#  BMT Independence

Octavio Arizmendi  Saul Rogelio Mendoza  Josué Vazquez-Becerra

###### Abstract

We introduce the notion of BMT independence, allowing us to take arbitrary
mixtures of boolean, monotone, and tensor independence and generalizing the
notion of BM independence of Wysoczanski. Pair-wise independence relations are
encoded through a directed graph, which in turn determines the way mixed
moments must be computed. Corresponding Central and Poisson-Type Limit
Theorems are provided along with an explicit construction to realize BMT
independent random variables as bounded operators on certain Hilbert space.

###### Contents

  1. 1 Introduction
  2. 2 Set Partitions and Graphs
    1. 2.1 Set partitions
    2. 2.2 Digraphs
    3. 2.3 The kernel notation
  3. 3 BMT Independence
    1. 3.1 Notions of Independence
    2. 3.2 BMT independence
    3. 3.3 Weak BM independence
    4. 3.4 Consistency
  4. 4 Construction of BMT algebras of operators
  5. 5 BMT Central Limit Theorem
    1. 5.1 Central Limit Theorem
    2. 5.2 Further properties of CLT
  6. 6 Poisson Limit Theorem
  7. 7 Concluding Remarks

##  1 Introduction

This paper tries to extend some notions in non-commutative probability, where
the fundamental framework is a _non-commutative probability space_ , this is,
a pair $(\mathcal{A},\varphi)$ where $\mathcal{A}$ is a complex algebra with
multiplicative identity $1_{\mathcal{A}}$ and
$\varphi:\mathcal{A}\to\mathbb{C}$ is a linear functional so that
$\varphi(1_{\mathcal{A}})=1$.

In this framework, from the probabilistic viewpoint, a fundamental notion is
that of independence and the generality of this framework allows considering
multiple notions of independence. In order to give a sense of this notion, and
in the seek to try to find a way to decide which of these independences one
should look at, Ben Ghorbal and Schürmann [4] axiomatized the notion of a
universal products. For each of these universal products of probability spaces
one can associate a notion of independence. They showed that there are three
notions of independence satisfying the axioms coming from the universal
products. These are Boolean, free and tensor independence. The story was
complemented by the work of Muraki [18], when he generalized the notion of
universal product, introducing _natural products_ , by removing the
_commutativity axiom_ (now known as symmetry axiom), which states, in simple
words, that for algebras $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, that fact
that $\mathcal{A}_{1}$ is independent of $\mathcal{A}_{2}$, implies that
$\mathcal{A}_{2}$ should be independent of $\mathcal{A}_{1}$ .

In this new setting, Muraki, showed in [18], that there are only five natural
notions of independence, these are, the free, tensor, Boolean, monotone and
antimonotone independences. In this paper we will be interested in tensor,
Boolean and monotone (or antimonotone 111From the fact that $a$ monotone
independent to $b$ if and only if $b$ independent antimonotone to $a$, we
usually work only with monotone independence (implying the corresponding for
the antimonotone independence). ) cases.

While from the above, one may think of free, Boolean, monotone and tensor
independence as parallel theories with no interaction between them, in this
paper we will be interested in mixtures of them. This is, of course, not the
first paper that considers such a setting. In 2007, Wysoczanski [24]
introduced BM independence, a generalization of monotone and Boolean
independence, giving a framework where some mixtures (following a partial
order) of those independences can be represented, giving, in addition, a
central limit theorem whose limit is a symmetric distribution that has as
particular cases the symmetric Bernoulli and the arcsine distribution. The
work [9] gives a framework that combines Boolean and free independences. On a
similar vein, the called $\Lambda-$freeness defined in [14] mixes free and
tensor independence in an algebraic framework. Following the last work,
Speicher and Wysoczanski ([22]) represented any mixture of tensor and free
variables in terms of a symmetric matrix $\epsilon$ with non-diagonal entries
either $0$ or $1$, where $\epsilon_{ij}=0$ represent that algebras
$\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ are free, and $\epsilon_{ij}=1$
represent that algebras $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ commute and
are tensor independent. Finally, the work of Lenczewski [10], considers
$\Lambda$-Boolean independence and $\Lambda$-monotone independence which mix
tensor independence with Boolean independence and with monotone independence,
respectively.

In this paper we create a framework that generalizes BM independence,
$\Lambda$-Boolean and $\Lambda$-monotone independence, allowing to consider
mixtures of tensor, Boolean and monotone independent algebras. This appears to
be the first time 3 of the natural independences are combined. We call this
new notion of independence BMT (Boolean, monotone and tensor) independence.

To describe such notion we need a graph telling the information about pairwise
independence. The idea is that, vertices correspond to random variables (or
algebras), and the relation between two of them is as follows: Vertices joined
with a directed edge will correspond to monotone independence between
elements, vertices joined by an undirected edge will correspond to tensor
independence, and unjoined vertices will correspond to Boolean independence.
As an example, if we have the variables $\\{X_{1},X_{2},X_{3},X_{4},X_{5}\\}$
and its independence graph $G$ is the one in the Figure 1, we want the
following relations.

  * •

$X_{5}$ is Boolean independent from $X_{1},X_{2},X_{3},X_{4}$

  * •

$X_{4}$ is Boolean independent from $X_{3}$ and $X_{1}$.

  * •

$X_{2}$ is classical independent from $X_{4}$.

  * •

$X_{2}$ is monotonically independent from $X_{3}$.

  * •

$X_{1}$ monotonically independent from $X_{2}$ and from $X_{3}$.

$X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$ Figure 1: Independence graph of
$(X_{1},X_{2},X_{3},X_{4},X_{5})$.

The main question is, of course, how to define the whole joint distribution,
involving not only pairwise relations, and if this joint distribution is
representable in a $C^{\ast}$-probability space. To be clear, BMT variables
will have a graph $G$ telling us the independence relations between pairs of
variables/algebras, and in the framework we develop, we will have a way for
computing all the mixed moments between these variables.

The main contributions of this paper are, firstly, to give a consistent way to
calculate such mixed moment, see Definition 3.4, and secondly, to give a
concrete construction which satisfies such notion of independence.

Now, for each of the five notions of independence, Boolean, Free, monotone and
tensor, there exists a central limit theorem associated with it. Under the
tensor independence hypothesis, the limit distribution is the normal
distribution; with free independence, the limit is the semicircle
distribution; the symmetrical Bernoulli distribution corresponds to Boolean
independence: while the arcsine distribution is the limit for the monotone
central limit theorem. Here we note that in the usual monotone central limit
theorem a total order is imposed in the algebras. In this framework, it is
possible to induce monotone and antimonotone independent variables without
assuming any specific order. Our third main contribution is proving a central
limit theorem for BMT independent variables with a sequence of independence
graphs, having as particular cases the monotone, tensor and Boolean central
limit theorems. Similarly, we prove s Poisson limit theorem or law of small
numbers.

This work is distributed as follows. In Section 2, we introduce important
definitions on graphs and partitions. We proving some new properties of these
objects, which will be used crucially in the rest of the paper. In section 3,
we introduce the notion of BMT independence with respect to a graph. We
provide the first properties and relation with other notions of independence.

In Section 4, we present a model where we deploy the variables whose pairing
independence relations are given by a graph $G$.

In Section 5, we present the BMT central limit theorem. We give examples,
including Boolean, Free and Monotone CLT’s. We also considers some relations
between different graphs. Similarly, in Section 6 we consider a Poisson Limit
theoerem. We conclude with some open questions and remarks.

##  2 Set Partitions and Graphs

In this section, we introduce the main combinatorial objects that are used in
our analysis of BMT independent random variables. First, we describe different
types of set partitions. Second, we review some terminology on directed
graphs. Finally, we define the kernel of a function subordinated to a directed
graph, which is the object that determines how joint moments of BMT
independent random variables must be computed.

###  2.1 Set partitions

######  Definition 2.1.

A _partition_ $\pi=\\{B_{1},B_{2},\ldots,B_{r}\\}$ of a non-empty set $S$ is a
set of non-empty and pair-wise disjoint subsets of $S$ whose union is $S$,
i.e., $B\subset S$ and $B\neq\emptyset$ for every $B\in\pi$, $B\cap
B^{\prime}\neq\emptyset$ implies $B=B^{\prime}$ for all $B,B^{\prime}\in\pi$,
and $\cup_{B\in\pi}B=S$.

We refer to the elements of a partition as _blocks_ , and the total number of
blocks in partition $\pi$ will be denoted by $\\#(\pi)$. Moreover, a block is
said to be _even_ if it has even cardinality and _odd_ otherwise. A partition
containing only even blocks is called _even_ , but if all of its blocks have
exactly two elements we will refer to it as a _pairing_.

######  Example 2.2.

The sets $\pi_{1}=\\{\\{1,3\\},\\{2,4,5,6\\}\\}$,
$\pi_{2}=\\{\\{1,3,6\\},\\{2\\},\\{4,5\\}\\}$, and $\pi_{3}=\\{\\{1,3\\}$
$\\{4,6\\},\\{2,5\\}\\}$ are all partitions of $\\{1,2,3,4,5,6\\}$. The
partitions $\pi_{1}$ and $\pi_{3}$ are both even, but while $\pi_{3}$ is a
pairing, $\pi_{1}$ is not. The partition $\pi_{2}$ is neither even nor odd
since it contains two odd blocks, $\\{2\\}$ and $\\{1,6,3\\}$, and one even
block, $\\{4,2\\}$.

The set of all partitions of $S$, the set of all even partitions of $S$, and
the set of all pairing partitions of $S$ are denoted by $P(S)$,
$P_{\text{even}}(S)$, and $P_{2}(S)$, respectively. We let $[m]$ and
$[\ell,m]$ denote the set of integers $\\{1,2,\ldots,m\\}$ and
$\\{\ell,\ell+1,\ldots,m\\}$, respectively, for any integers $0\leq\ell\leq
m$. When referring to set partitions of $[m]$, we will omit the square
brackets. For instance, we will write $P_{\text{even}}(m)$ instead of
$P_{\text{even}}([m])$.

Each partition $\pi\in P(m)$ can be represented graphically by writing the
numbers $1,2,\ldots,m$ on a line, drawing vertical lines above each number
with matching heights for numbers in the same block, and joining with a
horizontal line the vertical lines of the numbers that belong to the same
block, see Figure 2. This leads to the concepts of _nesting_ and _crossing_
blocks.

######  Definition 2.3.

Suppose we are given two distinct blocks $B=\\{b_{1}<b_{2}<\cdots<b_{r}\\}$
and $C=\\{c_{1}<c_{2}<\cdots<c_{s}\\}$ from a partition $\pi\in P(m)$. We say
that $B$ _is nested inside_ $C$ if there exists $k\in[1,r-1]$ such that
$c_{k}<b_{j}<c_{k+1}$ for every $b_{j}\in B$. Additionally, we say $B,C\in\pi$
_cross each other_ if there exist $b_{i},b_{j}\in B$ and $c_{k},c_{\ell}\in C$
such that $b_{i}<c_{k}<b_{j}<c_{\ell}$.

Every partition $\pi\in P(S)$ is equivalent to an equivalence relation
$\sim_{\pi}$ on $S$ where $k\sim_{\pi}{k^{\prime}}$ if and only if $k$ and
${k^{\prime}}$ belong to the same block of $\pi$. Hence, the equivalence
relation $\sim_{\pi}$ has the blocks of $\pi$ as equivalence classes. The set
of partitions $P(S)$ is partially ordered by refinement: we put
$\pi\leq\theta$, and say that $\pi$ is a _refinement_ of $\theta$, if every
block of $\pi$ is contained in some block of $\theta$. Notice that
$\pi\leq\theta$ if and only if $k\sim_{\pi}{k^{\prime}}$ implies
$k\sim_{\theta}{k^{\prime}}$ for all $k,{k^{\prime}}\in S$. In Example 2.2,
the partition $\pi_{3}$ is a refinement of $\pi_{1}$, and there is no other
refinement between $\pi_{1}$, $\pi_{2}$, and $\pi_{3}$.

(a) $\pi_{1}=\\{\\{1,3\\},\\{2,4,5,6\\}\\}$

(b) $\pi_{2}=\\{1,3,6\\},\\{2\\},\\{\\{4,5\\}\\}$

(c) $\pi_{3}=\\{\\{1,3\\},\\{2,5\\},\\{4,6\\}\\}$

Figure 2: Graphical representation for partitions $\pi_{1}$, $\pi_{2}$, and
$\pi_{3}$ from Example 2.2

###  2.2 Digraphs

######  Definition 2.4.

A _directed graph_ , or simply _digraph_ , is a pair $G=(V,E)$ where $V$ is a
non-empty set, called _vertex set_ , and $E$ is a (possibly empty) subset of
the Cartesian product $V\times V$, called _edge set_.

Given two digraphs $G_{1}=(V_{1},E_{2})$ and $G_{2}=(V_{2},E_{2})$, we say
that $G_{1}$ is a _subgraph_ of $G_{2}$, and denote this by $G_{1}\subset
G_{2}$, if $V_{1}\subset V_{2}$ and $E_{1}\subset E_{2}$. All digraphs in this
paper are assumed to be _simple_ , i.e., they contain no _loops_ —edges of the
form $(v,v)$. The following are some types of digraphs that will be considered
in this paper as they concern Boolean, monotone, and tensor independence:

  * •

_Empty digraph._ These are digraphs $G=(V,E)$ without edges, so $E$ is the
empty set.

  * •

_Complete digraph._ These are digraphs $G=(V,E)$ containing all possible
edges, so every ordered pair $(v,w)\in V\times V$ with $v\neq w$ belongs to
$E$.

  * •

_Digraph of a partial order._ These are digraphs $G=(V,E)$ where the vertex
set $V$ has a partial order $\preceq$ and the edge set $E$ contains an ordered
pair $(v,w)\in V\times V$ if and only if $v\prec w$, i.e., $v\preceq w$ and
$v\neq w$.

###  2.3 The kernel notation

Let $S$ and $V$ be non-empty sets. Given any function $\bm{i}:S\rightarrow V$,
we make the convention of taking $i_{k}=\bm{i}(k)$ for every $k\in S$.
Additionally, if $S=[m]$ for some integer $m\geq 1$, then each function
$\bm{i}:S\rightarrow V$ will be identified with the tuple
$(i_{1},i_{2},\ldots,i_{m})$.

######  Definition 2.5.

The _kernel_ of a function $\bm{i}:S\rightarrow V$ is the partition of $S$
determined by the equivalence relation $k\sim{k^{\prime}}$ if and only if
$i_{k}=i_{k^{\prime}}$. This partition is denoted by $\ker[\bm{i}]$.

######  Remark 2.6.

Notice that $\ker[\bm{i}]$ coincides with the partition of $S$ whose blocks
are all non-empty pre-images of $\bm{i}$, so we have $\ker[\bm{i}]=\\{\\{k\in
S\mid i_{k}=v\\}\neq\emptyset\mid v\in V\\}$. Furthermore, for any partition
$\pi\in P(S)$, the condition $\pi\leq\ker[\bm{i}]$ holds if and only if
$\bm{i}$ is constant when restricted to each block of $\pi$, i.e.,
$i_{k}=i_{\ell}$ whenever $k,\ell\in B$ and $B\in\pi$.

######  Example 2.7.

The function $\bm{i}:[6]\rightarrow[5]$ given by
$(i_{1},i_{2},i_{3},i_{4},i_{5},i_{6})=(4,1,3,4,1,4)$ has kernel
$\ker[\bm{i}]=\\{\\{1,4,6\\},\\{2,5\\},\\{3\\}\\}$. The function
$\bm{j}:[6]\rightarrow[5]$ given by
$(j_{1},j_{2},j_{3},j_{4},j_{5},j_{6})=(5,1,5,4,1,4)$ has kernel
$\ker[\bm{i}]=\\{\\{1,3\\},\\{2,5\\},\\{4,6\\}\\}$.

######  Definition 2.8.

The _kernel_ of a function $\bm{i}:S\rightarrow V$ _subordinated_ to a digraph
$G=(V,E)$ is the partition of $S$ determined by the equivalence relation
$k\sim k^{\prime}$ if and only if $i_{k}=i_{k^{\prime}}$ and
$(i_{\ell},i_{k})$ is an edge of $G$ whenever $i_{k}\neq i_{\ell}$ and either
$k<\ell<k^{\prime}$ or $k^{\prime}<\ell<k$. We denote this partition by
$\ker_{G}[\bm{i}]$.

######  Remark 2.9.

The partition $\ker_{G}[\bm{i}]$ is a refinement of the partition
$\ker[\bm{i}]$. Moreover, the second condition in the equivalence relation
defining $\ker_{G}[\bm{i}]$ only concerns $G_{\bm{i}}$ —the restriction of $G$
to the vertices $i_{1},i_{2},\ldots,i_{m}$— and not the whole graph $G$.
Hence, if $G_{\bm{i}}$ is the complete graph, the second condition is
immediately satisfied and $\ker_{G}[\bm{i}]$ is defined only by the relation
$k\sim k^{\prime}$ whenever $i_{k}=i_{k^{\prime}}$, which is exactly the
definition of $\ker[\bm{i}]$, yielding $\ker_{G}[\bm{i}]=\ker[\bm{i}]$ in this
case. Therefore, $\ker_{G}[\bm{i}]$ is not only a refinement but also a
generalization of $\ker[\bm{i}]$.

We have given a sufficient but not necessary condition for
$\ker_{G}[\bm{i}]=\ker[\bm{i}]$ to hold in Remark 2.9. This equality plays an
important role in our analysis of BMT random variables, so it will be
convenient to establish an equivalent condition to it in terms of digraphs. To
this end, we introduce the following.

######  Definition 2.10.

The _nesting-crossing graph_ of a partition $\pi\in P(m)$ is the digraph
$G_{\pi}=(V_{\pi},E_{\pi})$ with vertex set $V_{\pi}=\pi$ and edge set
$E_{\pi}=\left\\{(B,C)\in\pi\times\pi:B\neq C\text{ and either
}C\text{}\right.$ $\left.\text{is nested inside }C\text{ or }B\text{ and
}C\text{ cross each other}\right\\}$. Additionally, given a function
$\bm{i}:[m]\to[N]$ with $\ker[\bm{i}]=\pi$, we let $G_{\pi(\bm{i})}$ denote
the graph obtained from $G_{\pi}$ after relabeling each vertex $W\in V_{\pi}$
as $i_{k}$ with $k\in W$.

