi:=(B_{1},\ldots,B_{k})$ to indicate
that the blocks are ordered by their minimal elements:
$\min(B_{j})<\min(B_{j+1})$ for $1\leq j\leq k-1$. We denote by
$\mathcal{P}(p)$ the set of all partitions $\pi$ of $[p]$, and by $|B|$ the
cardinality of a block $B$ in $\pi$. If $|B|=1$, then the block $B$ consists
of one element and is called _singleton_. The numbers of blocks in $\pi$ will
be denoted by $b(\pi)$ and the number of singletons of $\pi$ by $s(\pi)$. One
says that a partition $\pi$ has a _crossing_ if there exists two distinct
blocks $B_{i}$ and $B_{j}$ in $\pi$, and elements $u_{1},u_{2}\in
B_{i},v_{1},v_{2}\in B_{j}$ such that $u_{1}<v_{1}<u_{2}<v_{2}$. Otherwise,
$\pi$ has _no crossings_ and it is called a noncrossing partition. The set of
all _noncrossing partitions_ of $[p]$ will be denoted by $\mathcal{NC}(p)$ (or
by $\mathcal{NC}(p,k)$ for noncrocssing partitions in $\mathcal{NC}(p)$ with
exactly $k$ blocks).
Pictorially, a partition $\pi\in\mathcal{P}(p)$ will be presented by drawing
the integers 1 to $p$ (from right to left) on a horizontal line and then
joining the elements of a blocks by lines above; in particular singletons are
marked by vertical lines. For instance, the graphical representation of the
partition

$\pi=(\\{1\\},\\{2,4\\},\\{3,5\\},\\{6,11\\},\\{7,9,10\\},\\{8\\})\in\mathcal{P}(11),$

is the following figure:

1110987654321

This partition has two singletons $\\{1\\}$ and $\\{8\\}$ and a crossing
between the blocks $\\{2,4\\}$ and $\\{3,5\\}$. Of course, singletons do not
produce crossings.

######  Definition 2.3 (Inner and outer blocks of a noncrossing partition).

A block $B_{j}\in\pi\in\mathcal{NC}(p)$ is called inner if there exists
another block $B_{i}$ such that

$\min B_{i}<\min B_{j}\leq\max B_{j}<\max B_{i},$

where $\min B$ (resp. $\max B$) denotes the minimal (resp. maximal) element of
the block $B$. Otherwise, $B_{j}$ is called an outer block.

Observe that in the noncrossing partition every block is either inner or
outer.

It is convenient to define a partial order $\preceq_{\pi}$ on the blocks of a
noncrossing partition $\pi=(B_{1},\ldots,B_{k})\in\mathcal{NC}(p,k)$. Namely,
we will write $B_{i}\preceq_{\pi}B_{j}$ if $B_{j}$ is inside $B_{i}$, that is

$\min B_{i}<\min B_{j}\leq\max B_{j}<\max B_{i}.$

A noncrossing partition $\pi=(B_{1},\ldots,B_{k})\in\mathcal{NC}(p,k)$ is
called a _pair partition_ (notation: $\pi\in\mathcal{NC}_{2}(p)$) if
$|B_{j}|=2$ for all $1\leq j\leq k$ (each block $B_{i}$ contains exactly two
elements). On the other hand, a partition $\pi\in\mathcal{NC}(p)$ is called a
_noncrossing partition with pair or singleton blocks_ if $|B|\in\\{1,2\\}$ for
all $B\in\pi$; the set of all such partitions will be denoted by
$\mathcal{NC}_{2}^{1}(p)$. We distinguish two subsets of
$\mathcal{NC}_{2}^{1}(p)$ which will appear in the combinatorial descriptions
of our Poisson type limit distribution in the next section:

  * •

The set $\mathcal{NC}_{2.o}^{1,i}(p)\subset\mathcal{NC}_{2}^{1}(p)$, in which
the partitions have no inner pair blocks and no outer singletons, i.e. pair
blocks must be outer and singletons must be inner.

  * •

The set $\mathcal{NC}_{2}^{1,i}(p)\subset\mathcal{NC}_{2}^{1}(p)$ in which the
partitions have no outer singletons (i.e. singletons must be inner blocks, but
pair blocks can be either inner or outer).

Let $(\xi_{p},\ldots,\xi_{1})$ be a sequence of elements of ${\mathbb{X}}$.
For $\xi\in\\{\xi_{p},\ldots,\xi_{1}\\}$, we define a subset $B(\xi):=\\{1\leq
j\leq p:\xi_{j}=\xi\\}\subset[p]$, and denote by $i=i(\xi):=\min B(\xi)$ being
the minimal element. If the cardinality of the set
$\\{\xi_{p},\ldots,\xi_{1}\\}$ is $k$, then we obtain $k$ disjoint subsets
$B_{1},\ldots,B_{k}$, which form a partition of $[p]$. Re-ordering the blocks
to get that $\min(B_{i})<\min(B_{i+1})$ for $1\leq i\leq k-1$, we obtain then
a partition $\pi=(B_{1},\ldots,B_{k})\in\mathcal{P}(p)$. In this case, we say
that the partition $\pi$ is adapted to the sequence $(\xi_{p},\ldots,\xi_{1})$
and we denote this by $(\xi_{p},\ldots,\xi_{1})\sim\pi$. Observe that the
partition adapted to a sequence is unique.

For a finite subset $\mathbf{J}\subset\mathbf{I}$ and a partition
$\pi=(B_{1},\ldots,B_{k})\in\mathcal{P}(p)$ with $k$ blocks, we define the
label function $\mathtt{L}:\pi\longrightarrow\mathbf{J}$ by
$\mathtt{L}(B_{i})\in\mathbf{J}$ for $1\leq i\leq k$. If we consider
$(\xi_{p},\ldots,\xi_{1})\sim\pi$ as above, then for
$\xi\in\\{\xi_{p},\ldots,\xi_{1}\\}$, $\mathtt{L}(B(\xi)):=\xi$ will be called
the label of the block $B(\xi)$, and
$(\mathtt{L}(B_{k}),\ldots,\mathtt{L}(B_{1}))$ will be called the label
sequence of the partition $\pi$.

######  Remark 2.4.

If a partition $\pi=(B_{1},\ldots,B_{k})\in\mathcal{P}(p)$ is adapted to a
sequence $(\xi_{p},\ldots,\xi_{1})$, then the set
$\\{\xi_{p},\ldots,\xi_{1}\\}$ of elements and the set
$\\{\mathtt{L}(B_{k}),\ldots,\mathtt{L}(B_{1})\\}$ of labels are equal.
Moreover, the label sequence $(\mathtt{L}(B_{k}),\ldots,\mathtt{L}(B_{1}))$ is
uniquely defined.

Following [2], let $\pi=(B_{1},\ldots,B_{k})\in\mathcal{NC}_{2}^{1}(p)$ be a
given partition and let $B_{i}$ and $B_{j}$ be two blocks, which for any other
block $B_{l}\in\pi$, satisfy

$\text{if $B_{i}\preceq_{\pi}B_{l}\preceq_{\pi}B_{j}$ then $B_{i}=B_{l}$ or
$B_{j}=B_{l}$}.$

Then we say that the block $B_{j}$ is a direct successor of the block $B_{i}$;
equivalently, the block $B_{i}$ is a direct predecessor of the block $B_{j}$.

######  Example 2.5.

Consider $\pi=(B_{1},B_{2},B_{3},B_{4},B_{5})\in\mathcal{NC}_{2}(8)$ with
$B_{1}=\\{1,8\\},B_{2}=\\{2,4\\},B_{3}=\\{3\\},B_{4}=\\{5,7\\}$ and
$B_{5}=\\{6\\}$. Then, graphically,

$\pi=$$B_{1}$$B_{4}$$B_{2}$$B_{5}$$B_{3}$.$8$.$7$.$6$.$5$.$4$.$3$.$2$.$1$

and

  * •

$B_{1}$ is a direct predecessor of $B_{2}$ and $B_{4}$;

  * •

$B_{2}$ and $B_{4}$ are direct successors of $B_{1}$;

  * •

$B_{3}$ is a direct successor of $B_{2}$;

  * •

$B_{5}$ is a direct successor of $B_{4}$.

For a given partition $\pi\in\mathcal{P}(p)$, we define the partition
$\tilde{\pi}$ as that obtained from $\pi$ by removing all singletons,
$\tilde{\pi}$ will be called the reduced partition of $\pi$. Obviously, for
$\pi\in\mathcal{NC}_{2}(p)$, we have $\tilde{\pi}=\pi$, but in general, for
$\pi\in\mathcal{NC}_{2}^{1}(p)$ $\tilde{\pi}$ is a subpartition of $\pi$. For
instance,

if $\pi=$$\in\mathcal{NC}_{2}^{1}(11)$ then
$\tilde{\pi}=$$\in\mathcal{NC}_{2}(6).$

######  Definition 2.6 (bm-order on noncrossing partitions with pair or
singleton blocks).

Let $(\xi_{p},\ldots,\xi_{1})$ be a given sequence of elements from a
partially ordered set $(\mathbb{X},\preceq)$, such that
$(\xi_{p},\ldots,\xi_{1})\sim\pi=(B_{1},\ldots,B_{k})\in\mathcal{NC}_{2}^{1}(p)$,
and let $\mathtt{L}:\pi\longrightarrow(\mathbb{X},\preceq)$ be a label
function. We say that the sequence $\xi:=(\xi_{p},\ldots,\xi_{1})$ defines bm-
order on the partition $\pi$ (notation $\xi\trianglelefteq\pi$) if for all
$1\leq i\neq j\leq k$ the following conditions hold:

  1. (1)

If $|B_{j}|=2$ and $B_{i}\prec_{\pi}B_{j}$, then
$\mathtt{L}(B_{i})\preceq\mathtt{L}(B_{j})$;

  2. (2)

If $|B_{j}|=1,B_{i}\prec_{\pi}B_{j}$ and $B_{j}$ is a direct successor of
$B_{i}$, then $\mathtt{L}(B_{i})=\mathtt{L}(B_{j})$.

We say that the sequence defines strict bm-order on the partition $\pi$
(notation $\xi\vartriangleleft\pi$) if for all $1\leq i\neq j\leq k$ the
condition $(1)$ reads as: $B_{i}\prec_{\pi}B_{j}$ implies that
$\mathtt{L}(B_{i})\prec\mathtt{L}(B_{j})$.

Note that, in a bm-ordered partition a singleton block must have the same
label as its direct predecessor and pair blocks are labeled increasingly.

Remark 2.4 suggest the following:

######  Definition 2.7.

Let $\xi:=(\xi_{p},\ldots,\xi_{1})\in\mathbb{X}^{p}$ be a given sequence with
the adapted partition $\pi=(B_{1},\ldots,B_{k})\in\mathcal{NC}_{2}^{1}(p)$ and
$\mu:=(\mathtt{L}(B_{k}),\ldots,\mathtt{L}(B_{1}))$ be the label sequence of
blocks $B_{1},\ldots,B_{k}\in\pi$. If $\xi\trianglelefteq\pi$ (resp.
$\xi\vartriangleleft\pi$), we say that the label sequence $\mu$ defines bm-
order (resp. defines strict bm-order) on $\pi$ and we will use the notation
$\mu\trianglelefteq_{\mathtt{L}}\pi$ (resp.
$\mu\vartriangleleft_{\mathtt{L}}\pi$).

