The Five Natural Notions of Independence
Arturo Jaramillo Gil
University of Kansas
September 30
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
1 / 25
Overview
Overview
In a “Classical Probability Space”, consider bounded random variables
X1 , . . . , Xn ∈ L∞ (Ω, P) (Compact support implies that the joint
distribution of X1 , . . . Xn is determined by their mixed moments).
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
2 / 25
Overview
Overview
In a “Classical Probability Space”, consider bounded random variables
X1 , . . . , Xn ∈ L∞ (Ω, P) (Compact support implies that the joint
distribution of X1 , . . . Xn is determined by their mixed moments). This
means, evaluating the operation E on words (i.e. products of some of the
Xi ’s) determines their distribution.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
2 / 25
Overview
Overview
In a “Classical Probability Space”, consider bounded random variables
X1 , . . . , Xn ∈ L∞ (Ω, P) (Compact support implies that the joint
distribution of X1 , . . . Xn is determined by their mixed moments). This
means, evaluating the operation E on words (i.e. products of some of the
Xi ’s) determines their distribution.
For example, to know the joint law of (X , Y ), when X , Y are independent,
we only require to use the property
E [X n Y m ] = E [X n ] [Y m ] .
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
2 / 25
Overview
Overview
In a “Classical Probability Space”, consider bounded random variables
X1 , . . . , Xn ∈ L∞ (Ω, P) (Compact support implies that the joint
distribution of X1 , . . . Xn is determined by their mixed moments). This
means, evaluating the operation E on words (i.e. products of some of the
Xi ’s) determines their distribution.
For example, to know the joint law of (X , Y ), when X , Y are independent,
we only require to use the property
E [X n Y m ] = E [X n ] [Y m ] .
Goal: Find analogous descriptions of the laws of X , Y when X , Y are
non-conmutative random objects.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
2 / 25
Preliminaries
Non-commutative Probability Spaces
Definition
A non-commutative probability space is a pair (A, τ ) where A is a unital
algebra over C and τ : A → C is a linear functional such that τ (1A ) = 1.
An element a ∈ A is called a (non-commutative) random variable.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
3 / 25
Preliminaries
Non-commutative Probability Spaces
Definition
A non-commutative probability space is a pair (A, τ ) where A is a unital
algebra over C and τ : A → C is a linear functional such that τ (1A ) = 1.
An element a ∈ A is called a (non-commutative) random variable.
τ will play the role of the classical expectation E, and we will be interested
in objects of the type
τ (aim1 1 . . . aimk k )
If τ (ab) = τ (ba) for all a, b ∈ A, we say that (A, τ ) is tracial.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
3 / 25
Preliminaries
Non-commutative Probability Spaces
We define the (algebraic) distribution of a1 , . . . , an , the linear functional
µa1 ,...,an : C hX1 , . . . , Xn i → C → given by
1
k
. . . Xim
µa1 ,...,an (Xim
) := τ (aim1 1 . . . aimk k ),
1
k
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
4 / 25
Preliminaries
Non-commutative Probability Spaces
We define the (algebraic) distribution of a1 , . . . , an , the linear functional
µa1 ,...,an : C hX1 , . . . , Xn i → C → given by
1
k
. . . Xim
µa1 ,...,an (Xim
) := τ (aim1 1 . . . aimk k ),
1
k
Definition
A non-commutative probability space (A, τ ) such that A is a C ∗ -algebra
and that τ is positive (i.e. τ (a∗ a) ≥ 0 for all a ∈ A) is called a C ∗
probability space. If a random variable a ∈ A satisfies aa∗ = a∗ a is called
normal, and if it satisfies a∗ = a, is called Hermitian (or self-adjoint). If a
random variable can be written as a = xx ∗ it is called positive, and we
write a ≥ 0.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
4 / 25
Preliminaries
Non-commutative Probability Spaces
Theorem
If a ∈ (A, τ ) is normal (resp. self-adjoint), then there exists a a probability