######  Remark 2.11.

The edge set $E_{\pi}$ can be equivalently defined as the set of all ordered
pairs $(B,C)\in\pi\times\pi$ with $B\neq C$ such that there exists $\ell\in B$
with $\min C<\ell<\max C$. The latter is also equivalent to the existence of
elements $k,k^{\prime}\in C$ and $\ell\in B$ with $k<\ell<k^{\prime}$ .

######  Example 2.12.

Consider $\pi=\\{\\{1,5\\},\\{2,3,7\\},\\{4\\},\\{6\\}\\}$ and take
$B_{1}=\\{1,5\\}$, $B_{2}=\\{2,3,7\\}$, $B_{3}=\\{4\\}$, and $B_{4}=\\{6\\}$.
The nesting-crossing graph $G_{\pi}$ has then vertex set
$V_{\pi}=\\{B_{1},B_{2},B_{3},$ $B_{4}\\}$ and edge set
$E_{\pi}=\\{(B_{1},B_{2}),(B_{2},B_{1}),(B_{3},B_{1}),(B_{3},B_{2}),(B_{4},B_{2})\\}$.
Moreover, the function $\bm{i}=(1,8,8,4,1,5,8)$ satisfies $\ker[\bm{i}]=\pi$,
so the graph $G_{\pi(\bm{i})}$ has vertex set $V_{\pi(\bm{i})}=\\{1,4,5,8\\}$
and edge set $E_{\pi(\bm{i})}=\\{(1,8),(8,1),(4,1),(4,8),(5,8)\\}$. See Figure
3.

Figure 3: Graphical representation of the partition
$\pi=\\{\\{1,5\\},\\{2,3,7\\},\\{4\\},\\{6\\}\\}$, the function
$\bm{i}=(1,8,8,4,1,5,8)$, the nesting-crossing graph $G_{\pi}$, and the graph
$G_{\pi(\bm{i})}$.

######  Lemma 2.13.

Let $\bm{i}:[m]\to V$ be a function with $\ker[\bm{i}]=\pi$ for some partition
$\pi\in P(m)$. Then for any digraph $G=(V,E)$ the equality
$\ker_{G}[\bm{i}]=\ker[\bm{i}]$ holds if and only if $G_{\pi(\bm{i})}$ is a
subgraph of $G$.

Since the vertex set of $G_{\pi(\bm{i})}$ is always a subset of the vertex set
of $G$, and the definition of $\ker_{G}[\bm{i}]$ and $\ker[\bm{i}]$ implies
that the equality $\ker_{G}[\bm{i}]=\ker[\bm{i}]$ holds if and only if
$(i_{\ell},i_{k})$ is an edge of $G$ whenever $k<\ell<k^{\prime}$, $i_{k}\neq
i_{\ell}$, and $i_{k}=i_{k^{\prime}}$, the previous lemma is an immediate
consequence of the next proposition.

######  Proposition 2.14.

Suppose $\pi$ is a partition in $P(m)$. Then, for any function
$\bm{i}:[m]\to[N]$ with $\ker[\bm{i}]=\pi$, the edge set $E_{\pi(\bm{i})}$ of
the graph $G_{\pi({\bm{i}})}$ satisfies

$E_{\pi(\bm{i})}=\left\\{(i_{\ell},i_{k})\mid\ i_{k}\neq i_{\ell}\text{ and
there exists }k^{\prime}\in[m]\text{ with }k<\ell<k^{\prime},\
i_{k}=i_{k^{\prime}}\right\\}.$

###### Proof.

Put $\widetilde{E}=\left\\{(i_{\ell},i_{k})\mid\ i_{k}\neq i_{\ell}\text{ and
there exists }k^{\prime}\in[m]\text{ with }k<\ell<k^{\prime},\
i_{k}=i_{k^{\prime}}\right\\}$. Suppose first $(i_{\ell},i_{k})$ is an edge in
$E_{\pi({\bm{i}})}$ and let $C$ and $B$ be the blocks of $\pi$ with $\ell\in
C$ and $k\in B$. Since $\ker[\bm{i}]=\pi$ and $(i_{\ell},i_{k})\in
E_{\pi({\bm{i}})}$, we must have $i_{k}\neq i_{\ell}$. Moreover, the pair
$(C,B)$ must be an edge in $E_{\pi}$, and hence either $C$ is nested inside
$B$ or $C$ and $B$ cross each other. Now, if $C$ is nested inside $B$, then we
can take $k^{\prime}=\min B$ and $k^{\prime\prime}=\max B$ to get
$(i_{\ell},i_{k})=(i_{\ell},i_{k^{\prime}})$, $i_{k^{\prime}}\neq i_{\ell}$,
$k^{\prime}<\ell<k^{\prime\prime}$, and $i_{k}=i_{k^{\prime}}$. On the other
hand, if $C$ and $B$ cross each other, there must exist
$k^{\prime},k^{\prime\prime}\in B$ and $\ell\in C$ with
$k^{\prime}<\ell<k^{\prime\prime}$; additionally, we get
$(i_{\ell},i_{k})=(i_{\ell},i_{k^{\prime}})$, $i_{k^{\prime}}\neq i_{\ell}$,
and $i_{k}=i_{k^{\prime}}$. In any case, we obtain that $(i_{\ell},i_{k})$
belongs to $\widetilde{E}$.

Suppose now $(i_{\ell},i_{k})$ belongs to $\widetilde{E}$ and let
$k^{\prime}\in[m]$ with $k<\ell<k^{\prime}$ and $i_{k}=i_{k^{\prime}}$. Take
$C$ and $B$ as above. Since $\ker[\bm{i}]=\pi$, we have $k^{\prime}$ belongs
to $B$; moreover, $i_{k}\neq i_{\ell}$ implies the blocks $B$ and $C$ are
distinct. And hence, either $C$ is nested inside $B$ or $C$ and $B$ cross each
other due to $k<\ell<k^{\prime}$. In any case, we obtain $(C,B)$ is an edge in
$E_{\pi}$, and therefore $(i_{\ell},i_{k})$ belongs to $E_{\pi(\bm{i})}$ since
$i_{\ell}$ and $i_{k}$ are the replacements of $C$ and $B$, respectively, in
the construction of $G_{\pi(\bm{i})}$ from $G_{\pi}$. ∎

##  3 BMT Independence

This is one of the main sections of the paper. We first recall some notions
from non-commutative probability, in particular, the notions of boolean,
monotone, and tensor independence. We then introduce the notion of BMT
independence, which provides a framework that allows for arbitrary mixtures of
boolean, monotone, and tensor independence and, consequently, generalizing the
notion of BM independence.

###  3.1 Notions of Independence

######  Definition 3.1.

A _non-commutative probability space_ is a pair $(\mathcal{A},\varphi)$ where
$\mathcal{A}$ is a complex algebra with multiplicative identity
$\mathbf{1}_{\mathcal{A}}$ and $\varphi:\mathcal{A}\to\mathbb{C}$ is a linear
functional so that $\varphi(\mathbf{1}_{\mathcal{A}})=1$. The elements of
$\mathcal{A}$ are called _random variables_.

Suppose $a\in\mathcal{A}$ is a random variable in a non-commutative
probability space $(\mathcal{A},\varphi)$. The value $\varphi(a^{k})$ is
called the _k-th moment_ of $a$. Moreover, the sequence
$(\varphi(a^{k}))_{k=1}^{\infty}$ of all moments of $a$ is called the
_(algebraic) distribution_ of $a$. We say that a probability measure $\mu$ on
the real line $\mathbb{R}$ is the _analytical distribution_ of $a$ if for
every integer $k\geq 1$ we have

$\varphi(a^{k})=\int_{-\infty}^{+\infty}t^{k}d\mu(t).$

######  Definition 3.2.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. An
infinite sequence $a_{1},a_{2},a_{3}\ldots$ contained in $\mathcal{A}$ is said
to _converge in moments_ to a random variable $a\in\mathcal{A}$ (resp., $\mu$)
if for every integer $k\geq 1$ we have

$\lim_{N\to\infty}\varphi(a^{k}_{N})=\varphi(a^{k})\quad\left(\text{resp.,
}=\int_{-\infty}^{+\infty}t^{k}d\mu(t)\right).$

For any sequence of random variables $a_{1},a_{2},\ldots,a_{m}\in\mathcal{A}$
and any set $B=\\{k_{1}<k_{2}<\cdots<k_{r}\\}\subset[m]$, we let

$(a_{k})|_{B}=a_{k_{1}}a_{k_{2}}\cdots a_{k_{r}}=\prod_{k\in
B}^{\rightarrow}a_{k}.$

Suppose $(\mathcal{A}_{i})_{i\in I}$ is a family of sub-algebras of
$\mathcal{A}$ and
$a_{1}\in\mathcal{A}_{i_{1}},\ldots,a_{m}\in\mathcal{A}_{i_{m}}$ for some
indexes $i_{1},\ldots,i_{m}\in I$. We say $a_{1}a_{2}\cdots a_{m}$ is _an
alternating product of elements of_ $(\mathcal{A}_{i})_{i\in I}$ if the
indexes $i_{k}$ satisfy that $i_{1}\neq i_{2}$, $i_{2}\neq i_{3}$, $\ldots$,
$i_{m-1}\neq i_{m}$.

######  Definition 3.3.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. A family
$(\mathcal{A}_{i})_{i\in I}$ of subalgebras of $\mathcal{A}$ is said to be

  1. (B)

_boolean independent_ if for any alternating product $a_{1}\cdots a_{m}$ of
elements of $(\mathcal{A}_{i})_{i\in I}$ we have

$\varphi(a_{1}\cdots a_{m})=\varphi(a_{1})\cdots\varphi(a_{m})$

  2. (M)

_monotone independent_ if $I$ has a linear order $<$ and for any alternating
product $a_{1}\cdots a_{m}$ of elements of $(\mathcal{A}_{i})_{i\in I}$ with
$a_{j}\in\mathcal{A}_{i_{j}}$ we have

    1. M.1

$\varphi(a_{1}\cdots a_{m})=\varphi(a_{k})\varphi(a_{1}\cdots
a_{k-1}a_{k+1}\cdots a_{n})$ if $i_{k-1}<i_{k}>i_{k+1}$ for some $k\in[2,m-1]$

    2. M.2

$\varphi(a_{1}\cdots a_{m})=\prod_{\ell=1}^{m}\varphi(a_{\ell})$ if
$i_{1}>\cdots>i_{k-1}>i_{k}<i_{k+1}<\cdots<i_{m}$ for some $k\in[m]$.

  3. (T)

_tensor independent_ if for any (not necessarily alternating) product
$a_{1}\cdots a_{m}$ of elements of $(\mathcal{A}_{i})_{i\in I}$ with
$a_{j}\in\mathcal{A}_{i_{j}}$ we have

$\varphi(a_{1}\cdots a_{m})\ \
=\prod_{B\in\ker({\bm{i}})}\varphi\left(\prod_{k\in B}^{\to}a_{k}\right)$

where $\prod_{k\in B}^{\rightarrow}a_{k}:=a_{k_{1}}a_{k_{2}}\cdots a_{k_{r}}$
provided $B=\\{k_{1}<k_{2}<\cdots<k_{r}\\}$.

###  3.2 BMT independence

We now present the main definition of this paper, the notion of BMT
independence, which relies upon $\ker_{G}[\bm{i}]$, the kernel of a function
subordinated to a digraph $G$, see Definition 2.8.

######  Definition 3.4.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. Suppose
$(\mathcal{A}_{i})_{i\in I}$ is a family of sub-algebras of $\mathcal{A}$ and
$G=(I,E)$ is a digraph on the set of indices $I$. The family
$(\mathcal{A}_{i})_{i\in I}$ is said to be _BMT independent_ with respect to
the pair $(\varphi,G)$ if for every integer $m\geq 1$ and variables
$a_{1}\in\mathcal{A}_{i_{1}},a_{2}\in\mathcal{A}_{i_{2}},\ldots,a_{m}\in\mathcal{A}_{i_{m}}$
we have

$\varphi(a_{1}a_{2}\cdots a_{m})\ \
=\prod_{B\in\ker_{G}[\bm{i}]}\varphi[(a_{k})|_{B}].$

The above digraph $G=(I,E)$ gives the pair-wise independence relations between
sub-algebras from boolean, mononote, and tensor. For any two distinct sub-
algebras $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$, we have that
$\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ are boolean independent if neither
$(i,j)$ nor $(j,i)$ is an edge of $G$, $\mathcal{A}_{i}$ is monotone
independent from $\mathcal{A}_{j}$ if $(i,j)$ is an edge of $G$ but $(j,i)$ is
not, and $\mathcal{A}_{i}$ and $\mathcal{A}_{j}$ are tensor independent if
both $(i,j)$ and $(j,i)$ are edges of $G$. Thus, our notion of BMT
independence has boolean, mononote, and tensor independence as particular
cases. This is proved next.

######  Proposition 3.5.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. Suppose
$(\mathcal{A}_{i})_{i\in I}$ is a family of sub-algebras BMT independent with
respect to a digraph $G=(I,E)$. Thus, we have

  1. (i)

the algebras $(\mathcal{A}_{i})_{\in I}$ are tensor independent if $G$ is the
graph complete,

  2. (ii)

the algebras $(\mathcal{A}_{i})_{i\in I}$ are boolean independent if $G$ is
the empty graph, and

  3. (iii)

the algebras $(\mathcal{A}_{i})_{i\in I}$ are monotone independent if $I$ has
a total order $<$ and $G$ is the digraph of $<$, i.e., $(i,j)$ is an edge of
$G$ if and only if $j<i$.

###### Proof.

We will prove _(i)_ , _(ii)_ , and _(iii)_ separately. To this end, let us
assume $a_{1}\cdots a_{m}$ is a (not necessarily alternating) product of
elements of $(\mathcal{A}_{i})_{i\in I}$ with $a_{k}\in\mathcal{A}_{i_{k}}$.

Proof of _(i)_ . If $G$ is the complete graph, then
$\ker_{G}[\bm{i}]=\ker[\bm{i}]$ for any $\bm{i}=(i_{1},\ldots,i_{m})$, see
Remark 2.9. Thus, by Definitions 3.3 and 3.4, the algebras
$(\mathcal{A}_{i})_{\in I}$ are tensor independent since

$\varphi(a_{1}\cdots a_{m})\ \
=\prod_{B\in\ker_{G}[\bm{i}]}\varphi((a_{k})|_{B})\ \ =\ \
\prod_{B\in\ker[\bm{i}]}\varphi((a_{k})|_{B}).$

Proof of _(ii)_ . Suppose now $G$ is the empty graph. Note that if
$a_{1}\cdots a_{m}$ is an alternating product, then
$\ker_{G}[\bm{i}]=\\{\\{1\\},\\{2\\},\dots,\\{m\\}\\}$. Indeed, $a_{1}\cdots
a_{m}$ is alternating only if $i_{1}\neq i_{2}$, $i_{2}\neq i_{3}$, $\ldots$,
$i_{m-1}\neq i_{m}$, and hence $i_{k}=i_{k^{\prime}}$ only if $k=k^{\prime}$
or $k+1\leq k^{\prime}$. But then, since $G$ contains no edges, the second
condition defining $\ker_{G}[\bm{i}]$ is never satisfied when $k+1\leq
k^{\prime}$, see Definition 2.8. And therefore, $\ker_{G}[\bm{i}]$ contains
only singletons $\\{k\\}$ as equivalence classes. Thus, Definitions 3.3 and
3.4 imply the algebras $(\mathcal{A}_{i})_{\in I}$ are boolean independent
since

$\varphi(a_{1}\cdots a_{m})\ \
=\prod_{B\in\ker_{G}[\bm{i}]}\varphi((a_{k})|_{B})\ \ =\ \
\varphi(a_{1})\varphi(a_{2})\cdots\varphi(a_{m}).$

Proof of _(iii)_ . Suppose that $\leq$ is a total order of the set $I$ with
associated digraph $G=(I,E)$ where $E=\\{(i,j)\in I\times I\mid j<i\\}$.
Assume $a_{1}\cdots a_{m}$ is an alternating product. We will show that M.1
and M.2 from Definition 3.3 hold.

For M.1, we have $i_{k-1}<i_{k}>i_{k+1}$ for some $k\in[2,m-1]$. Then, neither
$(i_{k-1},i_{k})$ nor $(i_{k+1},i_{k})$ is an edge of $G$, and hence $\\{k\\}$
is a singleton in $\ker_{G}[\bm{i}]$, see Definition 2.8. Thus, by BMT
independence, we obtain

$\varphi(a_{1}a_{2}\cdots a_{m})\ \ =\ \
\varphi(a_{k})\prod_{\begin{subarray}{c}B\in\ker_{G}[\bm{i}]\\\
B\neq\\{k\\}\end{subarray}}\varphi((a_{\ell})|_{B}).$

Let $\bm{i}^{\prime}=(i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{m})$. It is
enough to prove that $B\in\ker_{G}[\bm{i}]$ and $B\neq\\{k\\}$ if and only if
$B\in\ker_{G}[\bm{i}^{\prime}]$ since BMT independence also gives

$\varphi(a_{1}\cdots a_{k-1}a_{k+1}\cdots a_{m})\ \
=\prod_{B\in\ker_{G}[\bm{i}^{\prime}]}\varphi((a_{\ell})|_{B}).$

Take $\pi=\ker_{G}[\bm{i}]$ and $\theta=\ker_{G}[\bm{i}^{\prime}]$. Due to
Definition 2.8, we need to show that $r\sim_{\pi}r^{\prime}$ and
$r\sim_{\theta}r^{\prime}$ are equivalent for any $r,r^{\prime}\in[m]$ with
$r,r^{\prime}\neq k$. However, $r\sim_{\pi}r^{\prime}$ implies
$r\sim_{\theta}r^{\prime}$ already, so just the other implication is needed.