Our proofs of Poisson type limit theorems on discrete bm-Fock spaces
eventually reduce to the study of the following combinatorial sets.

######  Definition 2.8.

For $\pi\in\mathcal{NC}_{2}^{1}(p,k)$ and $\rho\in\Pi_{d}$ we define the
following sets of sequences from $\mathbf{I}$:

  1. (1)

the set

$\mathrm{bmo}(\pi,\rho):=\\{\xi=(\xi_{1},\ldots,\xi_{p})\in[0,\rho]_{{\mathbf{I}}}^{p}:\xi\trianglelefteq\pi\\}$

of all sequences of elements which satisfy $0\preceq\xi_{j}\preceq\rho$,
$\xi_{j}\in\mathbf{I}$ ($1\leq j\leq p$), and which define bm-order on $\pi$;

  2. (2)

the set

$\mathrm{BMO}(\pi,\rho):=\\{\mu=(\mu_{1},\ldots,\mu_{k}):\mu_{i}=\mathtt{L}(B_{i})\in[0,\rho]_{{\mathbf{I}}},\quad
1\leq i\leq k\quad and\quad\mu\trianglelefteq_{\mathtt{L}}\pi\\}$

of all label sequences of elements which satisfy $0\preceq\mu_{j}\preceq\rho$,
$\mu_{j}\in\mathbf{I}$ ($1\leq j\leq k$), and define bm-order
$\pi\in\mathcal{NC}_{2}^{1}(p,k)$;

  3. (3)

the set

$\mathtt{BMO}(\pi,\rho):=\\{\mu=(\mu_{1},\ldots,\mu_{k}):\mu_{i}=\mathtt{L}(B_{i})\in[0,\rho]_{{\mathbf{I}}},\quad
1\leq i\leq k\quad and\quad\mu\vartriangleleft_{\mathtt{L}}\pi\\}$

of all label sequences of elements which satisfy $0\preceq\mu_{j}\preceq\rho$,
$\mu_{j}\in\mathbf{I}$ ($1\leq j\leq k$), and define strict bm-order on
$\pi\in\mathcal{NC}_{2}^{1}(p,k)$.

######  Remark 2.9.

If $\pi\in\mathcal{NC}_{2}(p)$ is a pair partition, then the sets
$\mathrm{BMO}(\pi,\rho)$ and $\mathtt{BMO}(\pi,\rho)$ are equal.

We conclude this subsection by recalling the following result from [12] which
plays a significant role in this study.

######  Theorem 2.10 ([12]).

Let $\pi\in\mathcal{NC}(p,k)$ be a noncrossing partition of $p$ elements with
$1\leq b(\pi)=k\leq p$ blocks, then for each positive symmetric cone $\Pi_{d}$
which we consider, there exists the limit

$\lim_{\rho\xrightarrow[]{\Pi}\infty}\frac{|\mathtt{BMO}(\pi,\rho)|}{\mathtt{v}(\rho)^{k}}=V(\pi),$

where the function $V(\pi):=V_{\Pi_{d}}(\pi)$ depends on the cone $\Pi_{d}$
and the volume characteristic sequence $(\gamma_{n}(\Pi_{d}))_{n\geq 1}$, and
satisfies the following recursive formula:

$V(\pi)=\left\\{\begin{array}[]{ll}1&\hbox{if $b(\pi)=1$ or $\pi=\emptyset$
,}\\\ \gamma_{b(\pi)}\cdot\prod\limits_{i=1}^{k}V(\pi_{i})&\hbox{if
$\put(0.0,0.0){$\pi=$}$ \put(1.5,0.0){\line(0,1){2.5}} \put(1.6,1.0){ $...$}
\put(3.0,0.0){\line(0,1){2.5}} \put(3.1,1.0){ $\pi_{1}$}
\put(4.5,0.0){\line(0,1){2.5}} \put(4.9,1.0){$...$}
\put(6.0,0.0){\line(0,1){2.5}} \put(6.1,1.0){ $\pi_{2}$}
\put(7.5,0.0){\line(0,1){2.5}} \put(7.6,1.0){\Huge$...$}
\put(9.2,0.0){\line(0,1){2.5}} \put(9.3,1.0){ $\pi_{k}$}
\put(10.7,0.0){\line(0,1){2.5}} \put(11.1,1.0){$...$}
\put(12.3,0.0){\line(0,1){2.5}} \put(1.5,2.5){\line(1,0){10.8}}
\put(12.5,0.0){ ,} \par}\\\ \prod\limits_{i=1}^{k}V(\pi_{i}),&\hbox{if
$\pi=\pi_{1}\cup\pi_{2}\cup\ldots\cup\pi_{k}$.}\end{array}\right.$

Here, the notation in the second case is understood as a partition $\pi$ with
one outer block with $m$ element (vertical lines) and inside it there are $k$
arbitrary partitions, $\pi_{1},\ldots,\pi_{k}$. In addition, by
$\pi=\pi_{1}\cup\pi_{2}\cup\ldots\cup\pi_{k}$, we mean a disjoint union of
sub-partitions with exactly one outer block (which could be as well a
singleton).

##  3\. Main results

###  3.1. Discrete bm-Fock space and related operators

Let us consider a family $\\{\mathcal{H}_{\xi}:\
\xi\in{\mathbf{I}}\subset\Pi_{d}\\}$ of Hilbert spaces, indexed by the
discrete set $\mathbf{I}$, with orthonormal basis
$\\{e_{\xi}^{m}\in\mathcal{H}_{\xi}:m=0,1,2,3,\cdots\\}$. We assume that
$\Omega:=e_{\xi}^{0}$ for all $\xi\in{\mathbf{I}}$ is a common unit vector,
called the _vacuum vector_.

Following the ideas of Muraki [8, 9], for every $n\in{\mathbb{N}}$, we define

$\varXi_{n}:=\\{(\rho_{n},\ldots,\rho_{1})\in\mathbf{I}^{n}:\quad\rho_{n}\succ\rho_{n-1}\succ\cdots\succ\rho_{1}\\}.$

The _discrete bm-Fock space_ , denoted by $\mathcal{F}_{bm}^{d}(\mathbf{I})$,
is the Hilbert space spanned by the _vacuum vector_ $\Omega$ and the simple
tensors of the form $h_{\rho_{n}}\otimes\cdots\otimes h_{\rho_{1}}$, where
$h_{\rho}\in\mathcal{H}_{\rho},h_{\rho}\bot\Omega$ and
$(\rho_{n},\ldots,\rho_{1})\in\varXi_{n}$; the scalar product is given by

$\langle h_{\rho},h_{\eta}\rangle=0\text{ if }\rho\neq\eta$

and

$\langle h_{\rho_{n}}\otimes\cdots\otimes
h_{\rho_{1}},f_{\rho_{m}}\otimes\cdots\otimes
f_{\rho_{1}}\rangle=\delta_{mn}\prod_{i=1}^{n}\langle
h_{\rho_{i}},f_{\rho_{i}}\rangle,$

where $\langle h_{\rho_{i}},f_{\rho_{i}}\rangle$ is the scalar product in
$\mathcal{H}_{\rho_{i}}$. Note that the discrete bm-Fock space
$\mathcal{F}_{bm}^{d}(\mathbf{I})$ is the _bm-product_ [17] of a family of
Hilbert spaces $\\{\mathcal{H}_{\xi},\xi\in\mathbf{I}\\}$, indexed by elements
in $\mathbf{I}\subset\Pi_{d}$.

In the following, we define the creation, the annihilation and the
conservation operators on the discrete bm-Fock space
$\mathcal{F}_{bm}^{d}(\mathbf{I})$, and show their properties. Let
$g_{\xi}\in\mathcal{H}_{\xi}$ be a unit vector, and let
$(\rho_{n},\ldots,\rho_{1})\in\varXi_{n}$ be an index sequence such that
$h_{\rho_{j}}\in\mathcal{H}_{\rho_{j}}$ for $1\leq j\leq n$.

  1. (1)

The creation operator $A_{g_{\xi}}^{+}$ is defined as follows:
$A_{g_{\xi}}^{+}\Omega=g_{\xi}$,
$A_{g_{\xi}}^{+}(h_{\rho_{n}}\otimes\cdots\otimes
h_{\rho_{1}})=\left\\{\begin{array}[]{ll}g_{\xi}\otimes
h_{\rho_{n}}\otimes\cdots\otimes h_{\rho_{1}}&\hbox{if $\xi\succ\rho_{n}$,}\\\
0&\hbox{ otherwise.}\end{array}\right.$

  2. (2)

The annihilation operator $A_{g_{\xi}}^{-}$ is defined by
$A_{g_{\xi}}^{-}\Omega=0$,
$A_{g_{\xi}}^{-}(h_{\rho_{n}}\otimes\cdots\otimes
h_{\rho_{1}})=\left\\{\begin{array}[]{ll}\langle
g_{\xi},h_{\rho_{n}}\rangle\cdot h_{\rho_{n-1}}\otimes\cdots\otimes
h_{\rho_{1}}&\hbox{if $\xi=\rho_{n}$,}\\\ 0&\hbox{otherwise
.}\end{array}\right.$

  3. (3)

The conservation operator $A_{g_{\xi}}^{\circ}$ is defined by
$A_{g_{\xi}}^{\circ}\Omega=0,$
$A_{g_{\xi}}^{\circ}(h_{\rho_{n}}\otimes\cdots\otimes
h_{\rho_{1}})=\left\\{\begin{array}[]{ll}\langle
g_{\xi},h_{\rho_{n}}\rangle\cdot h_{\rho_{n}}\otimes\cdots\otimes
h_{\rho_{1}}&\hbox{if $\xi=\rho_{n}$,}\\\
0&\hbox{otherwise.}\end{array}\right.$

It follows that these operators are bounded and that $A_{g_{\xi}}^{+}$ and
$A_{g_{\xi}}^{-}$ are mutually adjoint (i.e.
$(A_{g_{\xi}}^{+})^{*}=A_{g_{\xi}}^{-}$). Moreover, the conservation operator
$A_{g_{\xi}}^{\circ}$ is self-adjoint. In addition, we also have the following
commutation relations, for the unit vectors $g_{\xi}\in\mathcal{H}_{\xi}$ and
$g_{\eta}\in\mathcal{H}_{\eta}$

(6)
$A_{g_{\eta}}^{+}A_{g_{\xi}}^{+}=A_{g_{\xi}}^{-}A_{g_{\eta}}^{-}=0\quad\quad(\xi\succeq\eta)$

and

(7)
$A_{g_{\xi}}^{-}A_{g_{\eta}}^{\circ}=A_{g_{\xi}}^{\circ}A_{g_{\eta}}^{+}=A_{g_{\xi}}^{-}A_{g_{\eta}}^{+}=A_{g_{\xi}}^{\circ}A_{g_{\eta}}^{\circ}=0\quad\quad(\eta\neq\xi).$

For a fixed family of unit vectors $\\{g_{\xi}\in\mathcal{H}_{\xi}:\
\xi\in\mathbf{I}\\}$, we denote

$A_{\xi}^{\varepsilon}:=A_{g_{\xi}}^{\varepsilon}\text{ for
}\varepsilon\in\\{\circ,-,+\\}.$

Let $\mathcal{A}$ be the $C^{*}$-algebra of all bounded operators on
$\mathcal{F}_{bm}^{d}(\mathbf{I})$ and let $\varphi(A):=\langle
A\Omega,\Omega\rangle$ for $A\in\mathcal{A}$ be the _vacuum state_. For
$\xi\in{\mathbf{I}}$, let
$\mathcal{A}_{\xi}:=\text{alg}\\{A_{\xi}^{+},A_{\xi}^{-}\\}$ be the
$*$-algebra generated by $A_{\xi}^{+}$ and $A_{\xi}^{-}$. In a similar way as
in [20] one shows the following theorem.