measure µa in C (resp. R), with compact support such that
Z
z k z l µa (dz) = τ (ak (a∗ )l ).
C
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
5 / 25
The Five Natural Notions of Independence
Natural products
Given a non-commutative probability space (A, τ ) and two sub-algebras
A1 , A2 ⊂ A, the condition of A1 , A2 being independent should allow us to
recover τ |hA1 ,A2 i in terms of τ1 := τ |A1 and τ |A2 , through some
computational rule τ1 ~ τ2 . Here hA1 , A2 i denotes the subalgebra
generated by A1 and A2 .
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
6 / 25
The Five Natural Notions of Independence
Natural products
Given a non-commutative probability space (A, τ ) and two sub-algebras
A1 , A2 ⊂ A, the condition of A1 , A2 being independent should allow us to
recover τ |hA1 ,A2 i in terms of τ1 := τ |A1 and τ |A2 , through some
computational rule τ1 ~ τ2 . Here hA1 , A2 i denotes the subalgebra
generated by A1 and A2 .
The fact that the algebras of random variables are non-commutative,
opens the possibility to define other natural rules for computing mixed
moments. For instance, if a, b belong to different algebras, the mixed
moments of a, b could be given by
τ (an1 b m1 . . . ank b mk ) = τ (an1 )τ (b m1 ) · · · τ (ank )τ (b mk ).
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
6 / 25
The Five Natural Notions of Independence
Natural products
Let K denote the class of algebraic probability spaces. For subspaces
A1 , A2 , we define their free product. This is the algebra spanned by words
made out of letters coming alternatingly from A1 and A2 , that is
M
A1 t A2 :=
Aε1 ⊗ · · · ⊗ Aεn ,
ε∈A
where A is the set of infinite sequences ε = (ε1 , . . . , εn ) of some length
n ≥ 1, with εi ∈ {1, 2} and εi 6= εi+1 .
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
7 / 25
The Five Natural Notions of Independence
Natural products
Let K denote the class of algebraic probability spaces. For subspaces
A1 , A2 , we define their free product. This is the algebra spanned by words
made out of letters coming alternatingly from A1 and A2 , that is
M
A1 t A2 :=
Aε1 ⊗ · · · ⊗ Aεn ,
ε∈A
where A is the set of infinite sequences ε = (ε1 , . . . , εn ) of some length
n ≥ 1, with εi ∈ {1, 2} and εi 6= εi+1 . The product is given by
concatenation (in case the end of the first word is in the same algebra
than the first word of the second one we multiply such elements before
concatenation)
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
7 / 25
The Five Natural Notions of Independence
Natural products
Given two algebraic probability spaces (A1 , τ1 ) and (A2 , τ2 ), we would like
to define the product functional τ1 ~ τ2 on A1 t A2 , built up, in some
“natural” way (i.e. satisfying some basic axioms), by our functionals τ1 , τ2 .
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
8 / 25
The Five Natural Notions of Independence
Natural products
Given two algebraic probability spaces (A1 , τ1 ) and (A2 , τ2 ), we would like
to define the product functional τ1 ~ τ2 on A1 t A2 , built up, in some
“natural” way (i.e. satisfying some basic axioms), by our functionals τ1 , τ2 .
Definition
A natural product in K is a function
((A1 , τ1 ), (A2 , τ2 )) 7→ (A1 t A2 , τ1 ~ τ2 ) from K × K to K, that satisfies
the following properties
Associativity: (τ1 ~ τ2 ) ~ τ3 = τ1 ~ (τ2 ~ τ3 ) =: (τ1 ~ τ2 ) ~ τ3 .
Universality: Fro any morphisms j1 : B1 → A1 , j2 : B2 → A2 , we
have
(τ1 ◦ j1 ) ~ (τ2 ◦ j2 ) = (τ1 ~ τ2 ) ◦ (j1 q j2 ),
where j1 q j2 is the unique morphism making the diagram commute.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
8 / 25
The Five Natural Notions of Independence
Natural products
Definition
Extension:
(τ1 ~ τ2 ) ◦ i1 = τ1 ,
and
(τ1 ~ τ2 ) ◦ i2 = τ2
Normalization: For all a ∈ A1 , b ∈ A2 we have
(τ1 ~ τ2 )[i1 (a)i2 (b)] = (τ1 ~ τ2 )[i2 (b)i1 (a)] = τ1 [a]τ2 [b].
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
9 / 25
The Five Natural Notions of Independence
Natural products
Definition
The tensor (⊗), Boolean () and free (?) products over K are given by the
following calculation rules for a1 ⊗ · · · ⊗ an ∈ A1 t A2