Suppose $r\sim_{\theta}r^{\prime}$ with $r<r^{\prime}$ and $r,r^{\prime}\neq
k$. Thus, we obtain $i_{r}=i_{r^{\prime}}$ and $i_{r}<i_{\ell}$ whenever
$r<\ell<r^{\prime}$, $i_{\ell}\neq i_{r}$, and $\ell\neq k$. To get
$r\sim_{\pi}r^{\prime}$, the restriction $\ell\neq k$ in the latter condition
needs to be lifted. This is immediate if $r^{\prime}<k$ or $k<r$ or
$i_{k}=i_{r}$. Thus, without loss of generality, we can assume
$r<k<r^{\prime}$ and $i_{k}\neq i_{r}$. Now, if $i_{r}=i_{k-1}$, then
$i_{r}<i_{k}$ by hypothesis. On the other hand, if $i_{r}\neq i_{k-1}$, we
must have $r<k-1<r^{\prime}$, and hence $i_{r}<i_{k-1}<i_{k}$. This shows the
restriction $\ell\neq k$ is not needed, and hence $r\sim_{\pi}r^{\prime}$.

For M.2, we have $i_{1}>\cdots>i_{k-1}>i_{k}<i_{k+1}<\cdots<i_{m}$ for some
$k\in[m]$. Notice $\pi=\ker_{G}[\bm{i}]$ contains only singletons. Indeed, if
$r\neq r^{\prime}$ and either $1\leq r,r^{\prime}\leq k$ or $k\leq
r,r^{\prime}\leq m$, then either $i_{r}<i_{r^{\prime}}$ or $i_{r}<i_{r}$, and
hence $r\nsim_{\pi}r^{\prime}$ since $i_{r}\neq i_{r^{\prime}}$. Thus, for any
$r,r^{\prime}\in[m]$, we have that $r\sim_{\pi}r^{\prime}$ only if
$r=r^{\prime}$ or $r<k<r^{\prime}$ and $i_{r}=i_{r^{\prime}}$. However, the
latter case implies $i_{k}<i_{r}$ and hence $r\nsim_{\pi}r^{\prime}$.
Therefore, $\ker_{G}[\bm{i}]=\\{\\{1\\},\\{2\\},\ldots,\\{m\\}\\}$, and BMT
independence gives

$\varphi(a_{1}a_{2}\cdots
a_{m})=\varphi(a_{1})\varphi(a_{2})\cdots\varphi(a_{m}).$

∎

###  3.3 Weak BM independence

A main motivation for this paper is the notion of _BM independence_ ,
introduced and investigated by J. Wysoczanski in [24, 25] as a generalization
of monotone and boolean independence.

######  Definition 3.6.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. A family
$(\mathcal{A}_{i})_{i\in I}$ of subalgebras of $\mathcal{A}$ is said to be
_bm-independent_ if the set $I$ has a partial order $\preceq$ and for any
alternating product $a_{1}\cdots a_{n}$ of elements of
$(\mathcal{A}_{i})_{i\in I}$ with $a_{k}\in\mathcal{A}_{i_{k}}$ the following
two hold:

  *     * (BM1)

if $i_{k-1}\prec i_{k}\succ i_{k-1}$ or $i_{k+1}\nsim i_{k}\succ i_{k+1}$ or
$i_{k-1}\prec i_{k}\nsim i_{k+1}$ for some $k\in[2,m-1]$, then

$a_{1}\cdots a_{m}=\varphi(a_{k})a_{1}\cdots a_{k-1}a_{k+1}\cdots a_{m}.$

  *     * (BM2)

if $i_{1}\succ\cdots\succ i_{k}\nsim\cdots\nsim i_{k+\ell}\prec\cdots\prec
i_{n}$ for some $k\in[m]$ and $\ell\in[0,m-k]$, then

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

However, as we will show next, our notion of BMT independence contains a
weaker version of BM independence as a particular case.

######  Definition 3.7.

The family of sub-algebras $(\mathcal{A}_{i})_{i\in I}$ from the previous
definition is called _weak bm-independent_ if any alternating product
$a_{1}\cdots a_{n}$ satisfies (BM2) and (weak BM1), namely, if $i_{k-1}\prec
i_{k}\succ i_{k+1}$ or $i_{k-1}\nsim i_{k}\succ i_{k+1}$ or $i_{k-1}\prec
i_{k}\nsim i_{k+1}$ for some $k\in[2,m-1]$, then

$\varphi(a_{1}\cdots a_{n})=\varphi(a_{k})\varphi(a_{1}\cdots
a_{k-1}a_{k+1}\cdots a_{n})$

It follows from the linearity of $\varphi$ that BM independence implies weak
BM independence.

######  Remark 3.8.

Note that if $G=(I,E)$ is the digraph of a partial order $(I,\preceq)$, then
$i\prec j$ if and only if $(j,i)\in E$ and $(i,j)\notin E$, and $i\nsim j$ if
and only if $(j,i)\notin E$ and $(i,j)\notin E$.

######  Theorem 3.9.

If $G=(I,E)$ is the digraph of a partial order $(I,\preceq)$, then BMT
independence and weak BM independence coincide.

###### Proof.

Let $(I,\preceq)$ be a partial order with digraph $G=(I,E)$ where
$E=\\{(i,j)\in I\times I\mid j\prec i\\}$. Suppose $(\mathcal{A},\varphi)$ is
a non-commutative probability space and $(\mathcal{A}_{i})_{i\in I}$ is a
family of sub-algebras of $\mathcal{A}$. Consider an arbitrary alternating
productof $a_{1}\cdots a_{m}$ of elements of $(\mathcal{A})_{i\in I}$ with
$a_{k}\in\mathcal{A}_{i_{k}}$ and take $\bm{i}=(i_{1},\ldots,i_{m})$.

Assume first that $(\mathcal{A})_{i\in I}$ are BMT independent with respect to
$(\varphi,G)$. Suppose $i_{k-1}\prec i_{k}\succ i_{k+1}$ or $i_{k-1}\nsim
i_{k}\succ i_{k+1}$ or $i_{k-1}\prec i_{k}\nsim i_{k+1}$. To show that weak
BM1 holds, one follows a similar argument as in _(iii)_ from Proposition 3.5
to conclude it is enough to prove that $r\sim_{\theta}r^{\prime}$ implies
$r\sim_{\pi}r^{\prime}$ for any $r,r^{\prime}\in[m]$ with $r,r^{\prime}\neq k$
where $\pi=\ker_{G}[\bm{i}]$, $\theta=\ker_{G}[\bm{i}^{\prime}]$, and
$\bm{i}=(i_{1},\ldots,i_{k-1},i_{k+1},\ldots,i_{m})$.

Suppose $r\sim_{\theta}r^{\prime}$ with $r<r^{\prime}$ and $r,r^{\prime}\neq
k$. Thus, we obtain $i_{r}=i_{r^{\prime}}$ and $i_{r}\prec i_{\ell}$ whenever
$r<\ell<r^{\prime}$, $i_{\ell}\neq i_{r}$, and $\ell\neq k$, see Definition
2.8 and Remark 3.8. Similarly to _(iii)_ from Proposition 3.5, we just need to
lift the restriction $\ell\neq k$ when $r<k<r^{\prime}$ and $i_{k}\neq i_{r}$.
Now, if $i_{r}=i_{k-1}$ or $i_{r^{\prime}}=i_{k+1}$, then $i_{r}\prec i_{k}$
by hypothesis. On the other hand, if $i_{r}\neq i_{k-1}$ and
$i_{r^{\prime}}\neq i_{k+1}$, we must have $r<k-1<k+1<r^{\prime}$, and hence
$i_{r}\prec i_{k-1}$ and $i_{r}\prec i_{k+1}$, yielding $i_{r}\prec i_{k}$ by
transitivity. Thus, weak BM1 is satisfied.

Suppose $i_{1}\succ\cdots\succ i_{k}\nsim\cdots\nsim
i_{k+\ell}\prec\cdots\prec i_{m}$. To show that BM2 holds, we will prove
$\ker_{G}[\bm{i}]$ contains only singletons. Put $\pi=\ker_{G}[\bm{i}]$ and
take $r,r^{\prime}\in[m]$ with $r\sim_{\pi}r^{\prime}$ and $r\leq r^{\prime}$.
Thus, we have $i_{r}=i_{r^{\prime}}$ and $i_{r}\prec i_{\ell}$ whenever
$r<\ell<r^{\prime}$ and $i_{\ell}\neq i_{r}$. We will show that $r$ and
$r^{\prime}$ must be equal by contradiction. Suppose $r<r^{\prime}$. Note that
$r+2\leq r^{\prime}$ since $a_{1}\cdots a_{m}$ is alternating. Now, if
$r<k+l$, then $i_{r+1}\neq i_{r}$ and $r<r+1<r^{\prime}$ with $i_{r}\nprec
i_{r+1}$. Hence, we obtain $k+\ell\leq r$. On the other hand, if
$k<r^{\prime}$, then $i_{r^{\prime}-1}\neq i_{r}$ and
$r<r^{\prime}-1<r^{\prime}$ with $i_{r^{\prime}-1}\nsucc i_{r^{\prime}}$,
yielding $r^{\prime}\leq k$ and contradicting $r<r^{\prime}$. Therefore, we
must have $r=r^{\prime}$ and so
$\ker_{G}[\bm{i}]=\\{\\{1\\},\ldots,\\{m\\}\\}$. It follows from BMT
independence that $\varphi(a_{1}a_{2}\cdots
a_{m})=\varphi(a_{1})\varphi(a_{2})\cdots\varphi(a_{m}).$ This proves
$(\mathcal{A})_{i\in I}$ are weak BM independent.

Assume now that $(\mathcal{A})_{i\in I}$ are weak BM independent. Notice that
in the first part we actually proved the following two properties: (1)
$\ker_{G}[\bm{i}]=\\{\\{k\\}\\}\cup\ker_{G}[\bm{i}^{\prime}]$ if $i_{k-1}\prec
i_{k}\succ i_{k+1}$ or $i_{k-1}\nsim i_{k}\succ i_{k+1}$ or $i_{k-1}\prec
i_{k}\nsim i_{k+1}$ for some $k\in[2,m-1]$ and (2)
$\ker_{G}[\bm{i}]=\\{\\{1\\},\ldots,\\{m\\}\\}$ if $i_{1}\succ\cdots\succ
i_{k}\nsim\cdots\nsim i_{k+\ell}\prec\cdots\prec i_{m}$ for some $k\in[m]$ and
$\ell\in[0,m-k]$. One can verify straightforward that the negation of the
hypothesis in (1) is that for every $k\in[2,m-1]$ one has $i_{k-1}\nsim
i_{k}\nsim i_{k+1}$ or $i_{k-1}\nsim i_{k}\prec i_{k+1}$ or $i_{k-1}\succ
i_{k}\nsim i_{k+1}$ or $i_{k-1}\prec i_{k}\prec i_{k+1}$ or $i_{k-1}\succ
i_{k}\succ i_{k+1}$. It then follows that any alternating sequence
$i_{1},\ldots,i_{m}$ satisfies either (1) or (2). Therefore, the BMT
independence of $(\mathcal{A})_{i\in I}$ with respect to $(\varphi,G)$ is
obtained from applying induction on the set of all integers $n\geq 1$ such
that $\varphi(a_{1}\cdots
a_{n})=\prod_{B\in\ker_{G}[\bm{i}]}\varphi((a_{k})|_{B})\text{ for any
alternating product }a_{1}\cdots a_{n}$ and applying weak BM1 or BM2
accordingly. ∎

###  3.4 Consistency

To conclude this section, we show that BMT independence is consistent in the
following way. If two families of algebras are BMT independent and each
element in the first family is tensor (resp., monotone, Boolean) independent
from any element in the second family, then the whole families are tensor
(resp., monotone, Boolean) independent, regardless of the independence
relations within each family.

To illustrate this, let us consider algebras
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$ that are BMT independent
with respect to a digraph $G$ containing exactly two edges from
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ to $\mathcal{A}_{3}$ and possibly more
edges connecting $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ as follows:

$\mathcal{A}_{1}$$\mathcal{A}_{2}$$\mathcal{A}_{3}$

Thus $\mathcal{A}_{1}$ is monotone independent from $\mathcal{A}_{3}$ and
$\mathcal{A}_{2}$ is also monotone independent from $\mathcal{A}_{3}$. One can
then ask what the independence relation between $\mathcal{A}_{3}$ and
$\langle\mathcal{A}_{1},\mathcal{A}_{2}\rangle$, the algebra generated by
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, should be. Consistency for BMT
independence means that $\mathcal{A}_{3}$ and
$\langle\mathcal{A}_{1},\mathcal{A}_{2}\rangle$ must be monotone independent
no matter what the relation between $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
is.

Before we prove that BMT independence is consistent, we need the following two
technical lemmas.

######  Lemma 3.10.

Suppose $(\mathcal{A}_{i})_{i\in I}$ is a family of sub-algebras BMT
independent with respect to a digraph $G=(I,E)$. Let $a_{1}a_{2}\cdots a_{m}$
be a product (not necessarily alternating) of elements of
$(\mathcal{A}_{i})_{i\in I}$ with $a_{k}\in\mathcal{A}_{i_{k}}$ and
$\bm{i}=(i_{1},i_{2},\ldots,i_{m})$. If there exist
$1\leq\ell<\ell^{\prime}\leq m$ so that $\ker_{G}[\bm{i}|_{S}]$ and
$\ker_{G}[\bm{i}|_{S^{c}}]$ are contained in $\ker_{G}[\bm{i}]$ with
$S=\\{\ell,\ell+1,\cdots,\ell^{\prime}-1\\}$, then

$\varphi(a_{1}\cdots a_{m})=\varphi(a_{\ell}\cdots
a_{\ell^{\prime}-1})\varphi(a_{1}\cdots a_{\ell-1}a_{\ell^{\prime}}\cdots
a_{m})$

###### Proof.

Since $\ker_{G}[\bm{i}]$, $\ker_{G}[\bm{i}|_{S}]$, and
$\ker_{G}[\bm{i}|_{S^{c}}]$ are partitions of $[m]$, $S$, and $S^{c}$,
respectively, and $S\cup S^{c}=[m]$, the assumption that both
$\ker_{G}[\bm{i}|_{S}]$ and $\ker_{G}[\bm{i}|_{S^{c}}]$ are subsets of
$\ker_{G}[\bm{i}]$ implies the partition $\ker_{G}[\bm{i}]$ is the disjoint
union of $\ker_{G}[\bm{i}|_{S}]$ and $\ker_{G}[\bm{i}|_{S^{c}}]$. Therefore,
the relation $\varphi(a_{1}\cdots a_{m})=\varphi(a_{\ell}\cdots
a_{\ell^{\prime}-1})\varphi(a_{1}\cdots a_{\ell-1}a_{\ell^{\prime}}\cdots
a_{m})$ follows directly from the definition of BMT independence. ∎

######  Lemma 3.11.

Let $G=(I,E)$ be a digraph. For any tuple $\bm{i}=(i_{1},i_{2},\ldots,i_{m})$
with $i_{k}\in I$ and any subset $S\subset[m]$, we have
$\ker_{G}[\bm{i}|_{S}]\subset\ker_{G}[\bm{i}]$ if and only if
$k_{1}\sim_{\ker_{G}[\bm{i}]}k_{2}$ for any $k_{1},k_{2}\in S$ with
$k_{1}\sim_{\ker_{G}[\bm{i}|_{S}]}k_{2}$ and
$k_{1}\nsim_{\ker_{G}[\bm{i}]}k_{2}$ for all $k_{1}\in S$ and
$k_{2}\in[m]\setminus S$.

###### Proof.

Put $\pi=\ker_{G}[\bm{i}]$ and $\pi_{S}=\ker_{G}[\bm{i}|_{S}]$. Suppose first
$k_{1}\sim_{\pi}k_{2}$ for any $k_{1},k_{2}\in S$ with
$k_{1}\sim_{\pi_{S}}k_{2}$ and $k_{1}\nsim_{\pi}k_{2}$ for $k_{1}\in S$ and
$k_{2}\in[m]\setminus S$. Let $V$ be an arbitrary block of $\pi_{S}$. Take any
$k\in V$ and let $W$ be the unique block of $\pi$ such that $k\in W$. We will
show that $V=W$. Note that $V\subset W$ since for any $k^{\prime}\in V\subset
S$ we have $k\sim_{\pi_{S}}k^{\prime}$, and hence $k\sim_{\pi}k^{\prime}$ by
hypothesis. Additionally, we have $W\subset S$ since
$k\sim_{\pi}k^{\prime\prime}$ for any $k^{\prime\prime}\in W$ and by
hypothesis $k_{1}\nsim_{\pi}k_{2}$ if $k_{1}\in S$ and $k_{2}\in[m]\setminus
S$. Now, for any $k^{\prime\prime}\in W\subset S$, we have
$k\sim_{\pi}k^{\prime\prime}$, and hence $i_{k}=i_{k^{\prime\prime}}$ and
$(i_{\ell},i_{k})\in E$ whenever $k<\ell<k^{\prime\prime}$ if $k\leq
k^{\prime\prime}$, or $k^{\prime\prime}<\ell<k$ if $k^{\prime\prime}\leq k$,
with $\ell\in[m]\supset S$, so $k\sim_{\pi_{S}}k^{\prime\prime}$ and
$k^{\prime\prime}\in V$. Thus, we get $V=W$. Therefore, since $V$ was
arbitrary, we obtain $\ker_{G}[\bm{i}|_{S}]\subset\ker_{G}[\bm{i}]$.