######  Theorem 3.1.

The algebras $\\{\mathcal{A}_{\xi},\ \xi\in{\mathbf{I}}\\}$ are bm-independent
in $(\mathcal{A},\varphi)$.

###  3.2. Poisson type limit theorems

In this subsection we will investigate the limits distributions of the
operators $S_{\rho}(\lambda)$ defined on the discrete bm-Fock space
$\mathcal{F}_{bm}^{d}(\mathbf{I})$ by

(8)
$S_{\rho}(\lambda):=\frac{1}{\sqrt{\mathtt{v}(\rho)}}\sum_{\xi\in[0,\rho]_{\mathbf{I}}}(A_{\xi}^{+}+A_{\xi}^{-})+\lambda\sum_{\xi\in[0,\rho]_{\mathbf{I}}}A_{\xi}^{\circ},$

as $\rho\xrightarrow[]{\Pi}\infty$ under the vacuum state $\varphi$. To this
end we define

$m_{p}(\lambda):=\lim_{\rho\xrightarrow[]{\Pi}\infty}\varphi\left((S_{\rho}(\lambda))^{p}\right),$

the limit of $p$-th moments for such operators.
For each $\xi\in\Pi_{d}$ and $g_{\xi}\in\mathcal{H}_{\xi}$ such that $\lVert
g_{\xi}\rVert=1$, as in the previous section we will use the notations
$A_{\xi}^{+}:=A_{g_{\xi}}^{+},A_{\xi}^{-}:=A_{g_{\xi}}^{-}$ and
$A_{\xi}^{\circ}:=A_{g_{\xi}}^{\circ}$. Furthermore we assign the numbers
$-1,0,+1$ to the variable $\varepsilon$ according to $\varepsilon=-,\circ,+$.

Now we present formulation of our main results, which are the analogues of the
classical Poisson limit theorem for bm-independent random variables, indexed
by elements of positive symmetric cones.

######  Theorem 3.2.

For any positive integer $p\geq 2$, one has

(9)
$m_{p}(\lambda)=\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}\lambda^{s(\pi)}\cdot
V(\tilde{\pi}),$

where

(10)
$V(\tilde{\pi})=\lim\limits_{\rho\xrightarrow[]{\Pi}\infty}\frac{|\mathrm{BMO}(\pi,\rho)|}{(\mathtt{v}(\rho))^{b(\tilde{\pi})}}.$

Here $b(\tilde{\pi})$ is the number of blocks in the reduced partition
$\tilde{\pi}$ of $\pi$ and the limit (10) exists for every partition
$\pi\in\mathcal{NC}_{2}^{1,i}(p)$. The combinatorial function
$V(\tilde{\pi}):=V_{\Pi_{d}}(\tilde{\pi})$ depends on the positive cone
$\Pi_{d}$ and its volume characteristic $\gamma_{n}(\Pi_{d})$. Moreover, it is
multiplicative and satisfies the following recursive formula

(11) $V(\tilde{\pi})=\left\\{\begin{array}[]{ll}1&\hbox{if $b(\tilde{\pi})=1$
or $\tilde{\pi}=\emptyset$}\\\ \gamma_{|\tilde{\pi}|}(\Pi_{d})\cdot
V(\pi^{\prime})&\hbox{if $\par\put(0.2,0.0){$\tilde{\pi}=$}$
\begin{picture}(5.0,2.0)\put(1.3,0.0){\line(0,1){1.5}}
\put(3.0,0.2){$\pi^{\prime}$} \put(5.0,0.0){\line(0,1){1.5}}
\put(1.3,1.5){\line(1,0){3.72}} \end{picture}}\\\
\prod\limits_{i=1}^{k}V(\pi_{i})&\hbox{if \hskip
3.98337pt$\tilde{\pi}=\pi_{1}\cup\pi_{2}\cup\cdots\cup\pi_{k}$,}\end{array}\right.$

###  3.3. Examples and remarks

The following examples and figures explain the rules for the calculations of
$V(\tilde{\pi})$ for a given partition $\pi\in\mathcal{NC}_{2}^{1,i}(p)$. For
example, consider the following partition $\pi$:

$\pi=\\{\\{1,15\\},\\{2\\},\\{3,9\\},\\{4,8\\},\\{5\\},\\{6\\},\\{7\\},\\{10,13\\},\\{11\\},\\{12\\},\\{14\\}\\}\in\mathcal{NC}_{2}^{1,i}(15)$

Then we can draw the following pictures of $\pi$ and $\tilde{\pi}$

$\pi=$ and $\tilde{\pi}=$

Recall that, for $\Pi_{2}={\mathbb{R}}_{+}^{2}$ and $\Pi_{1}=\Lambda_{1}^{1}$,
the volume characteristic sequence is $\gamma_{n}=\frac{1}{n^{2}}$, and for
$\Pi_{3}={\mathbb{R}}_{+}^{3}$, $\gamma_{n}=\frac{1}{n^{3}}$, while for
$\Pi_{2}=\Lambda_{2}^{1}$ and $\Pi_{2}={\mathtt{Symm}}_{2}^{+}(\mathbb{R})$,
$\gamma_{n}=\frac{24}{(3n-1)3n(3n+1)}$. Now we can compute $V(\tilde{\pi})$
for such cones using the recursive formula (11):

$1.\quad\Pi_{2}={\mathbb{R}}_{+}^{2}$ and $\Pi_{1}=\Lambda_{1}^{1}.$

V($\tilde{\pi}$)=V( )=$\frac{1}{16}\cdot$ V
(\begin{picture}(5.0,2.0)\par\put(0.19,0.0){\line(0,1){1.5}}
\put(0.19,1.5){\line(1,0){1.32}} \put(1.49,0.0){\line(0,1){1.5}}
\par\end{picture}

)$\cdot V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.5}}
\put(0.2,1.5){\line(1,0){1.52}} \put(0.5,0.0){\line(0,1){1.0}}
\put(1.4,0.0){\line(0,1){1.0}} \put(0.5,1.0){\line(1,0){0.9}}
\put(1.7,0.0){\line(0,1){1.5}} \end{picture}$)=$\frac{1}{16}\cdot
1\cdot\frac{1}{4}\cdot
V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.0}}
\put(1.1,0.0){\line(0,1){1.0}} \put(0.2,1.0){\line(1,0){0.9}}
\end{picture})=\frac{1}{64}$

$2.\quad\Pi_{3}={\mathbb{R}}_{+}^{3}.$

V($\tilde{\pi}$)=V(

)=$\frac{1}{64}\cdot$
V(\begin{picture}(5.0,2.0)\par\put(0.2,0.0){\line(0,1){1.5}}
\put(0.2,1.5){\line(1,0){1.32}} \put(1.5,0.0){\line(0,1){1.5}}
\par\end{picture}

)$\cdot V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.5}}
\put(0.2,1.5){\line(1,0){1.52}} \put(0.5,0.0){\line(0,1){1.0}}
\put(1.4,0.0){\line(0,1){1.0}} \put(0.5,1.0){\line(1,0){0.9}}
\put(1.7,0.0){\line(0,1){1.5}} \end{picture}$)=$\frac{1}{64}\cdot
1\cdot\frac{1}{8}\cdot
V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.0}}
\put(1.1,0.0){\line(0,1){1.0}} \put(0.2,1.0){\line(1,0){0.9}}
\end{picture})=\frac{1}{512}$

$3.\quad\Pi_{2}=\Lambda_{2}^{1}$ and
$\Pi_{2}={\mathtt{Symm}}_{2}^{+}(\mathbb{R})$.

V($\tilde{\pi}$)=V( )=$\frac{2}{143}\cdot$
V(\begin{picture}(5.0,2.0)\par\put(0.2,0.0){\line(0,1){1.5}}
\put(0.2,1.5){\line(1,0){1.32}} \put(1.5,0.0){\line(0,1){1.5}}
\par\end{picture}

)$\cdot V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.5}}
\put(0.2,1.5){\line(1,0){1.52}} \put(0.5,0.0){\line(0,1){1.0}}
\put(1.4,0.0){\line(0,1){1.0}} \put(0.5,1.0){\line(1,0){0.9}}
\put(1.7,0.0){\line(0,1){1.5}} \end{picture}$)=$\frac{2}{143}\cdot
1\cdot\frac{12}{105}\cdot
V(\begin{picture}(5.0,2.0)\put(0.2,0.0){\line(0,1){1.0}}
\put(1.1,0.0){\line(0,1){1.0}} \put(0.2,1.0){\line(1,0){0.9}}
\end{picture})=\frac{8}{505}$

For illustration, the first few moments
($m_{1}(\lambda),\ldots,m_{6}(\lambda)$) of the monotone Poisson distribution
(cf. [9]) in comparison with some of the bm-Poisson analogues are listed below
(observe that always $m_{1}(\lambda)=0,m_{2}(\lambda)=1$ and
$m_{3}(\lambda)=\lambda$).

Monotone Poisson distribution |  |  |  |  bm-case for $\Pi_{3}={\mathbb{R}}_{+}^{3}$
---|---|---|---|---
$0$ |  |  |  |  $0$
$1$ |  |  |  |  $1$
$\lambda$ |  |  |  |  $\lambda$
$\lambda^{2}+\frac{3}{2}$ |  |  |  |  $\lambda^{2}+\frac{9}{8}$
$\lambda^{3}+\frac{7}{2}\lambda$ |  |  |  |  $\lambda^{3}+\frac{19}{8}\lambda$
$\lambda^{4}+\frac{9}{2}\lambda^{2}+\frac{5}{2}$ |  |  |  |  $\lambda^{4}+\frac{15}{4}\lambda^{2}+\frac{31}{24}$

bm-cases for $\Pi_{2}=\mathbb{R}_{+}^{2}$ and $\Pi_{1}=\Lambda_{1}^{1}$ |  |  |  |  bm-cases for $\Pi_{2}=\Lambda_{2}^{1}$ and $\Pi_{2}={\mathtt{Symm}}_{2}^{+}(\mathbb{R})$
---|---|---|---|---
$0$ |  |  |  |  $0$
$1$ |  |  |  |  $1$
$\lambda$ |  |  |  |  $\lambda$
$\lambda^{2}+\frac{5}{4}$ |  |  |  |  $\lambda^{2}+\frac{39}{35}$
$\lambda^{3}+\frac{11}{4}\lambda$ |  |  |  |  $\lambda^{3}+\frac{12}{35}\lambda$
$\lambda^{4}+\frac{9}{2}\lambda^{2}+\frac{59}{36}$ |  |  |  |  $\lambda^{4}+\frac{129}{35}\lambda^{2}+\frac{443}{350}$

In the special case $\lambda=0$, the operators (8) become

$S_{\rho}(0)=\frac{1}{\sqrt{\mathtt{v}(\rho)}}\sum_{\xi\preceq\rho,\hskip
2.84544pt\xi\in\mathbf{I}}(A_{\xi}^{+}+A_{\xi}^{-}).$

Since $\\{A_{\xi}^{+}+A_{\xi}^{-},\xi\in{\mathbf{I}}\\}$ are bm-independent on
$(\mathcal{A},\varphi)$, then we obtain a new example of noncommutative “de
Moivre-Laplace theorem” which is a kind of central limit theorem. Namely the
limit distribution of the operator $S_{\rho}(0)$ at
$\rho\xrightarrow[]{\Pi}\infty$ under the vacuum state $\varphi$ is given by

$m_{2n+1}(0)=0\quad\text{ and }\quad
m_{2n}(0)=\sum_{\pi\in\mathcal{NC}_{2}(2n)}V(\pi).$

On the other hand, by the bm-central limit theorem [20] the even moments
$m_{2n}(0)$ can be written as follows

$m_{2n}(0)=g_{n}=\sum_{k=1}^{n}\gamma_{k}(\Pi_{d})g_{k-1}g_{n-k},\quad\quad
g_{0}=g_{1}=1.$

Hence, we have another combinatorial description for the bm-central limit
theorem associated with symmetric cones [20].
It is worthwhile to mention that it is also possible to extend the results of
this paper to two classes of positive cones studied in [11] which are non-
symmetric. Namely, the sectorial cones

$\Omega_{\bm{u}}^{n}:=\biggl{\\{}\sum_{j=1}^{n}a_{j}u_{j}:\bm{u}:=(u_{1},\ldots,u_{n}),a_{j}\geq
0,u_{j}\in\mathbb{R}^{n}\text{ and }u_{1},\ldots,u_{n}\text{ are linearly
independent}\biggr{\\}}$

and the circular cones

$C_{\theta}^{n}:=\\{(t;x)\in\mathbb{R}_{+}\times\mathbb{R}^{n}:||x||\leq
t\cdot\tan\theta\\}.$

In what follows we will consider the subspace of
$\mathcal{F}_{bm}^{d}(\mathbf{I})$

$\mathcal{F}_{0}:=\text{ span }\\{\Omega,g_{\xi_{n}}\otimes\cdots\otimes
g_{\xi_{1}}:\xi_{n}\succ\xi_{n-1}\succ\cdots\succ\xi_{1},\ n\in\mathbb{N}\\},$

and by
$\mathtt{B}_{\xi}^{\varepsilon}:=A_{\xi}^{\varepsilon}|_{{}_{\mathcal{F}_{0}}}$
for $\varepsilon\in\\{-,\circ,+\\}$ and $\xi\in\mathbf{I}$ we will denote the
restrictions of $A_{\xi}^{\varepsilon}$ to the subspace $\mathcal{F}_{0}$.
Observe that each operator $A_{\xi}^{\varepsilon}$ preserves the subspace
$\mathcal{F}_{0}$, and moreover

(12) $\mathtt{B}_{\xi}^{\circ}=\mathtt{B}_{\xi}^{+}\mathtt{B}_{\xi}^{-}\text{
is in }\mathcal{F}_{0}.$

Let $\mathcal{C}$ be the unital $*$-algebra of all bounded operators on
$\mathcal{F}_{0}$. Hence, if we define $\mathcal{C}_{\xi}$ to be the
$*$-algebra generated by $\mathtt{B}_{\xi}^{+},\mathtt{B}_{\xi}^{-}$ and
$\mathtt{B}_{\xi}^{\circ}$, we get the following crucial Corollary.

######  Corollary 3.3.

The algebras $\\{\mathcal{C}_{\xi},\xi\in{\mathbf{I}}\\}$ are bm-independent
with respect to $\varphi$ in $(\mathcal{C},\varphi).$

###  3.4. Proof of Theorem 3.2

The proof of Theorem 3.2 consists of several reductions and combinatorial
considerations. First we will show that the limit (9) can be reduced
combinatorially to a sum over noncrossing partitions $\pi$ with pair or inner
singleton blocks only. Next, we shall prove that the only terms which survive
in the limit are those which satisfy the bm-ordered labellings
$\mu\trianglelefteq_{\mathtt{L}}\pi$ for a partition $\pi$. Finally, we shall
show that the limit when $\rho\xrightarrow[]{\Pi}\infty$ is the ratio of the
cardinality $|\mathrm{BMO}(\pi,\rho)|$ and the Euclidean volume
$\mathtt{v}(\rho)^{b(\tilde{\pi})}$ for $\pi\in\mathcal{NC}_{2}^{1,i}(p)$,
where $\tilde{\pi}$ is the reduced partition of $\pi$. The rest of the proof
then goes in a similar manner as for Theorem 6.1 in [12].

####  3.4.1. Combinatorial reduction

We start the proof observing that $\varphi((S_{\rho}(\lambda))^{p})$ can be
written as

$\displaystyle\varphi((S_{\rho}(\lambda))^{p})$
$\displaystyle=\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{(\varepsilon_{p},\ldots,\varepsilon_{1})\in\\{-1,0,+1\\}^{p}}\hskip
5.69046pt\sum_{(\xi_{p},\ldots,\xi_{1})\in[0,\rho]_{\mathbf{I}}^{p}}\varphi(A_{\xi_{p}}^{\varepsilon_{p}}\cdots
A_{\xi_{1}}^{\varepsilon_{1}})\lambda_{\rho}^{\sum_{i=1}^{p}\delta_{0}(\varepsilon_{i})}$
(13)
$\displaystyle=\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{(\varepsilon_{p},\ldots,\varepsilon_{1})\in\\{-1,0,+1\\}^{p}}\hskip
5.69046pt\sum_{(\xi_{p},\ldots,\xi_{1})\in[0,\rho]_{\mathbf{I}}^{p}}\varphi(\mathtt{B}_{\xi_{p}}^{\varepsilon_{p}}\cdots\mathtt{B}_{\xi_{1}}^{\varepsilon_{1}})\lambda_{\rho}^{\sum_{i=1}^{p}\delta_{0}(\varepsilon_{i})}$

where

$\delta_{0}(\varepsilon_{i})=\left\\{\begin{array}[]{ll}1&\hbox{if
$\varepsilon_{i}=0$}\\\ 0&\hbox{otherwise,}\end{array}\right.$

and $\lambda_{\rho}=\lambda\cdot\sqrt{\mathtt{v}(\rho)}.$
For $p\in{\mathbb{N}}$, we will use the following notations
$\bm{\varepsilon}:=(\varepsilon_{p},\ldots,\varepsilon_{1})\in\\{-1,0,+1\\}^{p}$,
$\bm{\xi}:=(\xi_{p},\ldots,\xi_{1})\in[0,\rho]_{{\mathbf{I}}}^{p}$ and
$\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}}:=\mathtt{B}_{\xi_{p}}^{\varepsilon_{p}}\cdots\mathtt{B}_{\xi_{1}}^{\varepsilon_{1}}$.
By a simple induction argument on $p$, one obtains the following lemma which
describes a necessary conditions for non-vanishing of the mixed moments
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})$. The reader is refereed to
[2] for similar arguments in the setting of weakly monotone Fock space and to
[3] in the setting of interacting Fock space.

######  Lemma 3.4.

For a positive integer $p\geq 2$, let $\bm{\varepsilon}\in\\{-1,0,+1\\}^{p}$
and $\bm{\xi}\in[0,\rho]_{{\mathbf{I}}}^{p}$.
If $\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$, then the
sequence $\bm{\varepsilon}$ satisfies the following three conditions:

  1. (1)

$\varepsilon_{1}=+1,\varepsilon_{p}=-1,$

  2. (2)

$\sum\limits_{i=1}^{p}\varepsilon_{i}=0,$

  3. (3)

$\sum\limits_{i=1}^{k}\varepsilon_{i}\geq 0$ for $k=1,\ldots,p-1.$

From now on, by ${\mathtt{D}}_{p}$ we denote the set of all sequences
$\bm{\varepsilon}=(\varepsilon_{p},\ldots,\varepsilon_{1})\in\\{-1,0,+1\\}^{p}$
satisfying the conditions (1)-(3) in the statement of Lemma 3.4. According to
the same Lemma and using the above notations, (3.4.1) becomes

(14)
$\varphi((S_{\rho}(\lambda))^{p})=\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{\bm{\varepsilon}\in\mathtt{D}_{p}}\hskip
5.69046pt\sum_{\bm{\xi}\in[0,\rho]_{{\mathbf{I}}}^{p}}\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\lambda_{\rho}^{\sum_{i=1}^{p}\delta_{0}(\varepsilon_{i})}.$

For a fixed sequence $\bm{\varepsilon}\in\mathtt{D}_{p}$ and for each $1\leq
k\leq p-1$ ($p\geq 2$) such that $\varepsilon_{k}=+1$, by (2) and (3) in Lemma
3.4 one can define

(15) $T(k):=\min\\{l:l>k\quad
s.t.\quad\varepsilon_{k}+\cdots+\varepsilon_{l}=0\\}.$

By definition, $\varepsilon_{T(k)}=-1$. Furthermore, if there exists
$k<m<T(k)$ such that $\varepsilon_{m}=+1$, then

$\varepsilon_{k}+\cdots+\varepsilon_{m-1}>0\text{ and
}\varepsilon_{m}+\varepsilon_{m+1}+\cdots+\varepsilon_{T(k)}<0.$

This implies that

$k<m<T(m)<T(k).$

Therefore, if we define a partition

$\pi:=\\{\\{k,T(k)\\}:\varepsilon_{k}=+1\\}\cup\\{\\{j\\}:\varepsilon_{j}=0\\},$

we obtain $\pi\in\mathcal{NC}_{2}^{1}(p)$. In other words, for any sequence
$\bm{\varepsilon}\in\mathtt{D}_{p}$ we can uniquely associate a noncrossing
partition with blocks being a pair or a singleton, and we will often use the
identification
$\mathtt{D}_{p}\ni\bm{\varepsilon}\equiv\pi\in\mathcal{NC}_{2}^{1}(p)$.