Suppose now $\ker_{G}[\bm{i}|_{S}]\subset\ker_{G}[\bm{i}]$. Since
$\pi=\ker_{G}[\bm{i}]$ and $\pi_{S}=\ker_{G}[\bm{i}|_{S}]$ are the partitions
of $[m]$ and $S$ determined by the equivalence relation $k_{1}\sim k_{2}$ only
if $i_{k_{1}}=i_{k_{2}}$ and $(i_{\ell_{1}},i_{k_{1}})\in E$ whenever
$k_{1}<\ell_{1}<k_{2}$ with $\ell_{1}\in[m]$ and $\ell_{1}\in S$,
respectively, we have that if $k\nsim_{\pi_{S}}k^{\prime}$ for some
$k,k^{\prime}\in S$, then $k\nsim_{\pi}k^{\prime}$ due to $S$ being a subset
of $[m]$. Thus, it only remains to show that $k_{1}\nsim_{\pi}k_{2}$ for any
$k_{1}\in S$ and $k_{2}\in[m]\setminus S$. Take any $k\in S$ and let $V$ the
unique block of $\pi_{S}$ so that $k\in V$. By hypothesis, $V$ is also a block
of $\pi$, and hence, we obtain $k\nsim_{\pi}k^{\prime}$ for any
$k^{\prime}\in[m]\setminus V$, in particular, for any
$k^{\prime}\in[m]\setminus S$. ∎

######  Proposition 3.12.

Let $(\mathcal{A},\varphi)$ be a non-commutative probability space. Suppose
$(\mathcal{A}_{i})_{i\in I}$ is a family of subalgebras bmt independent with
respect to a digraph $G=(I,E)$. If $\\{I_{j}:j\in J\\}$ is a partition of $I$
into non-empty pairwise disjoint subsets and
$\mathcal{B}_{j}=\mathrm{alg}(\mathcal{A}_{i}:i\in I_{j})$, then

  1. (i)

$(\mathcal{B}_{j})_{j\in J}$ are tensor independent if $(i,i^{\prime})\in E$
whenever $i\in I_{j}$ and $i^{\prime}\in I_{j^{\prime}}$ where
$j,j^{\prime}\in J$ with $j\neq j^{\prime}$

  2. (ii)

$(\mathcal{B}_{j})_{j\in J}$ are boolean independent if $(i,i^{\prime})\notin
E$ whenever $i\in I_{j}$ and $i^{\prime}\in I_{j^{\prime}}$ where
$j,j^{\prime}\in J$ with $j\neq j^{\prime}$

  3. (iii)

$(\mathcal{B}_{j})_{j\in J}$ are monotone independent if $J$ has a total order
$<$ so that $(i,i^{\prime})\in E$ and $(i^{\prime},i)\notin E$ whenever $i\in
I_{j}$ and $i^{\prime}\in I_{j^{\prime}}$ where $j,j^{\prime}\in J$ with
$j^{\prime}<j$

###### Proof.

Let $b_{1}b_{2}\cdots b_{n}$ be alternating product of elements of
$(\mathcal{B}_{j})_{j\in J}$ where each $b_{r}\in\mathcal{B}_{j_{r}}$ is an
alternating product of elements of $(\mathcal{A}_{i})_{i\in I_{j_{r}}}$.
Hence, each $b_{r}$ is of the form

$b_{r}=a_{(m_{0}+\cdots+m_{r-1})+1}a_{(m_{0}+\cdots+m_{r-1})+2}\cdots
a_{(m_{0}+\cdots+m_{r-1})+m_{r}}$

with $a_{k}\in\mathcal{A}_{i_{k}}$, $i_{k}\in I_{j_{r}}$, and $i_{k}\neq
i_{k+1}$. Note that we have assumed $m_{0}=0$. Take
$\ell_{k}=m_{0}+m_{1}+m_{2}+\cdots+m_{k}$ and $m=\ell_{n}$. Thus, we have

$\varphi(b_{1}b_{2}\cdots b_{n})\ =\ \varphi(a_{1}a_{2}\cdots a_{m}).$

Proof of _(i)_ . Suppose $(i,i^{\prime})\in E$ whenever $i\in I_{j}$ and
$i^{\prime}\in I_{j^{\prime}}$ where $j,j^{\prime}\in J$ with $j\neq
j^{\prime}$. Put $\bm{j}=(j_{1},j_{2},\ldots,j_{n})$ and $S_{U}=\cup_{r\in
U}[l_{r-1}+1,l_{r}]$ for each $U\in\ker[\bm{j}]\in P(n)$. We will show that
$\pi_{U}=\ker_{G}[\bm{i}|_{S_{U}}]$ is a subset of $\pi=\ker_{G}[\bm{i}]$ for
each $U\in\ker[\bm{j}]$.

Take any $U\in\ker[\bm{j}]$. Suppose $k_{1}\sim_{\pi_{U}}k_{2}$ with
$k_{1},k_{2}\in S_{U}$ with $k_{1}<k_{2}$ and let $\ell\in[n]$ so that
$k_{1}<\ell<k_{2}$ and $i_{\ell}\neq i_{k_{1}}$. Let $r_{1},t\in[n]$ with
$r_{1}\leq t$ such that $k_{1}\in[\ell_{r_{1}-1}+1,\ell_{r_{1}}]$ and
$\ell\in[\ell_{t-1}+1,\ell_{t}]$, so we have $i_{k_{1}}\in I_{j_{r}}$ and
$i_{\ell}\in I_{j_{t}}$. Note that $\ell\in S_{U}$ if and only if
$j_{t}=j_{r_{1}}$. Thus, if $\ell\notin S_{U}$, then $(i_{\ell},i_{k_{1}})\in
E$ by hypothesis since $j_{t}\neq j_{r_{1}}$; on the other hand, if $\ell\in
S_{U}$, then $(i_{\ell},i_{k_{1}})\in E$ since $k_{1}\sim_{\pi_{U}}k_{2}$. In
any case, we obtain $k_{1}\sim_{\pi}k_{2}$. Suppose now $k_{1}\in S_{U}$ and
$k_{2}\in[m]\setminus S_{U}$. Let $r_{1},r_{2}\in[n]$ such that
$k_{1}\in[l_{r_{1}-1}+1,l_{r_{1}}]$ and $k_{2}\in[l_{r_{2}-1}+1,l_{r_{2}}]$,
so we have $i_{k_{1}}\in I_{j_{r_{1}}}$ and $i_{k_{2}}\in I_{j_{r_{2}}}$.
Since $k_{2}\notin S_{U}$, we have $j_{r_{2}}\neq j_{r_{1}}$, and hence
$i_{k_{1}}\neq i_{k_{2}}$ and $k_{1}\nsim_{\pi}k_{2}$ since $I_{j_{r_{1}}}$
and $I_{j_{r_{2}}}$ are disjoint. Thus, Lemma 3.11 implies
$\ker_{G}[\bm{i}|_{S_{U}}]$ is a subset of $\ker_{G}[\bm{i}]$.

Now, since $\ker_{G}[\bm{i}]$ and $\ker_{G}[\bm{i}|_{S_{U}}]$ are partitions
of $[m]$ and $S_{U}$, respectively, and $[m]=\cup_{U\in\ker[\bm{j}]}S_{U}$,
the fact that $\ker_{G}[\bm{i}|_{S_{U}}]$ is a subset of $\ker_{G}[\bm{i}]$
for every $U\in\ker[\bm{j}]$ implies the partition $\ker_{G}[\bm{i}]$ is the
disjoint union of $\ker_{G}[\bm{i}|_{S_{U}}]$ with $U\in\ker[\bm{j}]$. It
follows from BMT independence that

$\varphi(b_{1}b_{2}\cdots b_{n})\ \
=\prod_{U\in\ker(\bm{j})}\prod_{V\in\ker_{G}(\bm{i}|_{S_{U}})}\varphi\left((a_{k})|V\right)\
\ =\prod_{U\in\ker(\bm{j})}\varphi\left((b_{r})|U\right).$

Therefore, $(\mathcal{B}_{j})_{j\in J}$ are tensor independent.

Proof of _ii)_ . Suppose now that $(i,i^{\prime})\notin E$ whenever $i\in
I_{j}$ and $i^{\prime}\in I_{j^{\prime}}$ with $j,j^{\prime}\in J$ and $j\neq
j^{\prime}$. Let $(\ell,\ell^{\prime})=(\ell_{0}+1,\ell_{1}+1)$ and
$S=\\{\ell,\ell+1,\ldots,\ell^{\prime}-1\\}$. Let $\pi$, $\pi_{S}$, and
$\pi_{S^{c}}$ denote $\ker_{G}[\bm{i}]$, $\ker_{G}[\bm{i}|_{S}]$, and
$\ker_{G}[\bm{i}|_{S^{c}}]$, respectively. We will show that the partitions
$\ker_{G}[\bm{i}|_{S}]$ and $\ker_{G}[\bm{i}|_{S^{c}}]$ are contained in
$\ker_{G}[\bm{i}]$.

Since $S$ and $S^{c}$ are sub-intervals of $[m]$, if either
$k_{1}\sim_{\pi_{S}}k_{2}$ with $k_{1},k_{2}\in S$ or
$k_{1}\sim_{\pi_{S^{c}}}k_{2}$ with $k_{1},k_{2}\in S^{c}$, then
$k_{1}\sim_{\pi}k_{2}$. Supose now $k_{1}\in S$ and $k_{2}\in S^{c}$. Thus, we
have $i_{k_{1}}\in I_{j_{1}}$ and $i_{k_{2}}\in I_{j_{r}}$ for some $r\geq 2$.
If $r=2$, then $i_{k_{1}}\neq i_{k_{2}}$ since $I_{j_{1}}$ and $I_{j_{2}}$ are
disjoint due to $j_{1}\neq j_{2}$; on the other hand, if $r>3$, then either
$i_{k_{1}}\neq i_{k_{2}}$ or $i_{k_{1}}=i_{k_{2}}$ and
$k_{1}<\ell^{\prime}\leq\ell_{2}<k_{2}$ with
$(i_{\ell^{\prime}},i_{k_{1}})\notin E$ since $i_{\ell^{\prime}}\in I_{j_{2}}$
and $j_{1}\neq j_{2}$. In any case, we obtain $k_{1}\nsim_{\pi}k_{2}$. It
follows from Lemma 3.11 that $\ker_{G}[\bm{i}|_{S}]$ and
$\ker_{G}[\bm{i}|_{S^{c}}]$ are subsets of $\ker_{G}[\bm{i}]$.

Hence, we obtain $\varphi(b_{1}b_{2}\cdots
b_{n})=\varphi(b_{1})\varphi(b_{2}\cdots b_{n})$ by Lemma 3.10. Repeating the
same argument for
$(\ell,\ell^{\prime})=(\ell_{1}+1,\ell_{2}+1),\ldots,(\ell_{n-2}+1,\ell_{n-1}+1)$,
we obtain $\varphi(b_{1}b_{2}\cdots
b_{n})=\varphi(b_{1})\varphi(b_{2})\cdots\varphi(b_{n})$. Therefore,
$(\mathcal{B}_{j})_{j\in J}$ are boolean independent.

Proof of _iii)_ . Finally, suppose now that $J$ has a total order $<$ so that
$(i,i^{\prime})\in E$ and $(i^{\prime},i)\notin E$ whenever $i\in I_{j}$ and
$i^{\prime}\in I_{j^{\prime}}$ where $j,j^{\prime}\in J$ with $j^{\prime}<j$.
Assume first $j_{1}>\cdots>j_{r-1}>j_{r}<j_{r+1}<\cdots<j_{n}$ for some $r$.
The same recursive argument used for boolean independence above can be applied
in this case, first for
$(\ell,\ell^{\prime})=(\ell_{0}+1,\ell_{1}+1),\ldots,(\ell_{r-2}+1,\ell_{r-1}+1)$,
to get $\varphi(b_{1}b_{2}\cdots
b_{n})=\varphi(b_{1})\cdots\varphi(b_{r-1})\varphi(b_{r}\cdots b_{n})$, and
second for
$(\ell^{\prime},\ell)=(\ell_{n-1}+1,\ell_{n}),\ldots,(\ell_{r-1}+1,\ell_{r})$.
Note that we go in decreasing order from $n$ to $r$ in the latter case. We
then obtain

$\varphi(b_{1}b_{2}\cdots
b_{n})=\varphi(b_{1})\cdots\varphi(b_{r-1})\varphi(b_{r})\cdots\varphi(b_{n}).$

Assume now $j_{r-1}<j_{r}>j_{r+1}$ for some $r$. Let $\ell=\ell_{r-1}+1$,
$\ell^{\prime}=\ell_{r}+1$, and $S=\\{\ell,\ell+1,\ldots,\ell^{\prime}-1\\}$.
We will show that $\ker_{G}[\bm{i}|_{S}]$ and $\ker_{G}[\bm{i}|_{S^{c}}]$ are
subsets of $\ker_{G}[\bm{i}]$.

Since $S$ is a sub-interval of $[m]$, we obtain $k_{1}\sim_{\pi}k_{2}$ for any
$k_{1},k_{2}\in S$ with $k_{1}\sim_{\pi_{S}}k_{2}$. Suppose now $k_{1}\in S$
and $k_{2}\in S^{c}$. Thus, we have $i_{k_{1}}\in I_{j_{r}}$ and $i_{k_{2}}\in
I_{j_{t}}$ for some $t\in\\{1,\ldots,r-1,r+1,\ldots,n\\}$. If $t=r+1$, then
$i_{k_{1}}\neq i_{k_{2}}$ since $I_{j_{r}}$ and $I_{j_{r+1}}$ are disjoint due
to $j_{r}\neq j_{r+1}$; on the other hand, if $t>r+1$, then either
$i_{k_{1}}\neq i_{k_{2}}$ or $i_{k_{1}}=i_{k_{2}}$ and
$k_{1}<\ell^{\prime}\leq\ell_{r+1}<k_{2}$ with
$(i_{\ell^{\prime}},i_{k_{1}})\notin E$ since $i_{k_{1}}\in I_{j_{r}}$,
$i_{\ell^{\prime}}\in I_{j_{r+1}}$, and $j_{r+1}<j_{r}$. Similar arguments
work if $t\leq r-1$. In any case, we obtain $k_{1}\nsim_{\pi}k_{2}$. It then
follows from Lemma 3.11 that $\ker_{G}[\bm{i}|_{S}]$ is contained in
$\ker_{G}[\bm{i}]$.

To obtain that $\ker_{G}[\bm{i}|_{S^{c}}]$ is also contained in
$\ker_{G}[\bm{i}]$, it only remains to prove that $k_{1}\sim_{\pi}k_{2}$
whenever $k_{1}\sim_{\pi_{S^{c}}}k_{2}$ with $k_{1},k_{2}\in S^{c}$. So, let
us assume $k_{1},k_{2}\in S^{c}$ with $k_{1}\sim_{\pi_{S^{c}}}k_{2}$. If
$k_{1},k_{2}\leq\ell-1$ or $\ell^{\prime}\leq k_{1},k_{2}$, then
$k_{1}\sim_{\pi}k_{2}$ since $\\{1,2,\ldots,\ell-1\\}$ and
$\\{\ell^{\prime},\ell^{\prime}+1,\ldots,m\\}$ are sub-intervals of $[m]$.
Thus, without loss of generality, we can assume $k_{1}\leq\ell-l$ and
$\ell^{\prime}\leq k_{2}$. The condition $k_{1}\sim_{\pi_{S^{c}}}k_{2}$ means
$i_{k_{1}}=i_{k_{2}}$ and $(i_{\ell^{\prime\prime}},i_{k_{1}})\in E$ for
$k_{1}<\ell^{\prime\prime}<k_{2}$ with $i_{\ell^{\prime\prime}}\neq i_{k_{1}}$
and $\ell^{\prime\prime}\in S^{c}$. Hence, to obtain $k_{1}\sim_{\pi}k_{2}$,
we only need to show that $(i_{\ell^{\prime\prime}},i_{k_{1}})\in E$ for
$k_{1}<\ell^{\prime\prime}<k_{2}$ with $i_{\ell^{\prime\prime}}\neq i_{k_{1}}$
and $\ell^{\prime\prime}\in S$. Take $1\leq t\leq r-1$ so that $i_{k_{1}}\in
I_{j_{t}}$. Now, if $i_{k_{1}}=i_{\ell-1}$ and $\ell^{\prime\prime}\in S$,
then $j_{t}=j_{r-1}$ and $i_{\ell^{\prime\prime}}\in I_{j_{r}}$, and hence
$i_{k_{1}}\neq i_{\ell^{\prime\prime}}$ since $I_{j_{r-1}}$ and $I_{j_{r}}$
are disjoint and $(i_{\ell^{\prime\prime}},i_{k_{1}})\in E$ since
$j_{r-1}<j_{r}$. On the other hand, if $i_{k_{1}}\neq i_{\ell-1}$, then
$k_{1}<\ell-1<\ell^{\prime}\leq k_{2}$ with $\ell-1\in S^{c}$ and
$i_{\ell-1}\in I_{j_{r-1}}$, so we must have $(i_{\ell-1},i_{k_{1}})\in E$
since $k_{1}\sim_{\pi_{S^{c}}}k_{2}$; however, $(i_{\ell-1},i_{k_{1}})\in E$
only if $j_{t}\leq j_{r-1}$, so we obtain $j_{t}<j_{r}$, and hence
$(i_{\ell^{\prime\prime}},i_{k_{1}})\in E$ for any $\ell^{\prime\prime}\in S$
since $i_{\ell^{\prime\prime}}\in I_{j_{r}}$ and $i_{k_{1}}\in I_{j_{t}}$. In
any case, we get $k_{1}\sim_{\pi}k_{2}$, and therefore
$\ker_{G}[\bm{i}|_{S^{c}}]$ is a subset of $\ker_{G}[\bm{i}]$ by Lemma 3.11.

It follows from Lemma 3.10 that $\varphi(b_{1}\cdots
b_{n})=\varphi(b_{r})\varphi(b_{1}\cdots b_{r-1}b_{r+1}\cdots b_{n})$.
Therefore, $(\mathcal{B}_{j})_{j\in J}$ are monotone independent. ∎

######  Remark 3.13.

Associativity of Boolean, monotone, and tensor independence is recaptured by
consistency for BMT independence. To demonstrate this, let us come back to the
situation described above regarding
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}$. We add an edge from
$\mathcal{A}_{1}$ to $\mathcal{A}_{2}$, so the independence digraph becomes

$\mathcal{A}_{1}$$\mathcal{A}_{2}$$\mathcal{A}_{3}$

By consistency, Proposition 3.12, we know that $\mathcal{A}_{1}$ is monotone
independent from $\left<\mathcal{A}_{2},\mathcal{A}_{3}\right>$ and
$\mathcal{A}_{2}$ is monotone independent from $\mathcal{A}_{3}$. Moreover,
$\mathcal{A}_{1}$ is monotone independent from $\mathcal{A}_{2}$, and
$\left<\mathcal{A}_{1},\mathcal{A}_{2}\right>$ is monotone independent from
$\mathcal{A}_{3}$. Both situations are equivalent due to BMT independence, but
this is nothing else than associativity of monotone independence.

##  4 Construction of BMT algebras of operators

The purpose of this section is to provide with an analytic framework for BMT
random variables. Namely, we give an explicit construction to realize any
finite family of BMT independent random variables as bounded operators on a
Hilbert space. Additionally, we show that this construction produces random
variables that are BM independent and not just weak BM independent when the
corresponding independence digraph comes from a partial order.