######  Lemma 3.5.

For $p\geq 2$, let
$\bm{\varepsilon}\in\mathtt{D}_{p},\bm{\xi}\in[0,\rho]_{{\mathbf{I}}}^{p}$ and
$1\leq k\leq p-1$ such that $\varepsilon_{k}=+1$. If
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$ then
$\xi_{k}=\xi_{T(k)}.$

Proof: We use induction over $p\geq 2$:
For $p=2$, we have just one pair block $\\{1,T(1)=2\\}$, then (7) gives the
thesis.
For $p=3$, we have the sequences $\bm{\varepsilon}=(-1,0,+1)$ and
$\bm{\xi}=(\xi_{3},\xi_{2},\xi_{1})$. Hence, $T(1)=3$ and the assumption
$\varphi(\mathtt{B}_{\xi}^{\varepsilon})\neq 0$ yields
$\xi_{1}=\xi_{2}=\xi_{3}=\xi_{T(1)}$.
We assume that the lemma holds for all $r<p$ and that
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$. Consider

$\text{$1\leq k_{0}\leq p-1$ such that $\varepsilon_{k_{0}}=+1$ and
$T(k_{0})\leq T(k)$ for all $1\leq k\leq p-1$ with $\varepsilon_{k}=+1$},$

that is,

$T(k_{0}):=\min\\{T(k):1\leq k\leq p-1\\}.$

Then, if $|T(k_{0})-k_{0}|\geq 2$, we have $\varepsilon_{j}=0$ for all
$k_{0}<j<T(k_{0})$, and if there exists $1\leq i\leq k_{0}$, then
$\varepsilon_{i}\in\\{0,+1\\}$. Suppose initially that $T(k_{0})=p$, hence
$k_{0}=1$. In this case,

$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=\varphi(\mathtt{B}_{\xi_{p}}^{-}\mathtt{B}_{\xi_{p-1}}^{\circ}\cdots\mathtt{B}_{\xi_{2}}^{\circ}\mathtt{B}_{\xi_{1}}^{+})\neq
0.$

Hence,

$\xi_{1}=\xi_{2}=\cdots=\xi_{p-1}=\xi_{p}=\xi_{T(1)}.$

If instead $T(k_{0})<p$, the assumption
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$ yields

$\mathtt{B}_{\xi_{k_{0}}}^{+}(\mathtt{B}_{\xi_{k_{0}-1}}^{\varepsilon_{k_{0}-1}}\cdots\mathtt{B}_{\xi_{1}}^{+})\Omega\neq
0,$

thus

$\mathtt{B}_{\xi_{k_{0}}}^{+}(\mathtt{B}_{\xi_{k_{0}-1}}^{\varepsilon_{k_{0}-1}}\cdots\mathtt{B}_{\xi_{1}}^{+})\Omega=\mathtt{B}_{\xi_{k_{0}}}^{+}w=g_{\xi_{k_{0}}}\otimes
w\neq 0,$

where

$w=g_{\xi_{s_{l}}}\otimes\cdots\otimes g_{\xi_{s_{1}}}\otimes
g_{\xi_{1}}\text{ and
}\xi_{s_{1}},\ldots,\xi_{s_{l}}\in\\{\xi_{2},\ldots,\xi_{k_{0}-1}\\}\text{
with }\xi_{k_{0}}\succ\xi_{s_{l}}\succ\cdots\succ\xi_{s_{1}}\succ\xi_{1}.$

Consequently, since

$\mathtt{B}_{\xi_{T(k_{0})-1}}^{\circ}\cdots\mathtt{B}_{\xi_{k_{0}+1}}^{\circ}g_{\xi_{k_{0}}}\otimes
w\neq 0\Longrightarrow\xi_{T(k_{0})-1}=\cdots=\xi_{k_{0}+1}=\xi_{k_{0}},$

and

$\mathtt{B}_{\xi_{T(k_{0})}}^{-}(\mathtt{B}_{\xi_{T(k_{0})-1}}^{\circ}\cdots\mathtt{B}_{\xi_{k_{0}+1}}^{\circ})g_{\xi_{k_{0}}}\otimes
w=\mathtt{B}_{\xi_{T(k_{0})}}^{-}(\mathtt{B}_{\xi_{T(k_{0})}}^{\circ})^{n}g_{\xi_{k_{0}}}\otimes
w=\mathtt{B}_{\xi_{T(k_{0})}}^{-}g_{\xi_{k_{0}}}\otimes w\neq 0,$

where $n=T(k_{0})-1-k_{0}$.
Therefore, $\xi_{T(k_{0})}=\xi_{k_{0}}\text{ and
}\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=\varphi(\mathtt{B}_{\bm{\xi}^{\prime}}^{\bm{\varepsilon}^{\prime}})$,
where

$\bm{\varepsilon}^{\prime}=(\varepsilon_{p},\ldots,\varepsilon_{T(k_{0})+1},\varepsilon_{k_{0}-1},\ldots,\varepsilon_{1})\text{
and
}\bm{\xi}^{\prime}=(\xi_{p},\ldots,\xi_{T(k_{0})+1},\xi_{k_{0}-1},\ldots,\xi_{1}).$

Since $r=p-(n+2)<p$, the thesis follows after exploiting the induction
assumption for the sequences $\bm{\varepsilon}^{\prime}$ and
$\bm{\xi}^{\prime}$. $\Box$

######  Corollary 3.6.

Let $\bm{\varepsilon}\in\mathtt{D}_{p}$ and
$\bm{\xi}\in[0,\rho]_{\mathbf{I}}^{p}$. If
$\varphi(\mathtt{B}_{\bm{\varepsilon}}^{\bm{\xi}})\neq 0$, then the singletons
in the partition
$\pi=\\{\\{k,T(k)\\},\varepsilon_{k}=+1\\}\cup\\{\\{j\\},\varepsilon_{j}=0\\}$
which is associated with the sequence $\bm{\varepsilon}$, cannot be outer
blocks, i.e. $\pi\in\mathcal{NC}_{2}^{1,i}(p)$.

Using Lemma 3.5 and Corollary 3.6, the identity (14) becomes:

(16)
$\varphi((S_{\rho}(\lambda))^{p})=\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}\hskip
2.84544pt\sum_{[0,\rho]_{{\mathbf{I}}}^{p}\ni\bm{\xi}\sim\pi}\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\lambda^{s(\pi)},$

where the second summation is over all sequence
$\bm{\xi}=(\xi_{p},\ldots,\xi_{1})\in[0,\rho]_{{\mathbf{I}}}^{p}$ associated
with a given partition $\pi\in\mathcal{NC}_{2}^{1,i}(p)$.

####  3.4.2. Reduction to bm-ordered noncrossing partitions with pair or
singleton blocks:

In the next lemma we will show that the only terms which survive in the limit
(9) come from those sequences establishing a bm-order on noncrossing
partitions with pair or inner singleton blocks, i.e., those satisfying the two
conditions in Definition 2.6.

######  Lemma 3.7 (bm-order).

Let $\pi\in\mathcal{NC}_{2}^{1,i}(p)$ be a noncrossing partition with pair or
inner singleton blocks given by the sequences
$\bm{\varepsilon}\in\mathtt{D}_{p}$ and $\bm{\xi}\in[0,\rho]_{\mathbf{I}}^{p}$
such that $\bm{\xi}\sim\pi$. If
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$ then the sequence
$\bm{\xi}$ establishes a bm-order on $\pi$, i.e. $\bm{\xi}\unlhd\pi$.

Proof: By induction on $p\geq 2$:
For $p=2$, we have $\bm{\varepsilon}=(-1,+1),\bm{\xi}=(\xi_{2},\xi_{1})$ and
$\pi=\\{\\{1,2\\}\\}$. Then
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$ implies that
$\xi_{1}=\xi_{2}$ and $(\xi_{2},\xi_{1})$ establishes bm-order on
$\pi=\\{\\{1,2\\}\\}$.
For $p=3$, our sequences are $\bm{\varepsilon}=(-1,0,+1)$ and
$\bm{\xi}=(\xi_{3},\xi_{2},\xi_{1})$ with the associated partition
$\pi=\\{\\{1,3\\},\\{2\\}\\}$. Then

$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq
0\Longrightarrow\mathtt{B}_{\xi_{3}}^{-}\mathtt{B}_{\xi_{2}}^{0}\mathtt{B}_{\xi_{1}}^{+}\Omega\neq
0\Longrightarrow\xi_{1}=\xi_{2}=\xi_{3},$

and $\bm{\xi}=(\xi_{3},\xi_{2},\xi_{1})$ establishes bm-order on
$\pi=\\{\\{1,3\\},\\{2\\}\\}$.
Let us consider
$\bm{\xi}=(\xi_{p},\ldots,\xi_{1})\sim\pi\in\mathcal{NC}_{2}^{1,i}(p)$ and
$\bm{\varepsilon}=(\varepsilon_{p},\ldots,\varepsilon_{1})\in\mathtt{D}_{p}$.
We assume that the statement is true for each $q<p$ and
$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq 0$. We have the
following two cases, depending on the position of $T(1)$.

  1. (1)

$T(1)<p:$
Denote
$C:=\mathtt{B}_{\xi_{T(1)}}^{\varepsilon_{T(1)}}\cdots\mathtt{B}_{\xi_{1}}^{+}$
and
$D:=\mathtt{B}_{\xi_{p}}^{-}\cdots\mathtt{B}_{\xi_{T(1)+1}}^{\varepsilon_{T(1)+1}}$.
As $\xi_{1}=\xi_{T(1)}$ and $\varepsilon_{T(1)}=-1$, one gets

$C\Omega=\Omega\text{ and
}0\neq\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=\varphi(D).$

Then $\varepsilon_{T(1)+1}=+1$.
If we denote by
$\bm{\varepsilon}^{\prime}:=(\varepsilon_{T(1)},\ldots,\varepsilon_{1}),\bm{\xi}^{\prime}:=(\xi_{T(1)},\ldots,\xi_{1}),\bm{\varepsilon}^{\prime\prime}:=(\varepsilon_{p},\ldots,\varepsilon_{T(1)+1})$
and $\bm{\xi}^{\prime\prime}:=(\xi_{p},\ldots,\xi_{T(1)+1})$, then
$\bm{\varepsilon}^{\prime}\equiv\pi^{\prime}\in\mathcal{NC}_{2}^{1,i}(T(1))$
and
$\bm{\varepsilon}^{\prime\prime}\equiv\pi^{\prime\prime}\in\mathcal{NC}_{2}^{1,i}(p-T(1))$,
where $\bm{\xi}^{\prime}\sim\pi^{\prime}$ and
$\bm{\xi}^{\prime\prime}\sim\pi^{\prime\prime}$.
Hence,

$1=\varphi(C)=\varphi(\mathtt{B}_{\bm{\xi}^{\prime}}^{\bm{\varepsilon}^{\prime}})\neq
0\quad
and\quad\varphi(D)=\varphi(\mathtt{B}_{\bm{\xi}^{\prime\prime}}^{\bm{\varepsilon}^{\prime\prime}})\neq
0.$

Since $T(1)<p$ and $p-T(1)<p$, we can use the induction hypothesis for
$\varphi(\mathtt{B}_{\bm{\xi}^{\prime}}^{\bm{\varepsilon}^{\prime}})\neq 0$ to
obtain that $\bm{\xi}^{\prime}$ establishes bm-order on $\pi^{\prime}$, and
for
$\varphi(\mathtt{B}_{\bm{\xi}^{\prime\prime}}^{\bm{\varepsilon}^{\prime\prime}})\neq
0$ to obtain that $\bm{\xi}^{\prime\prime}$ establishes bm-order on
$\pi^{\prime\prime}$.
However, there is no relation between the sequence
$\bm{\xi}^{\prime}\sim\pi^{\prime}$ and the sequence
$\bm{\xi}^{\prime\prime}\sim\pi^{\prime\prime}$, i.e. for
$\eta\in\bm{\xi}^{\prime}$ and $\mu\in\bm{\xi}^{\prime\prime}$ all three
possibilities $\eta\succeq\mu,\eta\preceq\mu$ and $\eta\nsim\mu$ are allowed.
Therefore, $\bm{\xi}$ establishes bm-order on
$\pi=\pi^{\prime}\cup\pi^{\prime\prime}$.