Let $H_{1},\ldots,H_{N}$ be complex Hilbert spaces with distinguished unit
vectors $\xi_{i}\in H_{i}$. Let $I_{i}$ be the identity operator on $H_{i}$
and let $P_{i}:H_{i}\to H_{i}$ be the orthogonal projection defined by

$P_{i}(x)=\langle x,\xi_{i}\rangle_{i}\xi_{i}.$

We consider the non-commutative probability spaces $(B(H_{i}),\varphi_{i})$
where $B(H_{i})$ is the space of bounded linear operators on $H_{i}$ and
$\varphi_{i}:B(H_{i})\to\mathbb{C}$ is the vector-state given by

$\varphi_{i}(A)=\langle A\xi_{i},\xi_{i}\rangle_{i}.$

Let $H$ denote the tensor product of Hilbert spaces $H_{1}\otimes\cdots\otimes
H_{N}$ with inner product

$\langle h_{1}\otimes\cdots\otimes h_{N},h_{1}^{\prime}\otimes\cdots\otimes
h_{N}^{\prime}\rangle=\prod_{i=1}^{N}\langle h_{i},h_{i}^{\prime}\rangle_{i}$

and let us consider the non-commutative probability space $(B(H),\varphi)$
where $\varphi(T)=\langle T\xi,\xi\rangle$ with unit vector
$\xi=\xi_{1}\otimes\cdots\otimes\xi_{N}$.

######  Definition 4.1.

Given a digraph $G_{N}=(V_{N},E_{N})$ with $V_{N}=[N]$ we define the
$\ast$-homomorphism $\pi_{i}:B(H_{i})\to B(H)$ as

$\pi_{i}(A)=P_{i,1}\otimes\cdots\otimes P_{i,i-1}\otimes A\otimes
P_{i,i+1}\otimes\cdots\otimes P_{i,N}$

where $P_{i,j}=I_{j}$, if $(i,j)\in E_{N}$, and $P_{i,j}=P_{j}$, if
$(i,j)\notin E_{N}$.

######  Proposition 4.2.

The triple $(H,\pi_{i},\xi)$ is a representation of $(B(H_{i}),\varphi_{i})$,
i.e., $\pi_{i}:B(H_{i})\to B(H)$ is a $\ast$-homomorphism and $\xi$ is a unit
vector such that $\varphi_{i}(A)=\langle\pi_{i}(A)\xi,\xi\rangle$ for each
$A\in B(H_{i})$.

###### Proof.

For each $A,B\in B(H_{i})$ we have that $\pi_{i}(AB)=\pi_{i}(A)\pi_{i}(B)$ and
$\pi_{i}(A^{*})=\pi_{i}(A)^{*}$ since $(A_{1}\otimes\cdots\otimes
A_{N})(B_{1}\otimes\cdots\otimes B_{N})=(A_{1}B_{1}\otimes\cdots\otimes
A_{N}B_{N})$, $(A_{1}\otimes\cdots\otimes
A_{N})^{*}=(A_{1}^{*}\otimes\cdots\otimes A_{N}^{*})$ and
$P_{i,j}=P_{i,j}^{2}=P_{i,j}^{*}$. Thus, $\pi_{i}$ is a $*$-homomorphism.
Moreover, since $P_{i,j}\xi_{j}=\xi_{j}$ we obtain $\varphi_{i}(A)=\langle
A\xi_{i},\xi_{i}\rangle_{i}\prod_{j\neq i}\langle
P_{i,j}\xi_{j},\xi_{j}\rangle_{j}=\langle\pi_{i}(A)\xi,\xi\rangle$.∎

######  Remark 4.3.

Notice that for all $A,B\in B(H_{i})$ the projection $P_{i}$ satisfies

$\displaystyle\langle AP_{i}B\xi_{i},\xi_{i}\rangle_{i}=\langle
A\xi_{i},\xi_{i}\rangle_{i}\langle
B\xi_{i},\xi_{i}\rangle_{i}\quad\text{and}\quad P_{i}AP_{i}=\langle
A\xi_{i},\xi_{i}\rangle_{i}P_{i}.$

Indeed, since $P_{i}(x)=\langle x,\xi_{i}\rangle_{i}\xi_{i}$ we obtain
$\langle AP_{i}(B\xi_{i}),\xi_{i}\rangle_{i}=\langle A\langle
B\xi_{i},\xi_{i}\rangle\xi_{i},\xi_{i}\rangle_{i}$ and
$[P_{i}AP_{i}](x)=\langle x,\xi_{i}\rangle_{i}P_{i}(A\xi_{i})=\langle
A\xi_{i},\xi_{i}\rangle_{i}P_{i}(x)$ for all $x\in H_{i}$. In terms of the
functionals $\varphi_{i}$, these relations become

$\varphi_{i}(AP_{i}B)=\varphi_{i}(B)\varphi_{i}(A)\quad\text{and}\quad
P_{i}AP_{i}=\varphi_{i}(A)P_{i}.$

######  Theorem 4.4.

The family of $\ast$-subalgebras $\\{\pi_{i}(B(H_{i}))\\}_{i=1}^{N}$ is BMT
independent in $(B(H),\varphi)$ with respect to $G_{N}$.

###### Proof.

Let us denote $\mathcal{A}_{i}=\pi_{i}(B(H_{i}))$ for $i=1,\ldots,N$. We
consider the monomial $a_{1}a_{2}\cdots a_{n}$ with
$a_{j}\in\mathcal{A}_{i_{j}}$. Let $A_{j}\in B(H_{i_{j}})$ be such that
$a_{j}=\pi_{i_{j}}(A_{j})$ for $j=1,\ldots,n$, this is

$a_{j}=P_{i_{j},1}\otimes\cdots\otimes P_{i_{j},i_{j}-1}\otimes A_{j}\otimes
P_{i_{j},i_{j}+1}\otimes\cdots\otimes P_{i_{j},N}.$

Put $\bm{i}=(i_{1},\ldots,i_{n})$ and $m=\\#\ker[\bm{i}]$. We also define

$C=\\{c_{1}<\cdots<c_{m}\\}=\\{i_{j}:1\leq j\leq n\\}\quad\text{and}\quad
D=\\{d_{1}<\cdots<d_{N-m}\\}=[N]\setminus C.$

The product $a_{1}a_{2}\cdots a_{n}$ can be written as
$B_{1}\otimes\cdots\otimes B_{N}$ where
$B_{d_{k}}\in\\{I_{d_{k}},P_{d_{k}}\\}$ for $k=1,\ldots,N-m$ and $B_{c_{k}}$
is of the form

$Q_{r_{1}(k)}A_{s_{1}(k)}Q_{r_{2}(k)}A_{s_{2}(k)}\cdots
Q_{r_{l(k)}(k)}A_{s_{l(k)}(k)}Q_{r_{l(k)+1}(k)}$

for $k=1,\ldots,m$ with
$\\{s_{1}(k)<\cdots<s_{l(k)}(k)\\}=\\{j\in[n]:i_{j}=c_{k}\\}$ and
$Q_{r_{w}(k)}=I_{c_{k}}$, if $(i_{l},i_{s_{w}(k)})\in E_{N}$ for all
$s_{w-1}(k)<l<s_{w}(k)$, and $Q_{r_{w}(k)}=P_{c_{k}}$, otherwise. Note that
$Q_{r_{w}(k)}\in\\{I_{c_{k}},P_{c_{k}}\\}$ comes from the fact that the
variables $a_{j}$ not in $\mathcal{A}_{c_{k}}$ have either $P_{c_{k}}$ or
$I_{c_{k}}$ in their $c_{k}$-term in the tensor product that defines them, so
between $A_{s_{w-1}(k)}$ and $A_{s_{w}(k)}$ there is a product of
$P_{c_{k}}$’s and $I_{c_{k}}$’s which is $I_{c_{k}}$ only if all the elements
in between are $I_{c_{k}}$. Then, we have that

$\displaystyle\varphi(a_{1}a_{2}\cdots a_{n})$
$\displaystyle=\left(\prod_{k=1}^{N-m}\langle
B_{d_{k}}\xi_{d_{k}},\xi_{d_{k}}\rangle_{d_{k}}\right)\left(\prod_{k=1}^{m}\langle
B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}\right)=\prod_{k=1}^{m}\langle
B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}$

Note that $\langle B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}=\langle
A_{s_{1}(k)}Q_{r_{2}(k)}A_{s_{2}(k)}\cdots
Q_{r_{l(k)}(k)}A_{s_{l(k)}(k)}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}$. Indeed,
if $Q_{r_{1}(k)}=I_{c_{k}}$, since $Q_{r_{l(k)+1}(k)}\xi_{c_{k}}=\xi_{c_{k}}$
we obtain that

$\langle B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}=\langle
A_{s_{1}(k)}Q_{r_{2}(k)}A_{s_{2}(k)}\cdots
Q_{r_{l(k)}(k)}A_{s_{l(k)}(k)}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}};$

on the other hand, if $Q_{r_{1}(k)}=P_{c_{k}}$, Remark 4.3 and the fact that
$Q_{r_{l(k)+1}(k)}\xi_{c_{k}}=P_{c_{k}}\xi_{c_{k}}$ imply

$\langle B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}=\langle
A_{s_{1}(k)}Q_{r_{2}(k)}A_{s_{2}(k)}\cdots
Q_{r_{l(k)}(k)}A_{s_{l(k)}(k)}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}.$

Let us consider
$W_{k}=\\{w_{1}(k)<\cdots<w_{t(k)}(k)\\}=\\{w:Q_{r_{w}(k)}=P_{c_{k}}\\}$ for
each $k=1,\ldots,m$. It follows from the definition of $\varphi_{c_{k}}$ that

$\displaystyle\varphi_{c_{k}}(B_{c_{k}})$
$\displaystyle=\varphi_{c_{k}}\bigg{[}\left(\prod_{l=1}^{w_{1}(k)-1}A_{s_{l}(k)}\right)P_{c_{k}}\cdots
P_{c_{k}}\left(\prod_{l=w_{t(k)-1}(k)}^{w_{t(k)}(k)-1}A_{s_{l}(k)}\right)P_{c_{k}}\left(\prod_{l=w_{t(k)}(k)}^{l(k)}A_{s_{l}(k)}\right)\bigg{]}$
$\displaystyle=\varphi_{c_{k}}\left(\prod_{l=1}^{w_{1}(k)-1}A_{s_{l}(k)}\right)\cdots\varphi_{c_{k}}\left(\prod_{l=w_{t(k)-1}(k)}^{w_{t(k)}(k)-1}A_{s_{l}(k)}\right)\varphi_{c_{k}}\left(\prod_{l=w_{t(k)}(k)}^{l(k)}A_{s_{l}(k)}\right)$

where the second equality comes from Remark 4.3. Now, recall that
$\ker_{G_{N}}[\bm{i}]$ is a refinement of $\ker[\bm{i}]$ where $s_{w-1}(k)\sim
s_{w}(k)$ if $(i_{l},i_{s_{w}(k)})\in E_{N}$ for all $s_{w-1}(k)<l<s_{w}(k)$.
Then, the block $\\{s_{1}(k)<\cdots<s_{l(k)}(k)\\}\in\ker[\bm{i}]$
corresponding to the subalgebra $c_{k}$ is decomposed in
$\ker_{G_{N}}[\bm{i}]$ into $\\{s_{1}(k),\ldots,s_{w_{1}(k)-1}(k)\\}$,
$\ldots$, $\\{s_{w_{t(k)-1}(k)-1}(k),\ldots,s_{w_{t(k)}(k)-1}(k)\\}$,
$\\{s_{w_{t(k)}}(k),\ldots,s_{l(k)}\\}$. Thus, we obtain that

$\displaystyle\varphi(a_{1}a_{2}\cdots a_{n})$
$\displaystyle=\prod_{k=1}^{m}\langle
B_{c_{k}}\xi_{c_{k}},\xi_{c_{k}}\rangle_{c_{k}}=\prod_{V\in\ker_{G_{N}}[\bm{i}]}\varphi_{c_{k}}\left(\prod_{s\in
V}^{\rightarrow}A_{s}\right)$

where $c_{k}$ depends on each block $V$ being the common value of all $i_{j}$
with $j\in V$. For each $V\in\ker_{G_{N}}[i]$ we have

$\displaystyle\varphi_{c_{k}}\left(\prod_{s\in
V}^{\rightarrow}A_{s}\right)=\varphi\left[\pi_{c_{k}}\left(\prod_{s\in
V}^{\rightarrow}A_{s}\right)\right]=\varphi\left[\prod_{s\in
V}^{\rightarrow}\pi_{c_{k}}\left(A_{s}\right)\right]=\varphi\left[\prod_{s\in
V}^{\rightarrow}a_{s}\right]$

due to $\varphi_{c_{k}}=\varphi\circ\pi_{c_{k}}$, the fact that $\pi_{c_{k}}$
is an algebra homomorphism, and the definition of the $a_{s}$. Therefore, we
get

$\displaystyle\varphi(a_{1}a_{2}\cdots
a_{n})=\prod_{V\in\ker_{G_{N}}[\bm{i}]}\varphi\left[\prod_{s\in
V}^{\rightarrow}a_{s}\right]=\prod_{V\in\ker_{G_{N}}[\bm{i}]}\varphi\left[(a_{i})|V\right].$

This proves that the subalgebras $\\{\pi_{j}(B(H_{j}))\\}_{1\leq j\leq N}$ are
$BMT$-independent in $(B(H),\varphi)$ with respect to $G_{N}$. ∎

######  Remark 4.5.

If we consider the construction of the homomorphisms $\pi_{i}$ and $G$ is the
digraph of a partial order $(V_{N},\preceq)$, then we will have $\xi\prec\rho$
if and only if $P_{\rho,\xi}=I_{\xi}$ and $P_{\xi,\rho}=P_{\rho}$ and
$\xi\nsim\rho$ if and only if $P_{\rho,\xi}=P_{\xi}$ and
$P_{\xi,\rho}=P_{\rho}$.

We already prove in the Theorem 3.9 that when $G$ is the digraph of a partial
order on a finite partially ordered set, BMT subalgebras with respect to $G$
satisfy BM2. However, in general they do not satisfy BM1, but a weaker version
of it (the weak BM1 property). However, in the tensor model we have built, we
can implement BM independence when considering such $G$. We establish this
result in the following theorem giving another construction of a finite
collection of $BM$ subalgebras different than the given in [24].

######  Theorem 4.6.

Suppose $G=([N],E)$ is the digraph of a partial order $\preceq$ on $[N]$. Then
the family of $\ast$-subalgebras $\\{\pi_{i}(B(H_{i})\\}_{i=1}^{N}$ is BM
independent.

###### Proof.

Since the subalgebras $\\{\pi_{i}(B(H_{i}))\\}_{i=1}^{N}$ are BMT independent,
they satisfy weak BM1 and BM2, so we only need to show that BM1 holds. Take
$\mathcal{A}_{i}=\pi_{i}(B(H_{i}))$ for $i\in[N]$. Assume
$a_{1}\in\mathcal{A}_{\xi}$, $a_{2}\in\mathcal{A}_{\rho}$,
$a_{3}\in\mathcal{A}_{\eta}$ with $\xi,\rho,\eta\in[N]$ satisfying
$\xi\prec\rho\succ\eta$, $\xi\nsim\rho\succ\eta$, or $\xi\prec\rho\nsim\eta$.
Recall that each variable $a\in\mathcal{A}_{i}$ is of the form

$a=P_{i,1}\otimes\cdots\otimes P_{i,i-1}\otimes T\otimes
P_{i,i+1}\otimes\cdots\otimes P_{i,N}=\pi_{i}(T)$

for some $T\in B(H_{i})$. So, let $T_{1}\in B(H_{\xi})$, $T_{2}\in
B(H_{\rho})$, and $T_{3}\in B(H_{\eta})$ be such that
$a_{1}=\pi_{\xi}(T_{1})$, $a_{2}=\pi_{\rho}(T_{2})$ and
$a_{3}=\pi_{\eta}(T_{3})$. As in the proof of Theorem 4.4, we take
$B_{i},C_{i}\in B(H_{i})$ such that $a_{1}a_{2}a_{3}=B_{1}\otimes\cdots\otimes
B_{N}$ and $a_{1}a_{3}=C_{1}\otimes\cdots\otimes C_{N}$. To show BM1 holds, it
is enough to prove $B_{i}=C_{i}$ for $i\neq\rho$ and
$B_{\rho}=\varphi(a_{2})C_{\rho}$.

Observe that $B_{j}=P_{\xi,j}P_{\rho,j}P_{\eta,j}$ and
$C_{j}=P_{\xi,j}P_{\eta,j}$ for $j\ \neq\xi,\rho,\eta$. If $P_{\xi,j}=P_{j}$
or $P_{\eta,j}=P_{j}$, then $B_{j}=C_{j}=P_{j}$ because each operator
$P_{\xi,j}$, $P_{\rho,j}$, or $P_{\eta,j}$ is either the projection $P_{j}$ or
the identity $I_{j}$. On the other hand, if $P_{\xi,j}=P_{\eta,j}=I_{j}$, due
to the definition of the operators $P_{i,j}$ and since $G$ is the digraph
associated to $\preceq$, we have $j\prec\eta$ and $j\prec\xi$. But, by
hypothesis, either $\xi\prec\rho$ or $\eta\prec\rho$ holds, and hence
$j\prec\rho$ and $P_{\rho,j}=I_{j}$. Therefore, $B_{j}=C_{j}$ for
$j\neq\xi,\rho,\eta$.

To prove $B_{j}=C_{j}$ for $j=\xi,\eta$ and $B_{\rho}=\varphi(a_{2})C_{\rho}$,
we consider two cases: $\xi=\eta$ and $\xi\neq\eta$. Suppose first $\xi=\eta$.
Thus, none of the two conditions $\xi\nsim\rho\succ\eta$ or
$\xi\prec\rho\nsim\eta$ holds, so we must have $\xi\prec\rho$, and hence
$P_{\rho,\xi}=I_{\xi}$ and $P_{\xi,\rho}=P_{\rho}$. Moreover,
$B_{\xi}=T_{1}P_{\rho,\xi}T_{3}$, $C_{\xi}=T_{1}T_{3}$,
$B_{\rho}=P_{\xi,\rho}T_{2}P_{\xi,\rho}$, and
$C_{\rho}=P_{\xi,\rho}P_{\xi,\rho}$ since $a_{1}=\pi_{\xi}(T_{1})$,
$a_{2}=\pi_{\rho}(T_{2})$, and $a_{3}=\pi_{\eta}(T_{3})$. Hence,
$B_{\xi}=C_{\xi}$ due to $P_{\rho,\xi}=I_{\xi}$ and
$B_{\rho}=P_{\rho}T_{2}P_{\rho}=\varphi(a_{2})C_{\rho}$ due to
$P_{\xi,\rho}=I_{\rho}$ and Remark 4.3.