  2. (2)

$T(1)=p:$
As $\varepsilon_{p}=-1$, let us take $j=\min\\{i:2\leq i\leq
p,\varepsilon_{i}=-1\\}$. Note that if $j=2$, then $T(1)=2=p$, and if $j=p$
(we have one outer pair block $\\{1,p\\}$ and $(p-2)$ inner singletons), then
$\varepsilon_{i}=0$ and $\xi_{i}=\xi_{1}=\xi_{T(1)}$ for $i=2,\ldots,p-1$. In
both cases, the sequence $\bm{\xi}$ establishes bm-order on
$\pi\in\mathcal{NC}_{2}^{1,i}(p)$, and

$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=\varphi(\mathtt{B}_{\xi_{p}}^{-}\mathtt{B}_{\xi_{p-1}}^{\circ}\cdots
B^{\circ}_{\xi_{\xi_{2}}}\mathtt{B}_{\xi_{1}}^{+})=\varphi(\mathtt{B}_{\xi_{p}}^{-}\mathtt{B}_{\xi_{1}}^{+})=1.$

If instead $2<j<p$, then $\varepsilon_{l}\in\\{0,+1\\}$, for all $2\leq l\leq
j-1$, the following two cases appear:

    1. (a)

There exists $2\leq l\leq j-1$ such that $\varepsilon_{l}=0$. Here, we define

$l^{\prime}=\min\\{l:2\leq l\leq j-1,\varepsilon_{l}=0\\},$

and

$\varphi(\mathtt{B}_{\bm{\varepsilon}}^{\bm{\xi}})=\varphi(\mathtt{B}_{\xi_{p}}^{\varepsilon_{p}}\mathtt{B}_{\xi_{p-1}}^{\varepsilon_{p-1}}\cdots\mathtt{B}_{\xi_{j+1}}^{\varepsilon_{j+1}}\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}+1}}^{\varepsilon_{l^{\prime}+1}}\mathtt{B}_{\xi_{l^{\prime}}}^{\circ}\mathtt{B}_{\xi_{l^{\prime}-1}}^{+}\cdots
B^{+}_{\xi_{2}}B^{+}_{\xi_{1}}\Omega)\neq 0.$

Therefore,

$\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}+1}}^{\varepsilon_{l^{\prime}+1}}\mathtt{B}_{\xi_{l^{\prime}}}^{\circ}\mathtt{B}_{\xi_{l^{\prime}-1}}^{+}\cdots
B^{+}_{\xi_{2}}B^{+}_{\xi_{1}}\Omega\neq 0.$

By (6) and (7), one has

$\xi_{l^{\prime}}=\xi_{l^{\prime}-1}\succ\xi_{l^{\prime}-2}\cdots\succ\xi_{2}\succ\xi_{1}$

and

$\displaystyle\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}}}^{\circ}\mathtt{B}_{\xi_{l^{\prime}-1}}^{+}\cdots\mathtt{B}_{\xi_{2}}^{+}\mathtt{B}_{\xi_{1}}^{+}\Omega$
$\displaystyle=\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}+1}}^{\varepsilon_{l^{\prime}+1}}\mathtt{B}_{\xi_{l^{\prime}}}^{\circ}(g_{\xi_{l^{\prime}-1}}\otimes\cdots\otimes
g_{\xi_{1}})$
$\displaystyle=\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}+1}}^{\varepsilon_{l^{\prime}+1}}(g_{\xi_{l^{\prime}-1}}\otimes\cdots\otimes
g_{\xi_{1}})$
$\displaystyle=\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{\varepsilon_{j-1}}\cdots\mathtt{B}_{\xi_{l^{\prime}+1}}^{\varepsilon_{l^{\prime}+1}}\mathtt{B}_{\xi_{l^{\prime}-1}}^{+}\cdots\mathtt{B}_{\xi_{2}}^{+}\mathtt{B}_{\xi_{1}}^{+}\Omega\neq
0.$

Moreover, the action of $\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}}$ on $\Omega$
is the same as the action of
$\mathtt{B}_{\bm{\xi}^{\prime}}^{\bm{\varepsilon}^{\prime}}$ on $\Omega$,
where
$\bm{\varepsilon}^{\prime}=(\varepsilon_{p},\ldots,\varepsilon_{l^{\prime}+1},\varepsilon_{l^{\prime}-1},\ldots,\varepsilon_{1})\in\mathtt{D}_{p-1},\bm{\xi}^{\prime}=(\xi_{p},\ldots,\xi_{l^{\prime}+1},\xi_{l^{\prime}-1},\ldots,\xi_{1})\in[0,\rho]_{{\mathbf{I}}}^{p-1}$
and $\bm{\xi}^{\prime}\sim\pi^{\prime}\in\mathcal{NC}_{2}^{1,i}(p-1)$. We can
now use the induction hypothesis to obtain that $\bm{\xi}^{\prime}$
establishes bm-order on
$\pi^{\prime}=\pi\setminus\\{l^{\prime}\\}\in\mathcal{NC}_{2}^{1,i}(p-1)$.
Moreover, since the block singleton $\\{l^{\prime}\\}$ is a direct successor
of the pair block $\\{l^{\prime}-1,T(l^{\prime}-1)\\}$ and
$\xi_{l^{\prime}}=\xi_{l^{\prime}-1}\succ\cdots\succ\xi_{1}$, then
$\bm{\xi}\trianglelefteq\pi\in\mathcal{NC}_{2}^{1,i}(p).$

    2. (b)

For all $2\leq l\leq j-1,\varepsilon_{l}=+1$. In this case, $j=T(j-1)$. Hence,

$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})\neq
0\Longrightarrow\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{+}\cdots\mathtt{B}_{\xi_{2}}^{+}\mathtt{B}_{\xi_{2}}^{+}\Omega\neq
0$

and

$\xi_{j}=\xi_{j-1}\succ\xi_{j-2}\succ\cdots\succ\xi_{1}.$

Consequently,

$\displaystyle\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{+}\cdots\mathtt{B}_{\xi_{1}}^{+}\Omega$
$\displaystyle=\mathtt{B}_{\xi_{j}}^{-}\mathtt{B}_{\xi_{j-1}}^{+}(g_{\xi_{j-2}}\otimes\cdots\otimes
g_{\xi_{1}})$ $\displaystyle=g_{\xi_{j-2}}\otimes
g_{\xi_{j-3}}\otimes\cdots\otimes g_{\xi_{1}}$
$\displaystyle=\mathtt{B}_{\xi_{j-2}}^{+}\mathtt{B}_{\xi_{j-3}}^{+}\cdots\mathtt{B}_{\xi_{1}}^{+}\Omega,$

and

$\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=\varphi(\mathtt{B}_{\xi_{p}}^{-}\cdots\mathtt{B}_{\xi_{j+1}}^{\varepsilon_{j+1}}\mathtt{B}_{\xi_{j-2}}^{+}\cdots\mathtt{B}_{\xi_{1}}^{+})=\varphi(\mathtt{B}_{\bm{\xi}^{\prime}}^{\bm{\varepsilon}^{\prime}})\neq
0,$

where

$\bm{\xi}^{\prime}:=(\xi_{p},\ldots,\xi_{j+1},\xi_{j-2},\ldots,\xi_{1})\sim\pi^{\prime}\in\mathcal{NC}_{2}^{1,i}(p-2)$

and

$\bm{\varepsilon}^{\prime}:=(\varepsilon_{p},\ldots,\varepsilon_{j+1},\varepsilon_{j-2},\ldots,\varepsilon_{1})\in\mathtt{D}_{p-2},\text{
where }\varepsilon_{1}=\cdots=\varepsilon_{j-2}=+1.$

The induction assumption applied to the sequences $\bm{\varepsilon}^{\prime}$
and $\bm{\xi}^{\prime}$, implies

$\bm{\xi}^{\prime}\trianglelefteq\pi^{\prime}=\pi\setminus\\{j-1,j\\}\in\mathcal{NC}_{2}^{1,i}(p-2).$

Since $\xi_{j}=\xi_{j-1}\succ\xi_{j-2}\cdots\succ\xi_{1}$, we have
$\bm{\xi}\trianglelefteq\pi\in\mathcal{NC}_{2}^{1,i}(p)$.

$\Box$

######  Remark 3.8.

The above arguments show that, for a sequence
$\bm{\xi}=(\xi_{p},\ldots,\xi_{1})\trianglelefteq\pi\in\mathcal{NC}_{2}^{1,i}(p)$,
one has $\varphi(\mathtt{B}_{\bm{\xi}}^{\bm{\varepsilon}})=1$.

By the previous results, the proof of Theorem 3.2 reduces to ascertain that
for each $p\in{\mathbb{N}}$ the following limit exists

$\lim_{\rho\xrightarrow[]{\Pi}\infty}\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}\hskip
2.84544pt\sum_{\bm{\xi}\in\mathrm{bmo}(\pi,\rho)}\varphi(\mathtt{B}_{\xi_{p}}^{\varepsilon_{p}}\cdots\mathtt{B}_{\xi_{1}}^{\varepsilon_{1}})\lambda_{\rho}^{s(\pi)}=\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}V(\tilde{\pi})\lambda^{s(\pi)},$

where $V(\tilde{\pi})$ satisfies the recurrence relation given in (11). Remark
3.8 yields that

$\lim_{\rho\xrightarrow[]{\Pi}\infty}\frac{1}{\mathtt{v}(\rho)^{\frac{p}{2}}}\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}\hskip
2.84544pt\sum_{\bm{\xi}\in{\mathrm{bmo}}(\pi,\rho)}\varphi(\mathtt{B}_{\xi_{p}}^{\varepsilon_{p}}\cdots\mathtt{B}_{\xi_{1}}^{\varepsilon_{1}})\lambda_{\rho}^{s(\pi)}=\sum_{\pi\in\mathcal{NC}_{2}^{1,i}(p)}\hskip
2.84544pt\lim_{\rho\xrightarrow[]{\Pi}\infty}\frac{|\mathrm{bmo}(\pi,\rho)|}{\mathtt{v}(\rho)^{b(\tilde{\pi})}}\lambda^{s(\pi)},$

where $b(\tilde{\pi})=\frac{p-s(\pi)}{2}$ is the number of blocks of the
reduced partition $\tilde{\pi}\in\mathcal{NC}_{2}(p-s(\pi))$ obtained by
$\pi\in\mathcal{NC}_{2}^{1,i}(p)$ after extracting the singleton blocks.
Theorem 3.2 thus follows from the following.

######  Proposition 3.9.

Let $\pi\in\mathcal{NC}_{2}^{1,i}(p)$ be a noncrossing partition with pair or
inner singleton blocks. Then for each positive symmetric cone $\Pi_{d}$

$\lim_{\rho\xrightarrow[]{\Pi}\infty}\frac{|\mathrm{bmo}(\pi,\rho)|}{\mathtt{v}(\rho)^{b(\tilde{\pi})}}=V(\tilde{\pi}),$

where the function $V(\tilde{\pi}):=V_{\Pi_{d}}(\tilde{\pi})$ depends on the
cone $\Pi_{d}$ and can be recursively expressed by the volume characteristic
sequence $(\gamma_{n}(\Pi_{d}))_{n\geq 0}$ as in (11).

Proof: By Theorem 2.10, we need to prove that

(17) $|\mathrm{bmo}(\pi,\rho)|=|\mathtt{BMO}(\tilde{\pi},\rho)|.$

Let us consider the sequence $\bm{\xi}$ with the adapted partition
$\pi=(B_{1},\ldots,B_{\frac{p+s(\pi)}{2}})\in\mathcal{NC}_{2}^{1,i}(p;\frac{p+s(\pi)}{2})$,
where $B_{1},\ldots,B_{\frac{p+s(\pi)}{2}}$ are the $\frac{p+s(\pi)}{2}=k$
blocks of $\pi$, and denote by $\mu:=(\mu_{k},\ldots,\mu_{1})$ the related
label sequence. Since
$[0,\rho]_{\mathbf{I}}^{p}\ni\bm{\xi}\trianglelefteq\pi$, then
$[0,\rho]_{\mathbf{I}}^{k}\ni\mu\trianglelefteq_{\mathtt{L}}\pi$, and by
Remark 2.4, we obtain

$|\mathrm{bmo}(\pi,\rho)|=|\mathrm{BMO}(\pi,\rho)|.$

Moreover, the labels of the direct successors (singleton blocks) are the same
as those of its direct predecessors (pair blocks). Then

$|\mathtt{BMO}(\tilde{\pi},\rho)|=|\mathrm{BMO}(\pi,\rho)|=|\mathrm{bmo}(\pi,\rho)|.$

$\Box$

## Appendix

In this section, we present the results for the single operator
$S(\lambda):=A_{\xi}^{+}+A_{\xi}^{-}+\lambda A_{\xi}^{\circ}$, for $\lambda>0$
and $\xi\in{\mathbf{I}}$. In particular, the $p$-th moment
$a_{p}(\lambda):=\varphi((S(\lambda))^{p})$ and the associated probability
measure $\nu_{\lambda}$ are given. It is worthwhile to mention that they are
the same as those obtained by Muraki [9].

######  Proposition 3.10.

For any positive integer $p\geq 2$, one has

(18)
$a_{p}(\lambda)=\sum_{\pi\in\mathcal{NC}_{2,o}^{1,i}(p)}\lambda^{s(\pi)},$

where $\mathcal{NC}_{2,o}^{1,i}(p)\subset\mathcal{NC}_{2}^{1}(p)$ is the set
of all noncrossing partitions with pair or singleton blocks such that the pair
blocks must be outer and the singletons must be inner.
Furthermore, the following recursive formula holds

(19) $a_{0}(\lambda)=1,a_{1}(\lambda)=0,a_{p}(\lambda)=\lambda
a_{p-1}(\lambda)+a_{p-2}(\lambda),\quad\text{ for }\quad p\geq 2.$

######  Examples 3.11.

For $p=0,1,\ldots,6$ we can use the recursive formula (19), to obtain the
following moment sequence:

$a_{p}(\lambda)=\\{1,0,1,\lambda,\lambda^{2}+1,\lambda^{3}+2\lambda,\lambda^{4}+3\lambda^{2}+1\\}.$

The moment generating function and the Cauchy transform for the moments of
$S(\lambda)=A_{\xi}^{+}+A_{\xi}^{-}+\lambda A_{\xi}^{\circ}$ are given,
respectively, by

(20) $\mathrm{M}_{\lambda}(x)=\frac{1-\lambda x}{1-\lambda x-x^{2}},$

and

(21) $\mathrm{G}_{\lambda}(x)=\frac{x-1}{x^{2}-\lambda x-1}.$

Then, by the Stieltjes inverse formula, the probability distribution
$\nu_{\lambda}$ of $S(\lambda)$ under the vacuum state $\varphi$ is given as
follows

$\nu_{\lambda}=p_{1}\delta_{x_{1}}+p_{2}\delta_{x_{2}},$

where $x_{1}=\frac{\lambda}{2}+\frac{\sqrt{\lambda^{2}+4}}{2}$,
$x_{2}=\frac{\lambda}{2}-\frac{\sqrt{\lambda^{2}+4}}{2}$,
$p_{1}=\frac{1}{2}-\frac{\lambda}{2\sqrt{\lambda^{2}+4}}$,
$p_{2}=\frac{1}{2}+\frac{\lambda}{2\sqrt{\lambda^{2}+4}}$ and $\delta_{x}$
denotes the Dirac measure at a point $x$.

Furthermore, according to Corollary 3.3, one can show that the operators
$\\{S(\lambda)\\}_{\xi}$ are bm-independent with respect to $\varphi$ in
$(\mathcal{C},\varphi)$, and then the operators $\\{S(\lambda)\\}_{\xi}$ can
be considered as _bm-Bernoulli_ random variables.

######  Remark 3.12.

For $\lambda=1$ and $p\geq 1$, the moment sequence $a_{p}(1)$ are the shifted
Fibonacci numbers, given by

$a_{p}(1)=F(p-1).$

## Acknowledgements

Research partially supported by the National Agency for Academic Exchange
(NAWA) POLONIUM project PPN/BIL/2018/1/00197/U/00021 and by the Polish
National Science Center (NCN) grant 2016/21/B/ST1/00628.

## References

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# RX J0123.4-7321 - the story continues: major circumstellar disk loss and
recovery.

M. J. Coe1, A. Udalski2, J. A. Kennea3, P. A. Evans4
1Physics & Astronomy, The University of Southampton, SO17 1BJ, UK
2Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478
Warszawa, Poland
3Department of Astronomy and Astrophysics, The Pennsylvania State University,
525 Davey Lab, University Park, PA 16802, USA
4University of Leicester, Astrophysics Group, School of Physics & Astronomy,
University Road, Leicester LE1 7RH, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

RX J0123.4-7321 is a well-established Be star X-ray binary system (BeXRB) in
the Small Magellanic Cloud (SMC). Like many such systems the variable X-ray
emission is driven by the underlying behaviour of the mass donor Be star.
Previous work has shown that the optical and X-ray were characterised by
regular outbursts at the proposed binary period of 119 d. However around
February 2008 the optical behaviour changed substantially, with the previously
regular optical outbursts ending. Reported here are new optical (OGLE) and
X-ray (Swift) observations covering the period after 2008 which suggest an
almost total circumstellar disc loss followed by a gradual recovery. This
indicates the probable transition of a Be star to a B star, and back again.
However, at the time of the most recent OGLE data (March 2020) the
characteristic periodic outbursts had yet to return to their early state,
indicating that the disk still had some re-building yet to complete.

###### keywords:

stars: emission line, Be X-rays: binaries

††pubyear: 2021††pagerange: RX J0123.4-7321 - the story continues: major
circumstellar disk loss and recovery.–RX J0123.4-7321 - the story continues:
major circumstellar disk loss and recovery.

##  1 Introduction

BeXRB are a large sub-group of the well-established category of High Mass
X-ray Binaries (HMXB) characterised by being a binary system consisting of a
massive mass donor star, normally an OBe type, and an accreting compact
object, a neutron star. They are particularly prevalent in the Small
Magellanic Cloud (SMC) which contains the largest known collection of BeXRBs .
Whilst catalogues have been produced listing such systems in the SMC (for
example Coe & Kirk (2015); Haberl & Sturm (2016)), it clear that the complex
interactions between the two stars continues to produce unexpected surprises.
In particular, the behaviour of the mass donor B-type star is major driver in
the observed characteristics of such systems.

The source that is the subject of this paper, RX J0123.4-7321, was identified
early on as an X-ray source in the SMC (Haberl et al., 2000), but it took over
another decade before its identity as BeXRB was clearly established by Sturm
et al. (2013). In their paper they not only identified the X-ray source as a
partner to the Be star [M2002] SMC 81035, but also used optical photometry
from the OGLE III project to show that the binary period of the system was 119
d.

Reported here are the successive 10 years of OGLE IV data and 4 years of X-ray
observations from the Swift X-ray Telescope (XRT; Burrows et al. 2005) as part
of the S-CUBED project (Kennea et al., 2018). The new OGLE data show a major
change in the behaviour of the Be star as it appears to almost completely lose
its circumstellar disk. This represents a rarely seen transition of [M2002]
SMC 81035 from a classic Be star to a regular B-type star, and back again. All
within a decade. The X-ray data, which are concurrent with the last few years
of OGLE IV, reveal 15 detections of RX J0123.4-7321, with the detections
extremely well correlated with the phase of the optical outbursts. These
epochs are believed to correspond to times of periastron passage of the
neutron star, and the profiles of the optical & X-ray outbursts are very
similar, suggesting a common mechanism related to the periodic distortion of
the circumstellar disk.

##  2 Observations

###  2.1 OGLE

The OGLE project (Udalski et al., 2015) provides long term I-band photometry
with a cadence of 1-3 days. The star [M2002] SMC81035 was observed
continuously for nearly 2 decades until COVID-19 restrictions prevented any
further observations from March 2020. It is identified in the OGLE catalogue
as:

OGLE III (I band): smc121.2.50
OGLE IV (I band): smc733.26.24
OGLE III (V band) : smc121.2.v.18
OGLE IV (V band): smc733.26.v.13

The I band data are shown in their entirety in the top panel of Fig 1. The
OGLE III data were previously presented in Sturm et al. (2013) who reported
overwhelming evidence for the binary period of this system based upon the
sharp regular outburst features seen every 119 d. However this striking
feature ended around TJD 5000 just as the OGLE III project phase also came to
an end. Presented here are the OGLE IV data which add a further 10 years to
the story, and indicate a very dramatic change in behaviour pattern from OGLE
III. For most of this last decade the regular outbursts are missing completely
and only just start to re-emerge around TJD 8500, though not as strongly as
before. Nonetheless, the presence of these features ten years later permit a
small refinement to be made to the binary period ephemeris that was originally
presented in Sturm et al. (2013). The updated ephemeris for the time of the
optical outbursts, $T_{opt}$, is:

$T_{opt}=54387.1+N(119.59)~{}\textrm{MJD}$ (1)

The optical counterpart [M2002] SMC 81035 is reported to be a B0.7 IIIe star
by Massey (2002) and Ramachandran et al. (2019). The exact intrinsic colours
of such a spectral type are not readily tabulated, blackbut the (V-I) colour
of an almost identical B0.5 V star is given in Pecaut & Mamajek (2013) as
-0.338. They do not quote the colours of a giant of the same type, but this
difference is given in (Wegner, 1994) as 0.08. Thus it is expected that the
intrinsic colours of a B0.5III will be (V-I) = -0.258.

The OGLE dust maps of the SMC (Skowron et al., 2021) enable the precise
reddening correction to be made for such an object in the SMC and that is
E(V-I)=0.067. Thus the predicted observed colours of [M2002] SMC 81035 if it
were a B-type star B0.5II with no circumstellar disk will be (V-I) = -0.19.
The difference between a B0.5 and B1 star is less than 0.01 (Pecaut & Mamajek,
2013) so any difference between a B0.5 and B0.7 will be negligible for the
purpose of this discussion. Thus the calculated colour is indicated in the
lower panel of Fig 1 as the horizontal dashed line. From this figure it is
immediately clear that all the observed OGLE colours for this object lie
significantly above the baseline, strongly suggesting the presence of a
circumstellar disk adding further reddening to the observed colours.

Figure 1: Upper panel : OGLE III and IV data - the transition between the two
observing phases occurred around TJD 5000. Lower panel: OGLE (V-I) colour
measurements. The dashed horizontal line indicates the predicted observed
colour of a B0.5 III star in the SMC.

Clearly if the red excess in the colour of [M2002] SMC 81035 is interpreted as
the existence of a circumstellar disk, then the huge variations in the (V-I)
colour seen in Fig 1 must also be indicating significant changes in the disk
size over the last 20 years. From a substantive disk at the start of this
period, through to almost nothing around TJD 5000. Then the star went through
a series of re-growth phases to reclaim some of its original status as a
probable Be type. This is further discussed below.