Suppose now $\xi\neq\eta$. In this case, we have that
$B_{\xi}=T_{1}P_{\rho,\xi}P_{\eta,\xi}$, $C_{\xi}=T_{1}P_{\eta,\xi}$,
$B_{\eta}=P_{\xi,\eta}P_{\rho,\eta}T_{3}$, $C_{\eta}=P_{\xi,\eta}T_{3}$,
$B_{\rho}=P_{\xi,\rho}T_{2}P_{\eta,\rho}$, and
$C_{\rho}=P_{\xi,\rho}P_{\eta,\rho}$. Since $P_{i,j}=P_{j}$ if $i\nsim j$ or
$i\prec j$, then $P_{\xi,\rho}=P_{\rho}=P_{\eta,\rho}$ provided
$\xi\prec\rho\succ\eta$, $\xi\prec\rho\nsim\eta$ or $\xi\nsim\rho\succ\eta$.
Thus, Remark 4.3 gives
$B_{\rho}=P_{\rho}T_{2}P_{\rho}=\varphi_{\rho}(T_{2})P_{\rho}=\varphi(a_{2})C_{\rho}$.
For $B_{\xi}=C_{\xi}$ and $B_{\eta}=C_{\eta}$, let us consider three sub-
cases: $\xi\nsim\eta$, $\xi\prec\eta$ and $\xi\succ\eta$. First, if
$\xi\nsim\eta$, so we get $P_{\eta,\xi}=P_{\xi}$ and $P_{\xi,\eta}=P_{\eta}$,
and hence $B_{\xi}=T_{1}P_{\xi}=C_{\xi}$ and $B_{\eta}=P_{\eta}T_{3}=C_{\eta}$
since each $P_{\rho,\xi}$ and $P_{\rho,\eta}$ is either an identity $I_{j}$ or
a projection $P_{j}$. Second, if $\xi\prec\eta$, so we get
$P_{\eta,\xi}=I_{\xi}$ and $P_{\xi,\eta}=P_{\eta}$, and hence
$B_{\xi}=T_{1}P_{\rho,\xi}$ and $C_{\xi}=T_{1}$; moreover, since
$P_{\rho,\eta}$ is either the identity $I_{\eta}$ or the projection
$P_{\eta}$, we obtain $B_{\eta}=P_{\eta}T_{3}=C_{\eta}$. Note that if
$\eta\prec\rho$, then $\xi\prec\eta\prec\rho$, and hence
$P_{\rho,\xi}=I_{\xi}$ and $B_{\xi}=C_{\xi}$; on the other hand, if
$\eta\nsim\rho$, then we must have $\rho\succ\xi$, and hence
$P_{\rho,\xi}=I_{\xi}$ and $B_{\xi}=C_{\xi}$. Finally, if $\xi\succ\eta$,
follwing similar arguments to the case $\xi\prec\eta$, we get
$B_{\xi}=T_{1}P_{\xi}=C_{\xi}$ since $P_{\eta,\xi}=P_{\xi}$ and
$B_{\eta}=P_{\xi,\eta}T_{3}=C_{\eta}$ since $\rho\succ\eta$ and
$P_{\rho,\eta}=I_{\eta}$. ∎

Let us mention that the idea of using tensor products and rank 1 projection to
construct independent algebras of operators is not new. Lenczewski [13] gave a
tensor model for Boolean independent random variables, which was extended by
Franz to include monotone and anti-monotone notions of independence in [7].

More interestingly and related to our construction, Lenczewski gave a tensor
model of $\Lambda$-boolean (mixtures of Boolean and tensor) independence and
$\Lambda$-monotone (mixtures of monotone and tensor) independence, [10]. More
recently he also gave a construction in [13] for $c$-monotone independence
[6].

##  5 BMT Central Limit Theorem

In this section, we prove the Central Limit Theorem (CLT) for BMT independent
random variables together with some of its properties regarding the possible
limiting distributions. In particular, we recover the known CLTs for Boolean,
monotone, and tensor independence and give sufficient conditions for the non-
compactess of the limiting measure.

Through the entire section, it is assumed that we are given a non-conmutative
probability space $(\mathcal{A},\varphi)$ together with a sequence of random
variables $(a_{i})_{i=1}^{\infty}$ that are identically distributed with zero
mean and unit variance 222As it is customary in non-commutative probability,
the identically distributed assumption can be relaxed to having uniformly
bounded moments, i.e., $\sup_{i}|\varphi(a_{i}^{n})|<\infty$ for each integer
$n\geq 1$.). Each finite sequence $a_{1},a_{2},\ldots,a_{N}$ is assumed to be
BMT independent with respect to a digraph $G_{N}=(V_{N},E_{N})$ with
$V_{N}=[N]$ and $G_{N-1}\subset G_{N}$.

###  5.1 Central Limit Theorem

The Central Limit Theorem for BMT independent random variables refers then to
determining the limiting distribution of the normalized sum
$(a_{1}+\cdots+a_{N})/\sqrt{N}$. Concretely, for each integer $k\geq 1$, and
up to an error term of order $N^{-1/2}$, we compute the value of

$\varphi\left[\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{k}\right].$

A first step is to single out the monomial containing singletons, as in most
of the combinatorial proofs of the CLT.

######  Proposition 5.1 (Singleton condition).

Let $a_{1},\ldots,a_{N}$ be centered BMT independent random variables. Let
$\bm{i}:[m]\to[N]$ be a set of indices. If $\pi=\ker[\bm{i}]$ contains a
singleton, then $\varphi(a_{i_{1}}\cdots a_{i_{m}})=0.$

###### Proof.

Notice that this follows directly from the definition of BMT independence and
the fact that $\varphi(a_{i})=0$ for any $i$. Indeed, if $\\{k\\}$ is a
singleton in $\ker[\bm{i}]$, so is in $\ker_{G}[\bm{i}]$, which is a
refinement of the former, and thus

$\varphi(a_{i_{1}}\cdots
a_{i_{m}})=\varphi(a_{i_{k}})\prod_{\begin{subarray}{c}V\in\ker_{G}[\bm{i}]\\\
V\neq\\{k\\}\end{subarray}}\varphi\bigg{(}\prod_{k\in
V}^{\rightarrow}a_{i_{k}}\bigg{)}=0.$

∎

######  Proposition 5.2.

Suppose $\bm{i}:[2m]\to[N]$ is such that $\pi=\ker[\bm{i}]$ is pair partition
in $\mathcal{P}_{2}(2m)$ and let $G_{\bm{i}}$ denote the independence graph of
$a_{i_{1}},\ldots,a_{i_{2m}}$. Thus, we have

$\varphi(a_{i_{1}}\cdots a_{i_{m}})=\begin{cases}1,\text{ if
}G_{\pi(\bm{i})}\subseteq G_{\bm{i}},\\\ 0,\text{ otherwise.}\end{cases}$

###### Proof.

Note first that the condition $G_{\pi(\bm{i})}\subseteq G_{\bm{i}}$ is
equivalent to $E_{\pi(\bm{i})}\subseteq E_{\bm{i}}$ since the graphs
$G_{\pi(\bm{i})}$ and $G_{\bm{i}}$ have the same set of vertices. Now, the BMT
independence of $a_{1},\ldots,a_{N}$ implies

$\displaystyle\varphi(a_{i_{1}}\cdots
a_{i_{m}})=\prod_{V\in\ker_{G}[\bm{i}]}\varphi\bigg{(}\prod_{k\in
V}^{\rightarrow}a_{i_{k}}\bigg{)}$

where $\ker_{G}[\bm{i}]$ is a refinement of $\ker[\bm{i}]$. Now, if
$\ker_{G}[\bm{i}]\neq\ker[\bm{i}]$, then $\ker_{G}[\bm{i}]$ has a singleton
since $\ker[\bm{i}]$ is a pair partition. This in turns implies by Proposition
5.1, that $\varphi(a_{i_{k}})=0$, since $a_{i_{k}}$ are assumed to be
centered. On the other hand, if $\ker_{G}[\bm{i}]=\ker[\bm{i}]$, we obtain
$\varphi(a_{i_{1}}\cdots
a_{i_{m}})=\prod_{V\in\ker_{G}[\bm{i}]}\varphi(\prod_{k\in
V}^{\rightarrow}a_{i_{k}})=1$ since $\varphi(a^{2}_{i_{k}})=1$ for each $k$.
Finally, the condition $\ker_{G}[\bm{i}]=\ker[\bm{i}]$ is equivalent to the
condition $G_{\pi(\bm{i})}\subseteq G_{\bm{i}}$ by Lemma 2.13. ∎

######  Theorem 5.3 (BMT central limit theorem).

Suppose $a_{1},\ldots,a_{N}$ are centered variables with unit variance,
uniformly bounded moments of all orders, and independence graph $G_{N}$. Then
for any integer $m\geq 1$ we have

$\displaystyle\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\sum_{\pi\in
P_{2}(m)}\ \ N^{-m/2}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker({\bm{i}})=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}\quad+\quad O(N^{-1/2})\ .$ (1)

where $G_{\pi}$ is the nesting-crossing graph of $\pi$ and $G_{{\bm{i}}}$ is
independence graph fo $a_{i_{1}},a_{i_{2}},\ldots,a_{i_{2k}}$ for
${\bm{i}}=(i_{1},i_{2},\ldots,i_{m})$.

###### Proof.

Observe that

$\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\frac{1}{N^{m/2}}\sum\limits_{i_{1}=1}^{N}\cdots\sum\limits_{i_{m}=1}^{N}\left(\prod_{k=1}^{m}a_{i_{k}}\right).$

From the fact that $\bm{i}\sim\bm{j}$ if $\ker[\bm{i}]=\ker[\bm{j}]$ defines
an equivalence relation, we can split the set of functions $\bm{i}:[m]\to[N]$
into disjoint sets. Using this and the linearity of $\varphi$, we have that

$\varphi\left[\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}\right]=\frac{1}{N^{m/2}}\sum\limits_{\pi\in\mathcal{P}(m)}\sum\limits_{\begin{subarray}{c}\bm{i}:[m]\to[N]\\\
\ker[i]=\pi\end{subarray}}\varphi(a_{i_{1}}\cdots a_{i_{m}}).$

The variables $(a_{i})$ are BMT independent, so we get

$\varphi\left[\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}\right]=\frac{1}{N^{m/2}}\sum\limits_{\pi\in\mathcal{P}(m)}\sum\limits_{\begin{subarray}{c}\bm{i}:[m]\to[N]\\\
\ker[i]=\pi\end{subarray}}\prod_{V\in\ker_{G}[i]}\varphi((a_{i_{k}})|k\in V).$

From Proposition 5.1, if $\pi$ contains a singleton, then
$\prod_{V\in\ker_{G}[i]}\varphi((a_{i_{k}})|k\in V)=0$. So, only partitions
$\pi$ with at least two elements per block contribute in the sum above. Take
$C>0$ such that $C\geq\sup_{n\leq m}|\varphi(a_{i}^{n})|$. Since
$i_{k}=i_{k^{\prime}}$ for any $k,k^{\prime}\in V$ and $V\in\ker_{G}[\bm{i}]$,
we have that

$\prod_{V\in\ker_{G}[\bm{i}]}|\,\varphi((a_{i_{k}})|k\in V)\,|\leq
C^{|\ker_{G}[i]|}\leq C^{m}.$

Thus, for any partition $\pi\in\mathcal{P}(m)$, we obtain

$\displaystyle\left|{\sum\limits_{\begin{subarray}{c}i:[m]\to[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}\prod_{V\in\ker_{G}[i]}\varphi((a_{i_{k}})|k\in
V)}\right|$ $\displaystyle\leq
C^{m}\sum\limits_{\begin{subarray}{c}\bm{i}:[m]\to[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}1\leq C^{m}N^{|\pi|}.$

This implies

$\displaystyle N^{-m/2}\sum\limits_{\begin{subarray}{c}\pi\in\mathcal{P}(m)\\\
|\pi|<m/2\end{subarray}}\left|{\sum\limits_{\begin{subarray}{c}i:[m]\to[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}\prod_{V\in\ker_{G}[i]}\varphi((a_{i_{k}})|k\in
V)}\right|\leq\sum\limits_{\begin{subarray}{c}\pi\in\mathcal{P}_{\chi}(m)\\\
|\pi|<m/2\end{subarray}}C^{m}N^{|\pi|-m/2}=O(N^{-1/2}).$

Let us denote by $\tilde{\mathcal{P}}(m)$ the set of all partitions
$\pi\in\mathcal{P}(m)$ with no singletons. We have proved that only partitions
in $\tilde{\mathcal{P}}(m)$ give a non-zero contribution and if $|\pi|<m/2$
this contribution is of order $O(N^{-1/2})$. Thus noticing that $|\pi|\leq
m/2$ for any $\pi\in\tilde{\mathcal{P}}(m)$ with equality only if $\pi$ is a
pair partition we arrive to

$\displaystyle\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\sum_{\pi\in
P_{2}(m)}\ \ N^{-m/2}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker({\bm{i}})=\pi\end{subarray}}\varphi(a_{i_{1}}\cdots a_{i_{m}})+\quad
O(N^{-1/2})\ .$

Finally, Proposition 5.2 gives

$\displaystyle\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\sum_{\pi\in
P_{2}(m)}\ \ N^{-m/2}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker({\bm{i}})=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}\quad+\quad O(N^{-1/2})\ .$

∎

######  Remark 5.4.

The set of pairing partitions $P_{2}(m)$ is empty if $m$ is odd. Thus, Theorem
5.3 states that odd moments in the CLT for BMT random variables always vanish
as $N\to\infty$, and therefore the limiting distribution must be symmetric if
it exists.

Moreover, the even moments satisfy the Carleman’s condition, namely,
$\sum^{\infty}_{k=1}m_{2k}^{-1/2k}=+\infty$ where $m_{2k}$ denotes the $2k$-th
moment since Theorem 5.3 gives that each $m_{2k}$ is bounded by
$|\\#P_{2}(2k)|=1\cdot 3\cdots(2k-1)$. Consequently, the limiting
distribution, if it exists, is determined by moments, and convergence in
distribution for BMT central limit theorem is equivalent to convergence in
moments.

######  Corollary 5.5 (Boolean Central Limit theorem).

If the independence graph $G_{N}$ of the variables $a_{1},a_{2},\ldots,a_{N}$
is the null graph for every integer $N\geq 1$, then
$(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in moments as $N\to\infty$ to the
Bernoulli distribution $(\delta_{-1}+\delta_{+1})/2$.

###### Proof.

By Remark 5.4 is is enough to consider even moments, so let $m=2k$. For all
$\bm{i}:[m]\to[N]$, $G_{\mathbf{i}}$ is a graph with no edges. So,
$\mathbf{1}_{G_{\pi(\bm{i})}\subseteq G_{\bm{i}}}$ is $1$ only when
$G_{\pi(\bm{i})}$ has no edges, i.e. only when
$\pi=\\{\\{1,2\\},\ldots,\\{2k-1,2k\\}\\}$. Thus,

$\displaystyle\lim_{N\to\infty}\varphi\left(\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{2k}\right)$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N^{k}}\sum\limits_{\underset{\ker[i]=\\{\\{1,2\\},\ldots,\\{2k-1,2k\\}\\}}{\bm{i}:[m]\to[N]}}1$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N^{k}}N(N-1)\cdots(N-k+1)=1.$

∎

######  Corollary 5.6 (Tensor Central Limit theorem).

If the independence graph $G_{N}$ of the variables $a_{1},a_{2},\ldots,a_{N}$
is the complete graph, then $(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in
moments as $N\to\infty$ to the Gaussian distribution
$\frac{1}{\sqrt{2\pi}}\exp(-t^{2}/2)\,dt$.

###### Proof.

Again, we only consider $m=2k$. Now, for all $\bm{i}:[m]\to[N]$,
$G_{\mathbf{i}}$ is a complete graph. So,
$\mathbf{1}_{G_{\pi(\bm{i})}\subseteq G_{\mathbf{i}}}$ is one for all
$\pi\in\mathcal{P}_{2}(2k)$. Thus,

$\displaystyle\lim_{N\to\infty}\varphi\left(\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{2k}\right)$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N^{k}}\sum\limits_{\pi\in\mathcal{P}_{2}(2k)}\sum\limits_{\underset{\ker[\bm{i}]=\pi}{\bm{i}:[m]\to[N]}}1$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N^{k}}N(N-1)\cdots(N-k+1)\\#\mathcal{P}_{2}(2k)$
$\displaystyle=\\#\mathcal{P}_{2}(2k).$

∎

######  Corollary 5.7 (Monotone Central Limit theorem).

If the independence graph $G_{N}$ of the variables $a_{1},a_{2},\ldots,a_{N}$
has edge set $E_{N}=\\{(j,i)\in[N]^{2}:i<j\\}$, then
$(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in moments as $N\to\infty$ to the
arcsine distribution $\frac{1}{2\pi}\sqrt{4-t^{2}}\,dt$.

###### Proof.

Let $\pi\in\mathcal{P}_{2}(2k)$, and $G_{\pi}$ be its nesting-crossing graph.
Note that if $\pi$ has a crossing, $G_{\mathbf{i}}$ cannot be a subgraph of
$G_{\pi}$ for all $\mathbf{i}$, since $G_{\mathbf{i}}$ has no double edges
while $G_{\pi}$ does.

So let $\pi\in\mathcal{NC}_{2}(2k)$, with blocks $b_{1},\dots,b_{k}$. The
nesting-crossing graph is just a nesting graph. In this case in order that
$G_{\pi}\subset G_{\mathbf{i}}$ we need that $i_{l}<j_{m}$ whenever $b_{l}$ is
nested with $b_{m}$. To count such indices we such a unorder subset of indices
of size $k$, and count the number of ordering in the indices which satify this
condition. Thus the cardinality of the set
$\\{i\in[N]_{\pi}^{k}|G_{\pi}\subset G_{\mathbf{i}}\\}$ equals
$\binom{N}{k}*M(\pi)$, where $M(\pi)$ is the number of labellings
$L:\pi\to[n]$ such $L(b_{i})\leq L(b_{j})$ if $b_{i}$ is nested in $b_{j}$.

In the limit for a partition $\pi\in\mathcal{NC}_{2}(2k)$ we get that

$\lim_{N\to\infty}\frac{1}{N^{k}}\sum\limits_{\begin{subarray}{c}\bm{i}:[m]\to[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}\mathbf{1}_{G_{\pi}\subseteq
G_{\mathbf{i}}}=\lim_{N\to\infty}\frac{1}{N^{k}}\binom{N}{k}M(\pi)=\frac{M(\pi)}{k!}.$

Thus summing over all pair partitions we get

$\displaystyle\lim_{N\to\infty}\varphi\left(\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{2k}\right)$
$\displaystyle=\sum\limits_{\pi\in\mathcal{NC}_{2}(2k)}\lim_{N\to\infty}\frac{1}{N^{k}}\sum\limits_{\begin{subarray}{c}\bm{i}:[m]\to[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}\mathbf{1}_{G_{\pi}\subseteq G_{\mathbf{i}}}$
$\displaystyle=\sum\limits_{\pi\in\mathcal{NC}_{2}(2k)}\frac{M(\pi)}{k!}$
$\displaystyle=\frac{1}{k!}.\\#\mathcal{M}_{2}(2k)$

where $\mathcal{M}_{2}(2k)$ denote the set of monotone pairings, see [2, 11,
5].

∎

Not all sequences of graphs lead to a limiting distribution, even if
$G_{n-1}\subset G_{n}$ for all $n$. We present and example to show this.

######  Example 5.8.

Consider the following sequence of graphs:

  * •

$G_{0}=(\\{1\\},\emptyset)$.

  * •

For all $n$, $G_{2n}=G_{2n-1}\cup\tilde{G}_{2n-1}$, where $\tilde{G}_{2n-1}$
is a copy of $G_{2n-1}$

  * •

For all $n$, $G_{2n+1}=\hat{H}_{2n+1}$, $H_{2n+1}=G_{2n}\cup\tilde{G}_{2n}$,
where $\hat{H}_{2n+1}$ is the graph $H_{2n+1}$ where we add the edges between
the vertices of $G_{2n}$ and $\tilde{G}_{2n}$.

The first of these graphs are shown in Figure 4.

$G_{0}$$G_{1}$$G_{2}$$G_{3}$$G_{4}$ Figure 4:  The sequence of graphs $G_{i}$,
which do not satisfy a CLT.

Now we consider where Bernoulli variables, $(X_{i})^{\infty}_{i=1}$, and
suppose that $X_{1},\dots,X_{2^{n}}$ are BMT independent with respect to
$G_{n}$; this is consistent by the construction of $G_{n}$. We are interested
in the normalized sum

$Y_{n}=\frac{X_{1}+\cdots+X_{2^{n}}}{2^{n/2}}.$

It is easy to calculate the first three moments

$\displaystyle\phi(Y_{n})=0,\quad\phi(Y_{n}^{2})=1,\quad\phi(Y_{n}^{3})=0.$
(2)

For the fourth moment we calculate a recursion for $n$ odd or even.

$\displaystyle\phi(Y_{2n}^{4})$
$\displaystyle=\frac{1}{4}(\phi((Y_{2n-1})^{4})+\phi(({Y}_{2n-1})^{4})+2\phi((Y_{2n-1})^{2}))=\frac{1}{2}(\phi((Y_{2n-1})^{4})+1),$
$\displaystyle\phi(Y_{2n+1}^{4})$
$\displaystyle=\frac{1}{4}(\phi((Y_{n})^{4})+\phi(({Y}_{n-1})^{4})+6\phi((Y_{n})^{2}))=\frac{1}{2}(\phi((Y_{2n})^{4})+3).$

From where, $\phi(Y_{2n+2}^{4})=\frac{1}{4}\phi(Y_{2n}^{4})+\frac{5}{4}$ and
$\phi(Y_{2n+2}^{4})=\frac{1}{4}\phi(Y_{2n}^{4})+\frac{7}{4}.$ Hence, one sees
that

$\phi(Y_{2n}^{4})\to 5/3\text{ and }\phi(Y_{2n+1}^{4})\to 7/3.$

Thus we do not have convergence in moments and neither convergence in
distribution, by Remark 5.4.

###  5.2 Further properties of CLT

We have observed some properties of BMT central limit theorem such as
determinacy of moments and symmetry. Here, we want to consider also some
properties in relation with the associated independence graphs. Firstly, we
show that if two graphs differ by a small number of edges then they provide
the same limiting distribution for the CLT. Secondly, we consider the
boundedness of support in the limiting distribution and finally, we provide a
description and examples on how consistency is reflected into Boolean, tensor
or monotone convolutions.

######  Theorem 5.9 (Perturbation).

Let $a_{1},\ldots,a_{N},b_{1},\ldots,b_{N}$ be centered variables with unit
variance and uniformly bounded moments of all orders. Suppose $G_{N}$ and
$H_{N}$ are the independence graphs of $a_{1},\ldots,a_{N}$ and
$b_{1},\ldots,b_{N}$, respectively. If the symmetric difference of $G_{N}$ and
$H_{N}$ has order $o(N^{2})$, then

$\lim_{N\to\infty}\left[\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}-\varphi\left(\frac{b_{1}+\cdots+b_{N}}{\sqrt{N}}\right)^{m}\right]=0\qquad\forall\
m\geq 1.$

Consequently, $(a_{1}+\cdots+a_{N})/\sqrt{N}$ has a limiting distribution if
and only if $(b_{1}+\cdots+b_{N})/\sqrt{N}$ does; moreover, the two limiting
distributions coincide if any of them exists.

###### Proof.

First note that $G_{\pi(\bm{i})}=H_{\pi(\bm{i})}$ since this is a graph that
depends only on the partition $\pi\in P(m)$ and a tuple
$\bm{i}=(i_{1},\ldots,i_{m})$ with $\ker[\bm{i}]=\pi$ and not on $G_{N}$ nor
$H_{N}$. So, to avoid confusion, let us put
$F_{\pi(\bm{i})}=G_{\pi(\bm{i})}=H_{\pi(\bm{i})}$. Due to Theorem 5.3, it is
enough to show that

$0=\lim_{N\to\infty}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}N^{-m/2}\left(\bm{1}_{F_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}-\bm{1}_{F_{\pi(\bm{i})}\subseteq H_{\bm{i}}}\right)$

for each pairing partition $\pi\in P_{2}(m)$. Since
$\bm{1}_{F_{\pi(\bm{i})}\subseteq G_{\bm{i}}}-\bm{1}_{F_{\pi(\bm{i})}\subseteq
H_{\bm{i}}}=0$ unless $F_{\pi(\bm{i})}\subseteq G_{\bm{i}}$ and
$F_{\pi(\bm{i})}\not\subseteq H_{\bm{i}}$ or $F_{\pi(\bm{i})}\subseteq
G_{\bm{i}}$ and $F_{\pi(\bm{i})}\not\subseteq H_{\bm{i}}$, a condition that
holds only if $G_{\bm{i}}\not\subseteq H_{\bm{i}}$ or $H_{\bm{i}}\not\subseteq
G_{\bm{i}}$, we obtain

$\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\left|{\bm{1}_{F_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}-\bm{1}_{F_{\pi(\bm{i})}\subseteq
H_{\bm{i}}}}\right|\leq\left|{S_{N,\pi}}\right|$

where ${S}_{N,\pi}=\left\\{\bm{i}:[m]\rightarrow[N]:\ker[\bm{i}]=\pi\text{ and
}G_{\bm{i}}\not\subseteq H_{\bm{i}}\text{ or }H_{\bm{i}}\not\subseteq
G_{\bm{i}}\right\\}.$

Take $\pi=\\{V_{1},V_{2},\ldots,V_{r}\\}\in P_{2}(m)$ with $r$ the number of
blocks of $\pi$. For a given tuple $\bm{i}=(i_{1},\ldots,i_{m})$, the graphs
$G_{\bm{i}}$ and $H_{\bm{i}}$ have the same vertex set
$\\{i_{k}:k=1,2,\ldots,m\\}$. And hence, $\ker[\bm{i}]=\pi$ implies $\bm{i}\in
S_{N,\pi}$ only if $G_{\bm{i}}$ has at least one edge that $H_{\bm{i}}$ does
not or $H_{\bm{i}}$ has at least one edge that $G_{\bm{i}}$ does not. Thus,
for each $\bm{i}\in$, at least one of the graphs $G_{\bm{i}}$ and $H_{\bm{i}}$
shares at least one edge with $G_{N}\triangle H_{N}$ the symmetric difference
of $G_{N}$ and $H_{N}$. This implies all elements in ${S}_{N,\pi}$ can be
constructed in the following way: (1) pick two distinct blocks $V,W\in\pi$, an
edge $(j_{V},j_{W})\in E(G_{N}\triangle H_{N})$, and $r-2$ distinct values
$j_{U}\in[N]\setminus\\{j_{V},j_{W}\\}$ for $U\in\pi\setminus\\{V,W\\}$; (2)
take $\bm{i}=(i_{1},\ldots,i_{m})$ where $i_{k}=j_{U}$ provided $k\in U$. The
previous construction is not necessarily injective but it does define a set
containing ${S}_{N,\pi}$. So, we get

$\left|{S_{N,\pi}}\right|\leq r(r-1)\cdot\left|{E(G_{N}\triangle
H_{N})}\right|\cdot(N-2)(N-3)\cdots(N-(r-1))\leq
r(r-1)\cdot\left|{E(G_{N}\triangle H_{N})}\right|\cdot N^{r-2}$

The desired result then follows since $r\leq m/2$ and
$\lim_{N\to\infty}\left|{E(G_{N}\triangle H_{N})}\right|/N^{2}=0$ imply

$\lim_{N\to\infty}\frac{r(r-1)\cdot\left|{E(G_{N}\triangle H_{N})}\right|\cdot
N^{r-2}}{N^{m/2}}=0.$

∎

######  Corollary 5.10.

If the independence graph $G_{N}$ of the variables $a_{1},a_{2},\ldots,a_{N}$
has order $o(N^{2})$, then $(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in
moments as $N\to\infty$ to the Bernoulli distribution
$(\delta_{-1}+\delta_{+1})/2$.

######  Corollary 5.11.

If the independence graph $G_{N}$ of the variables $a_{1},a_{2},\ldots,a_{N}$
contains no copy of a complete bipartite graph, then
$(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in moments as $N\to\infty$ to the
Bernoulli distribution $(\delta_{-1}+\delta_{+1})/2$.

###### Proof.

This follows from Theorem 5.9 and the Kovari-Sós-Turán theorem, the latter
stating that for fixed integers $s\geq r\geq 2$ any $N$-vertex graph with at
least $CN^{2-\frac{1}{r}}$ edges contains a complete bipartite subgraph
$K_{r,s}$. ∎

######  Proposition 5.12.

Let for each $N$, let $a_{1},\ldots,a_{N},$ and $b_{1},\ldots,b_{N},$ be a
couple of tuples of i.d. selfadjoint random variables with mean zero and
variance one. Suppose that $G_{N}$ and $H_{N}$ are the independence graphs of
$a_{1},\ldots,a_{N}$ and $b_{1},\ldots,b_{N}$, respectively. If $G_{N}$ is a
subgraph of $J_{N}$, for all $N$ and if $(a_{1}+\cdots+a_{N})/\sqrt{N}$ and
$(b_{1}+\cdots+b_{N})/\sqrt{N}$ converge in moments to $X$ and $Y$,
respectively, then $\|X\|_{\infty}\leq\|Y\|_{\infty}$.

###### Proof.

If $Y$ has unbounded support, then the statement follows by definition.
Otherwise $\|Y\|_{\infty}$ is finite and thus determined by moments.

Notice that both limiting distributions do not depend on the choice of
$a_{i}$’s, and we may assume that $a_{i}$ has the same distribution as
$b_{i}$, for all $i$.

Now, for any index $\mathbf{i}=(i_{1},\dots,i_{k})$,

$\varphi(a_{i_{1}}\cdots a_{i_{k}})\ \,\leq\varphi(b_{i_{1}}\cdots a_{i_{k}})$

Consequently, the moments of $(b_{1}+\cdots+b_{N})/\sqrt{N}$ are larger than
the moments of $(a_{1}+\cdots+a_{N})/\sqrt{N}$. By taking limits we see that,
if we denote the even moments of $X$ by $(m_{2n}(X))_{n>0}$, and the moments
of Y by $(m_{2n}(Y))_{n}>0$, we have the inequality

$m_{2n}(X)\leq m_{2n}(Y),\quad\text{for all }n.$

This means that $(m_{2n}(X))^{1/n}\leq\|Y\|_{\infty}$ and thus, letting
$n\to\infty$ we see that $\|X\|_{\infty}\leq\|Y\|_{\infty}$. ∎

######  Theorem 5.13.

Let $M_{N}$ denote the greatest integer $M\geq 0$ so that the full graph
$K_{M}$ is contained if $G_{N}$. If $\liminf_{N\to\infty}M_{N}/N>0$, then the
limit distribution of $(a_{1}+\cdots+a_{N})/\sqrt{N}$ if it exists has non-
compact support.

###### Proof.

Suppose $(a_{1}+\cdots+a_{N})/\sqrt{N}$ has an analytical limiting
distribution, i.e., there exists a probability measure $\mu$ on the real line
so that

$\lim_{N\to\infty}\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{k}=\
\int_{-\infty}^{+\infty}t^{k}\,d\mu(t)=:m_{k}(\mu)$

for all integers $k\geq 1$. Note that if $\mu$ has compact support, so
$\mu((-\infty,-L)\cup(+L,+\infty))=0$ for some $L\geq 0$, then $\sup_{k\geq
1}\sqrt[k]{\left|{m_{k}(\mu)}\right|}\leq L<\infty$. Thus, to prove $\mu$ has
no compact support, it enough to show that $\sup_{k\geq
1}\sqrt[2k]{\left|{m_{2k}(\mu)}\right|}=+\infty$.

By hypothesis, for each integer $N\geq 1$, there exists a set
$J_{N}=\\{j_{1}<j_{2}<\cdots<j_{M_{N}}\\}\subset[N]$ so that $G_{N}$ contains
the full graph on $J_{N}$, so $(j_{k},j_{\ell})$ is an edge of $G_{N}$ for any
distinct $j_{k},j_{\ell}\in J_{N}$. Thus, if
$\bm{i}=(i_{1},i_{2},\ldots,i_{2k})$ satisfies $i_{r}\in J_{N}$ for
$r=1,2,\ldots,2k$, the graph $G_{\bm{i}}$ is the full graph since it is the
restriction of $G_{N}$ to vertex set $\\{i_{r}\mid r=1,2,\ldots,2k\\}\subset
J_{N}$, and hence $G_{\pi(\bm{i})}$ is always a subgraph of $G_{\bm{i}}$ with
$\pi=\ker[\bm{i}]$. So, for any pairing partition $\pi\in P_{2}(2k)$, we get

$\sum_{\begin{subarray}{c}\bm{i}:[2k]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq G_{\bm{i}}}\
\ \geq\sum_{\begin{subarray}{c}\bm{i}:[2k]\rightarrow J_{N}\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}=\ M_{N}(M_{N}-1)\cdots(M_{N}-k+1)$

where $k$ is the number of blocks in $\pi$ since it is a pairing. Put
$2C=\liminf_{N\to\infty}M_{N}/N>0$. Thus, for each $k\geq 1$, there exists
$\tilde{N}_{k}\geq 1$ so that $N\geq\tilde{N}_{k}$ implies $M_{N}\geq k$ and

$\left(\frac{M_{N}}{N}\right)\left(\frac{M_{N}-1}{N}\right)\cdots\left(\frac{M_{N}-k+1}{N}\right)\geq
C^{k}.$

Hence, Theorem 5.3 implies $m_{2k}(\mu)\geq\left|{P_{2}(2k)}\right|C^{k}$
since the last two inequalities give

$\sum_{\pi\in P_{2}(2k)}\ \
N^{-k}\sum_{\begin{subarray}{c}\bm{i}:[2k]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq G_{\bm{i}}}\
\ \geq\sum_{\pi\in P_{2}(2k)}\ \ C^{k}\ \ =\ \ \left|{P_{2}(2k)}\right|C^{k}$

for $N\geq\tilde{N}_{k}$. Finally,
$\left|{P_{2}(2k)}\right|=(2k-1)(2k-3)\cdots(1)$ implies $\sup_{k\geq
1}\sqrt[2k]{\left|{m_{2k}(\mu)}\right|}=+\infty$. ∎

Finally, we consider the role of consistency in the BMT Central Limit
Theorems.