###  2.2 S-CUBED

RX J0123.4-7321 was detected by the S-CUBED survey (Kennea et al., 2018), a
shallow weekly X-ray survey of the optical extent of the SMC by the Swift
X-ray Telescope (XRT; Burrows et al. 2005). Individual exposures in the
S-CUBED survey are typically 60s long, and occur weekly, although
interruptions can occur due to scheduling constraints. Starting with the
S-CUBED observation taken on MJD 57722 (29 Nov 2016), S-CUBED detected RX
J0123.4-7321on several subsequent occasions over the following 4-5 years and
internally numbered it SC379. The occasions on which it was detected are
listed in Table 1 and shown in the top panel of Fig 2, together with the flux
measured on each occasion. blackNon-detections have upper limits in the range
$(5-10)\times 10^{-12}{\rm erg}\cdot{\rm cm}^{-2}\cdot{\rm s}^{-1}$.

S-CUBED observations taken with the UV/Optical Telescope (UVOT; Roming et al.
2005) were also analysed, utilizing the standard uvotmaghist tool. The middle
panel of that figure shows the UVOT measurements in the uvw1 band (2600Å), it
also indicates all the occasions when the source was observed. Broadly
speaking the UV tracks the I-band, but not in the same detail. This is
partially because the UVOT measurements lack the same precision as the OGLE
data, but also because the circumstellar disc radiates predominately in the
red end of the spectrum. These UVOT observations seem to confirm that [M2002]
SMC 81035 has not yet returned to the earlier much brighter state, but has
levelled off at some intermediary phase.

Figure 2: Comparison of the X-ray, UV and optical detections. Upper panel S-CUBED X-ray detections of RX J0123.4-7321 in flux units of ${\rm erg}\cdot{\rm cm}^{-2}\cdot{\rm s}^{-1}$. Centre panel : UVOT magnitudes in uvw1 filter of [M2002] SMC 81035. Bottom panel : OGLE IV I-band data of [M2002] SMC 81035. The vertical red dashed lines indicate the dates of expected optical outburst given by Equation 1. Table 1: Table of X-ray detections of RX J0123.4-7321 by S-CUBED over the range 0.3–10 keV Date | X-ray flux | X-ray error | Binary phase
---|---|---|---
MJD |  $10^{-12}$ erg per (sq-cm sec) |
57722.37 | 2.76 | 1.34 | 0.88
57736.24 | 3.02 | 1.49 | 0.00
57966.38 | 3.50 | 1.54 | 0.93
57973.62 | 2.85 | 1.42 | 0.99
58078.46 | 7.96 | 2.54 | 0.86
58218.41 | 7.82 | 4.46 | 0.03
58225.97 | 3.54 | 2.03 | 0.09
58344.77 | 4.57 | 2.66 | 0.09
58344.78 | 3.63 | 2.08 | 0.09
58939.04 | 6.54 | 2.83 | 0.06
58953.16 | 1.85 | 1.35 | 0.18
59052.03 | 8.26 | 3.27 | 0.01
59059.00 | 3.45 | 2.00 | 0.07
59093.00 | 5.26 | 3.07 | 0.35
59296.41 | 3.52 | 2.04 | 0.05

Assuming no spectral changes, one can sum all the detections together to
obtain an average flux value of $(5.63\pm 0.90)\times 10^{-12}{\rm
erg}\cdot{\rm cm}^{-2}\cdot{\rm s}^{-1}$. Using a standard SMC distance of 62
kpc (Scowcroft et al., 2016) and correcting for absorption fixed at the value
derived from Willingale et al. (2013), this corresponds to a 0.5-10 keV
luminosity of $(2.5\pm 0.4)\times 10^{36}$ erg$\cdot{\rm s}^{-1}$.

##  3 Discussion

The OGLE V and I band behaviour is shown in Fig 1. It is obvious that a
fundamental change occurred in this emission pattern around TJD 5000. The
source changed from clear regular outburst behaviour to one showing long
timescale fluctuations. The (V-I) colour record is hugely important in
interpreting this change as it shows the bluest colour around the time of this
switch in behavioural patterns. It suggests that the Be star lost almost all
its circumstellar disk around TJD 5000–5100, and then took 5–6 years to
gradually re-build it. It is only around the end of the OGLE IV data that we
see the re-commencement of the OGLE III outburst phenomenon, but still only
weakly compared to the early years.

Fig 3 shows the colour changes occurring between OGLE III and IV as a function
of the I band magnitude.In this figure the pathway taken by [M2002] SMC 81035
over this $\sim$20 year period is indicated. The source begins this time
interval in the top right hand corner, at maximum brightness and maximum
redness. It then substantially changes over the course of OGLE III heading
towards the lower left hand corner - faintest & bluest emission almost
reaching the state of a B0.7 III star with no circumstellar disk. In other
words we infer that it has transitioned from a Be star to a normal B-type
star. This has occurred on a timescale of just $\sim$5 years.

During OGLE IV we then see the star reverse its position on the colour-
magnitude diagram, and in a series of, presumably, mass ejection phases re-
builds its circumstellar disk and it resumes its Be type classification. It is
interesting to note that the re-building process exhibits itself as large-
scale fluctuations on a timescale of $\sim$1000 d. Such a behaviour could be
described as a super-orbital modulation, such as that described by
Rajoelimanana et al. (2011) and Townsend & Charles (2020). In fact the
timescales observed here of a binary period of 119 d and a super-orbital
period of $\sim$1000 d would fit comfortably on Figure 1 of Townsend & Charles
(2020) in the region occupied by other Be stars in X-ray binary partnerships.

From Fig 1 it can be seen that during the time of regular strong outbursts
seen in OGLE III the colour (V-I) would change sharply by $\sim$0.07
magnitudes. This changes comes from an increase in the I band of 0.18
magnitudes ($\sim$20% flux increase), whilst the V band only increased by 0.11
magnitudes ($\sim$10% flux increase). Assuming an optically thick disk then
this is indicative of an increase in the surface area of a disk with an
average temperature significantly cooler than that of the parent star,
resulting in a redder overall appearance for the system.

The phases of X-ray detections are shown in Fig 4 and compared to the OGLE III
data folded with the ephemeris given in Equation 1. It is immediately clear
that the X-ray emission is strongly correlated in phase with the OGLE profile,
both in width and peak position. The gravitational pull of the arriving
neutron star distorts the surface area of the optically thick circumstellar
disk increasing the I band emission. At the same time the larger disk is
providing material to accrete on to the neutron star triggering the X-ray
emission.

To estimate the probable dimensions of the disk and neutron star orbit some
assumptions are needed. In the absence of any direct H$\alpha$ equivalent
width measurements it is necessary to assume that [M2002] SMC 81035 is a
typical Be star, and we can estimate the circumstellar disk size from its
spectral classification, B0.7III. Rivinius et al. (2013) quotes the size of
the H$\alpha$ emitting disk for a very similar B0.5IVe star as having a
typical radius of $7.1R_{*}$. This translates into a disk size of $(3-7)\times
10^{10}$ m for such a star with luminosity class in the range III - V. Also
assuming, for simplicity, that the neutron star is in a circular orbit, then
the period of 119 d permits an estimate of the orbital radius to be $2\times
10^{11}$ m. That is $\sim$10 times larger than the H$\alpha$ disk size, but as
Rivinius et al. (2013) points out the actual disk size may well extend beyond
the H$\alpha$ emitting region by a significant amount. But, broadly speaking,
this is what Smooth Particle Hydrodynamic simulations of such systems predict
(Okazaki & Negueruela, 2001; Brown et al., 2019) \- the neutron star orbiting
just outside the disk and constraining further disk expansion.

However, the orbital behaviour of the neutron star in this system clearly is
not circular, but rather it must exhibit a high degree of eccentricity. Fig 4
shows that both the optical and X-ray outburst behaviour is restricted to
binary phase range 0.8–0.2. Though it is not possible to quantify the degree
of eccentricity from those profiles alone, it is obviously very similar to
that seen in another system SXP 756 (Coe & Edge, 2004). That system has a much
longer binary period of 394 d, but the optical behaviour is remarkably similar
to the source in this paper. The optical outbursts are even sharper, extending
over a binary phase range of just 0.2. It is therefore highly likely that both
of these systems are clear examples of high eccentricity orbits. Townsend et
al. (2011) demonstrated a significant correlation between the binary period
and the eccentricity (where that had been reliably measured as part of an
orbital solution) for BeXRB systems. From their Fig 6 it can be predicted that
the eccentricity of RX J0123.4-7321 could be in the range 0.4-0.5. Such a high
eccentricity would comfortably explain the behaviour seen in this system.

Finally we note from Fig 2 that though X-ray detections are reported for
blackmany periastron passages of the neutron star, there are some missed
occasions. They are missed even though S-CUBED clearly carried out
observations at the appropriate times. This is almost certainly just
indicative that the X-ray flux levels emitted during these Type I outbursts
from RX J0123.4-7321 are close to the detection threshold of the XRT telescope
on Swift with just a 60 s exposure. A small reduction in the X-ray strength
could easily lead to a missed detection. In fact, Fig 2 clearly shows evidence
for X-ray activity during many binary cycles, even when the I-band flux had
diminished around MJD 58000-58400. So, though the circumstellar disk was
obviously greatly reduced at these times there was always enough material in
the vicinity of the neutron star orbit at periastron to trigger accretion.
This is further evidence that the eccentricity of this system must be very
high indeed, permitting the neutron star to approach exceptionally close to
the Be star each periastron passage.

Figure 3: The colour changes observed in the optical companion to RX
J0123.4-7321 from both OGLE III and IV data. The red points come from OGLE III
and the red arrow indicates the direction of change with time. Blue points are
from OGLE IV and the blue arrow indicates the colour evolution with time
during that phase. Figure 4: Upper panel : OGLE III data folded at the updated
binary ephemeris - see Equation 1. Lower panel : the X-ray detections of RX
J0123.4-7321 as a function of the same binary ephemeris. The X-ray flux is in
units of erg cm-2 s-1.

##  4 Conclusions

Though RX J0123.4-7321 was previously well established as a member of the
BeXRB family in the SMC, its behaviour has dramatically changed since it was
last studied. The new OGLE optical colour data (V-I), combined with the I-band
magnitude, strongly indicate a major disk loss episode in this Be star lasting
several years. This was followed by a slow recovery over a similar amount of
time as the circumstellar disk was gradually rebuilt. Throughout the last few
years of this process the X-ray observations reported here were able to detect
Type I outbursts around what is believed to be the time of periastron. Despite
the almost complete disk loss, these X-ray detections continued over a 4 year
period up to the present. They point strongly to a very eccentric orbit
bringing the neutron star in close to the Be star and thereby enabling the
accretion of material from the limited remains of the circumstellar disk. It
will be interesting to see if the characteristic sharp optical outbursts
return in the future once the COVID-19 restrictions permit the resumption of
OGLE monitoring.

## Acknowledgements

The OGLE project has received funding from the National Science Centre,
Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. PAE acknowledges UKSA
support. JAK acknowledges support from NASA grant NAS5-00136. This work made
use of data supplied by the UK Swift Science Data Centre at the University of
Leicester.

## Data Availability

All X-ray data are freely available from the NASA Swift archive. The optical
data in this article will be shared on any reasonable request to Andrzej
Udalski of the OGLE project.

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