######  Proposition 5.14.

Let $a_{1},\ldots,a_{M_{N}},a_{M_{N}+1},\ldots,a_{M_{N}+L_{N}}$ be centered
variables with unit variance, uniformly bounded moments of all orders, and
independence graph $G_{M_{N}+L_{N}}$ where $N=M_{N}+L_{N}$. Suppose
$(a_{1}+\cdots+a_{M_{N}})/\sqrt{M_{N}}$ and
$(a_{M_{N}+1}+\cdots+a_{M_{N}+L_{N}})/\sqrt{L_{N}}$ converge in moments to
$a_{M}$ and $a_{L}$, respectively. If $t=\lim_{N\to\infty}M_{N}/N$ exists,
then $(a_{1}+\cdots+a_{N})/\sqrt{N}$ converges in moments to $\sqrt{t}\cdot
a_{M}\,+\,\sqrt{1-t}\cdot a_{L}$ with $a_{M}$ and $a_{L}$

  1. (1)

boolean independent if $(i,j),(j,i)\notin E_{M_{N}+L_{N}}$ whenever
$i\in[M_{N}]$ and $j\in[M_{N}+L_{N}]\setminus[M_{N}]$

  2. (2)

monotone independent if $(j,i)\in E_{M_{N}+L_{N}}$ and $(i,j)\notin
E_{M_{N}+L_{N}}$ whenever $i\in[M_{N}]$ and
$j\in[M_{N}+L_{N}]\setminus[M_{N}]$

  3. (3)

tensor independent if $(i,j),(j,i)\in E_{M_{N}+L_{N}}$ whenever $i\in[M_{N}]$
and $j\in[M_{N}+L_{N}]\setminus[M_{N}]$

###### Proof.

Take $a_{N,1}=(a_{1}+\cdots+a_{M_{N}})/\sqrt{N}$ and
$a_{N,2}=(a_{M_{N}+1}+\cdots+a_{M_{N}+L_{N}})/\sqrt{N}$. Thus, for each
integer $m\geq 1$, we get

$\displaystyle\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\sum_{i_{1},\ldots,i_{m}=1}^{2}\varphi\,(a_{N,i_{1}}a_{N,i_{2}}\cdots
a_{N,i_{m}}).$

Now, if _(1) ((2),(3))_ holds, we have $a_{N,1}$ and $a_{N,2}$ boolean
(monotone, tensor, respectively) independent due to Proposition 3.12, and
hence the last equality implies

$\lim_{N\to\infty}\varphi\left(\frac{a_{1}+\cdots+a_{N}}{\sqrt{N}}\right)^{m}=\varphi\,(\sqrt{t}\cdot
a_{M}\,+\,\sqrt{1-t}\cdot a_{L})^{m}$

with $a_{M}$ and $a_{L}$ boolean (monotone, tensor, respectively) independent
since $a_{N,1}$ and $a_{N,2}$ converge in moments to $\sqrt{t}\cdot a_{M}$ and
$\sqrt{1-t}\cdot a_{L}$, respectively. ∎

######  Example 5.15.

Suppose the independence graph $G_{N}$ of the variables $a_{1},\cdots,a_{N}$
is the the complete bipartite graph $K_{M_{N},L_{N}}$ with $N=M_{N}+L_{N}$.
From Corollary 5.5, it follows that each
$(a_{1}+\cdots+a_{M_{N}})/\sqrt{M_{N}}$ and
$(a_{M_{N}+1}+\cdots+a_{M_{N}+L_{N}})/\sqrt{L_{N}}$ converge in moments to a
Bernoulli distribution $(\delta_{-1}+\delta_{+1})/2$. If additionally we have
$\lim_{N\to\infty}=t$ exists, then $0\leq t\leq 1$ and
$(a_{1}+\cdots+a_{N})/\sqrt{N}$ converge in moments to

$\left[\frac{1}{2}\delta_{-\sqrt{t}}+\frac{1}{2}\delta_{+\sqrt{t}}\right]\ast\left[\frac{1}{2}\delta_{-\sqrt{1-t}}+\frac{1}{2}\delta_{+\sqrt{1-t}}\right]$

where $\ast$ denotes the classical convolution of measures.

######  Example 5.16.

Suppose the independence graph $G_{N}$ of the variables $a_{1},\cdots,a_{N}$
is the the disjoint union of two complete graphs $K_{M_{N}}$ and $K_{L_{N}}$
with vertex sets $[M_{N}]$ and $[M_{N}+L_{N}]\setminus[M_{N}]$ ,respectively,
and $N=M_{N}+L_{N}$. From Corollary 5.6, it follows that each
$(a_{1}+\cdots+a_{M_{N}})/\sqrt{M_{N}}$ and
$(a_{M_{N}+1}+\cdots+a_{M_{N}+L_{N}})/\sqrt{L_{N}}$ converge in moments to a
normal distribution $\mathcal{N}(0,1)$. Thus, if $\lim_{N\to\infty}=t$ exists,
then $0\leq t\leq 1$ and $(a_{1}+\cdots+a_{N})/\sqrt{N}$ converge in moments
to

$\mathcal{N}(0,\sqrt{t})\,\uplus\,\mathcal{N}(0,\sqrt{1-t})$

where $\uplus$ denotes the Boolean convolution of measures.

######  Example 5.17.

Suppose $G_{N}$ is the Turán Graph $T(N,r_{N})$, a complete multi-partite
graph formed by partitioning a set of $N$ vertices into $r_{N}$ subsets with
sizes as equal as possible. This graph becomes the empty graph (when
$r_{N}=1$), the complete graph (when $r_{N}=N$), and a complete bipartite
graph (when $r_{N}=2$). If $r_{N}\to r<\infty$, when $N\to\infty$, then the
bmt central limit theorem associated to $T(N,r_{N})$ has as limit distribution
a convolution of $r$ Bernoulli random variables.

######  Remark 5.18.

Proposition 5.14 and last examples shows the number of edges does not
determine the support compactness of the limit distribution in Theorem 5.3.
Indeed, suppose $G_{N}=K_{\lfloor N/2\rfloor}\sqcup\emptyset_{\lceil
N/2\rceil}$ and $H_{N}=K_{\lfloor N/2\rfloor,\lceil N/2\rceil}$ and let
$n_{N}$ and $m_{N}$ denote the the number of edges of $G_{N}$ and $H_{N}$,
respectively. Then, we have

$n_{N}=\frac{\lfloor\frac{N}{2}\rfloor(\lfloor\frac{N}{2}\rfloor-1)}{2}\leq\lfloor\frac{N^{2}}{8}\rfloor\leq\lfloor\frac{N^{2}}{4}\rfloor\leq\lfloor\frac{N}{2}\rfloor\lceil\frac{N}{2}\rceil=m_{N}.$

Thus, the limit distribution associated to $H_{N}$ is compactly supported
(being the classical convolution of two Bernoulli distributions), while the
limit distribution associated to $G_{N}$ is not (being the boolean convolution
of a Gaussian distribution and a Bernoulli distribution).

##  6 Poisson Limit Theorem

Apart from the central limit theorem, the law of rare events also known as
Poisson limit theorem is probably the the most important theorem in
probability.

In this limit theorem, one considers sums of independent Bernoulli variables
with common parameter $\lambda/n$ and considers its limit. In BMT independence
we can have such limits. The combinatorics appearing are similar as the one
for the central limit theorem. Not surprisingly, the main difference is that
here we need to consider the set of all instead of pair partitions to all
partitions.

######  Theorem 6.1.

Suppose $a_{1},a_{2},\ldots,a_{N}$ BMT independent with respect to the graph
$G_{N}$. If each $a_{i}$ has Bernoulli distribution
$\mu_{N}=(1-\frac{\lambda}{N})\delta_{0}+\frac{\lambda}{N}\delta_{1}$, then
the moments of the sum $a_{1}+\cdots a_{N}$ satisfy

$\displaystyle\varphi\left[\left(a_{1}+\cdots+a_{N}\right)^{m}\right]\
\,=\sum_{\pi\in P(m)}\lambda^{\\#(\pi)}\
N^{-\\#(\pi)}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}\quad+\quad O(N^{-1})\ .$

###### Proof.

Expanding the product $(a_{1}+\cdots+a_{N})^{m}$ and using the fact that
$\ker[\bm{i}]=\ker[\bm{i^{\prime}}]$ defines an equivalence relation on
$\\{\bm{i}\mid\bm{i}:[m]\to[N]\\}$, we have

$\varphi({a_{1}+\cdots+a_{N}})^{m}\ \ =\sum_{\pi\in P(m)}\
\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\varphi(a_{i_{1}}a_{i_{2}}\cdots a_{i_{m}}).$

Notice that $\varphi(a^{k}_{i})=\lambda/N$ since each $a_{i}$ has Bernoulli
distribution
$\mu_{N}=(1-\frac{\lambda}{N})\delta_{0}+\frac{\lambda}{N}\delta_{1}$. So, for
each partition $\pi\in P(m)$, the BMT independence of
$a_{1},a_{2},\ldots,a_{N}$ gives

$\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\varphi(a_{i_{1}}a_{i_{2}}\cdots a_{i_{m}})\
\ =\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\prod_{V\in\ker_{G}[\bm{i}]}\varphi((a_{i_{k}})|V)\
\ =\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\ker_{G}[\bm{i}])}$

where $\\#(\ker_{G}[\bm{i}])$ denotes the number of blocks in the partition
$\ker_{G}[\bm{i}]$. Now, recall that $\ker_{G}[\bm{i}]$ is a refinement of
$\ker[{\bm{i}}]$, so we can write

$\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\ker_{G}[\bm{i}])}\
\ =\ \ \ \sum_{\begin{subarray}{c}\theta\in P(m)\\\
\theta\leq\pi\end{subarray}}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[\bm{i}]=\pi\\\
\ker_{G}[\bm{i}]=\theta\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\ker_{G}[\bm{i}])}.$

Moreover, for any partitions $\pi,\theta\in P(m)$ with $\theta\leq\pi$, we
have

$\left|{\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[\bm{i}]=\pi,\ker_{G}[\bm{i}]=\theta\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\ker_{G}[\bm{i}])}}\right|\leq\left(\frac{\left|{\lambda}\right|}{N}\right)^{\\#(\theta)}\left|{\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[\bm{i}]=\pi\end{subarray}}1}\right|\leq\left|{\lambda}\right|^{\\#(\theta)}N^{\\#(\pi)-\\#(\theta)}$

where $\\#(\pi)$ and $\\#(\theta)$ denote the number of blocks in $\pi$ and
$\theta$, respectively. Hence, since $\theta\leq\pi$ implies
$\\#(\theta)\geq\\#(\pi)$ with equality only if $\theta=\pi$, we obtain

$\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\ker_{G}[\bm{i}])}\
\ =\ \ \sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[\bm{i}]=\pi,\ker_{G}[\bm{i}]=\pi\end{subarray}}\left(\frac{\lambda}{N}\right)^{\\#(\pi)}\quad+\quad
O(N^{-1}).$

Therefore, due the equivalence of the conditions $G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}$ and $\ker[\bm{i}]=\ker_{G}[\bm{i}]$ for any
${\bm{i}}:[m]\rightarrow[N]$ with $\ker[{\bm{i}}]=\pi$, we get

$\varphi\left[\left(a_{1}+\cdots+a_{N}\right)^{m}\right]\ \,=\sum_{\pi\in
P(m)}\lambda^{\\#(\pi)}\
N^{-\\#(\pi)}\sum_{\begin{subarray}{c}{\bm{i}}:[m]\rightarrow[N]\\\
\ker[{\bm{i}}]=\pi\end{subarray}}\bm{1}_{G_{\pi(\bm{i})}\subseteq
G_{\bm{i}}}\quad+\quad O(N^{-1})\ .$

∎

Similarly as for the CLT we may deduce the Boolean, monotone and tensor
Poisson limit Theorems, as applications of the above theorem by using similar
arguments as in the proofs of Corollaries 5.5, 5.6 and 5.7. We leave the
details of the proof to the reader.

######  Corollary 6.2.

Let for all $N$, $a_{1},\ldots,a_{N}$ have a Bernoulli distribution
$\mu_{N}=(1-\frac{\lambda}{N})\delta_{0}+\frac{\lambda}{N}\delta_{1}$. Then as
$N\to\infty$

  1. 1.

If $a_{1},\ldots,a_{N}$ are BMT independent with respect to the complete graph
$G_{N}=K_{N}$, then the variable $a_{1}+\cdots a_{N}$ converges in moments to
a classical Poisson distribution.

  2. 2.

If $a_{1},\ldots,a_{N}$ are BMT independent with respect to the empty graph
$G_{N}=\emptyset_{N}$, then the variable $a_{1}+\cdots a_{N}$ converges in
moments to a Boolean Poisson distribution, [23].

  3. 3.

If $a_{1},\ldots,a_{N}$ are BMT independent with respect to the graph $G_{N}$
associated with a total order $<$, then the variable $a_{1}+\cdots a_{N}$
converges in moments to a monotone Poisson distribution, [16]

##  7 Concluding Remarks

We have started the development of new framework both analytical and algebraic
that enables us to investigate arbitrary mixtures of Boolean, monotone and
tensor independence. This setting provides a unified approach for these three
notions of independence through a digraph depicting pair-wise independence
relations and what we have called the kernel of function subordinated to a
digraph. Nonetheless, besides establishing Central and Poisson-Type Limit
Theorems, there are many interesting problems that remain open. We mention a
few in this last section.

  1. (1)

The ideas in this paper and those from [8] could be combined to obtain a
corresponding notion of BMFT independence. In [8], the authors introduced the
notion of $\mathcal{T}$_-tree_ independence, which allowed them to study
mixtures of Boolean, free, and monotone independence. A closer look at our
notion of the kernel subordinated to a digraph reveals that this object splits
mixed moments according to the commutation relations from tensor independence,
much in the spirit of the reduced non-crossing partitions from mixtures of
free and tensor independence in [22, 14]. A notion of BMFT independence seems
likely to arise from combining the maximal non-crossing partitions from
$\mathcal{T}$_-tree_ independence and the commutation relations encoded in the
kernel subordinated to a digraph.

  2. (2)

The CLT and Poisson Limit theorem hint on the possible set of partitions that
describe the combinatorics of BMT independence. It would be interesting if
such intuition can be further studied in a systematical way. One possible
direction and an important open question is to define cumulants with respect
to BMT independence.

  3. (3)

It has been shown that if we consider sequences of random variables where
commuting and anticommuting relations are taken randomly, one obtains a
deterministic limit in the CLT, which correspond to $q$-Gaussian distributions
[20], see also [3] for a generalization. It would be interesting if
considering random directed graphs as the independence graphs in the BMT-
central limit theorem one would get some interesting interpolation between,
Gaussian, arcsine and symmetric Bernoulli distributions.

  4. (4)

It would be desirable to classify all probability measures that arise as
limits of the CLT for BMT independent random variables. In particular, one
should determine the digraphs that recover and generalize the limiting
measures obtained in [25] for BM independence. These latter measures were
shown to be symmetric and compactly supported, they include the semi-circular
law and their even moments satisfy the generalized recurrence relation for
Catalan numbers, namely,

$g_{n}=\sum_{r=1}^{n}\gamma_{r}\,g_{r-1}g_{n-r}.$

Section 5 in this paper shows that non-compact measures appear in the CLT for
BMT independent random variables, so they are not covered by BM independence.

  5. (5)

Diving into the last remark, one would like to find out properties of digraphs
$G_{N}=(V_{N},E_{N})$ that determine the compactness of limiting measures in
the CLT for BMT independent random variables. This seems to be a non-trivial
problem since it amounts to analyze the growth as $N$ goes to infinity of the
number of sub-graphs of $G_{N}$ that are isomorphic to each nesting-crossing
graph $G_{\pi}$. Moreover, the edge set $E_{N}$ must be of order $N^{2}$
—i.e., $\lim_{N\to\infty}|E_{N}|/N^{2}>0$— to obtain a limiting measure
different from the Bernoulli distribution, but at this order both compact and
non-compact measure appear.

## References

  * [1] L. Accardi, A. Ben Ghorbal and N. Obata, _Monotone independence, comb graphs and Bose-Einstein condensation_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 7, No. 3, pp. 419-435, (2004).
  * [2] O. Arizmendi, T. Hasebe, F. Lehner and C. Vargas, _Relations between cumulants in noncommutative probability_ , Advances in Mathematics, 282, (2014).
  * [3] Blitvic, N. (2014). Two-parameter non-commutative central limit theorem. In Annales de l’IHP Probabilités et statistiques (Vol. 50, No. 4, pp. 1456-1473).
  * [4] A. Ben Ghorbal and M. Schürmann, _Non-commutative notions of stochastic independence_ , Math. Proc. Cambridge Philos. Soc., 133, pp. 531-561, (2002).
  * [5] Hasebe, T., & Saigo, H. (2011). The monotone cumulants. In Annales de l’IHP Probabilités et statistiques (Vol. 47, No. 4, pp. 1160-1170).
  * [6] Hasebe, T. (2011). Conditionally monotone independence I: Independence, additive convolutions and related convolutions. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(03), 465-516.
  * [7] Franz, U. (2003). Unification of Boolean, monotone, anti-monotone, and tensor independence and Lévy processes. Mathematische Zeitschrift, 243, 779-816.
  * [8] D. Jekel and W. Liu, _An operad of non-commutative independences defined by trees_ , Dissertationes Math., 553, pp. 1-100, (2020).
  * [9] J. A. Kula and Wysoczanski , _An example of a Boolean-free type central limit theorem_ , Probab. Math. Statist., 33, pp. 341-352, (2013).
  * [10] Lenczewski, R. (2005). On noncommutative independence. In Quantum Probability And Infinite Dimensional Analysis (pp. 320-336).
  * [11] Lenczewski, R., & Salapata, R. (2006). Discrete interpolation between monotone probability and free probability. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 9(01), 77-106.
  * [12] Lenczewski, R. (1998). Unification of independence in quantum probability. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 1(03), 383-405.
  * [13] Lenczewski, R. (2019). Conditionally monotone independence and the associated products of graphs. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 22(04), 1950023.
  * [14] W. Mlotkowski, _$\Lambda-$ free probability_, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 6, No. 3, pp. 337-371, (2003).
  * [15] N. Muraki, _Monotonic independence, monotonic central limit theorem and monotonic law of small numbers_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 4, No. 1, pp. 39-58, (2001).
  * [16] N. Muraki, Monotonic convolution and monotonic Lévy-Hinčin formula, preprint, 2000, 40pp.
  * [17] N. Muraki, _The five independences as quasi-universal products_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 5, No. 1, pp. 113-134, (2002).
  * [18] N. Muraki, _The five independences as natural products_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 6, No. 3, pp. 337-371, (2003).
  * [19] A. Nica and R. Speicher, _Lectures on the Combinatorics of Free Probability_ , London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, (2006).
  * [20] Speicher, R. (1992). A non-commutative central limit theorem. Mathematische Zeitschrift, 209(1), 55-66.
  * [21] R. Speicher, _On universal products_ , Free probability theory, Waterloo, 12, pp. 257-266, (1997).
  * [22] R. Speicher and J. Wysoczanski, _Mixtures of classical and free independence_ , Arch. Math., 107, pp. 445-453, (2016).
  * [23] R. Speicher and R. Woroudi, _Boolean convolution_ , 12. 267-279. 10.1090/fic/012/13, (1997).
  * [24] J. Wysoczanski, _BM independence and central limit theorems associated with symmetric cones_ , noncomm. harmonic anal. with appl. to probab., Banach center publications, Vol. 78, (2007).
  * [25] J. Wysoczanski, _BM independence and BM-central limit theorems associated with symmetric cones_ , Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 13, No. 3, pp. 461-488, (2010).