 

→
→
Y
Y
(1)
(2)
(τ1 ⊗ τ2 )[a1 ⊗ · · · ⊗ an ] = τ1 
ak  τ2 
al 
k∈V1
l∈V2



Y h (1) i
Y h (2) i
τ1 τ2 [a1 ⊗ · · · ⊗ an ] = 
τ1 ak  
τ2 al 
k∈V1
τ1 ? τ2 [a1 ⊗ · · · ⊗ an ] = −
X
I ⊂{1,...,n}
I ({1,...,n}
Arturo Jaramillo Gil (University of Kansas)
(τ1 ? τ2 )
"→
Y
k∈I
The Five Natural Notions of Independence
l∈V2
#!
ak
Y
−τεl
(ε )
al l
!
l ∈I
/
September 30
10 / 25
The Five Natural Notions of Independence
Natural products
Definition
The monotone (.) and anti-monotone (.) products over K are given by
the following calculation rules for a1 ⊗ · · · ⊗ an ∈ A1 t A2



→
h
i
Y
Y
(1)
(2)
(τ1 . τ2 )[a1 ⊗ · · · ⊗ an ] = τ1 
ak  
τ2 al 
k∈V1
l∈V2


(τ1 / τ2 )[a1 ⊗ · · · ⊗ an ] = 
Y
h
(1)
τ1 al
i
k∈V2
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence

 τ2 
→
Y

(2)
al 
l∈V2
September 30
11 / 25
The Five Natural Notions of Independence
Natural products
Definition
The monotone (.) and anti-monotone (.) products over K are given by
the following calculation rules for a1 ⊗ · · · ⊗ an ∈ A1 t A2



→
h
i
Y
Y
(1)
(2)
(τ1 . τ2 )[a1 ⊗ · · · ⊗ an ] = τ1 
ak  
τ2 al 
k∈V1
l∈V2


(τ1 / τ2 )[a1 ⊗ · · · ⊗ an ] = 
Y
h
(1)
τ1 al
i
k∈V2

 τ2 
→
Y

(2)
al 
l∈V2
Theorem
There are only five natural products in K: the tensor, Boolean, free,
monotone and anti-monotone products.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
11 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
Definition
We say that A1 , A2 ⊂ A are independent with respect to the product ~, if
τ |hA1 ,A2 i ◦ ι = τ1 ~ τ2 ,
where ι : A1 t A2 denotes the unique extension of the inclusions of A1
and A2 into the free product A1 t A2 , to hA1 , A2 i.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
12 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
Definition
We say that A1 , A2 ⊂ A are independent with respect to the product ~, if
τ |hA1 ,A2 i ◦ ι = τ1 ~ τ2 ,
where ι : A1 t A2 denotes the unique extension of the inclusions of A1
and A2 into the free product A1 t A2 , to hA1 , A2 i.
Remarks:
1
For tensor, Boolean and free products, independence is a symmetric
relation (not true for monotone and anti-monotone products).
2
By associativity, we can define independent of any finite collection of
algebras.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
12 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
We have the following rules for determining independence of a collection
of sub-algebras {Ai }i∈I of A.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
13 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
We have the following rules for determining independence of a collection
of sub-algebras {Ai }i∈I of A.
Tensor: for {i(ε1 ), . . . , i(εm )} ⊂ I and a1 ∈ Ai(ε1 ) , . . . , am ∈ Ai(εm ) ,
we have


n
Y
 Y


τ (a1 · · · am ) =
τ
a
k


j=1
Arturo Jaramillo Gil (University of Kansas)
1≤k≤m
εk =j
The Five Natural Notions of Independence
September 30
13 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
We have the following rules for determining independence of a collection
of sub-algebras {Ai }i∈I of A.
Tensor: for {i(ε1 ), . . . , i(εm )} ⊂ I and a1 ∈ Ai(ε1 ) , . . . , am ∈ Ai(εm ) ,
we have


n
Y
 Y


τ (a1 · · · am ) =
τ
a
k


j=1
1≤k≤m
εk =j
Boolean: we require
τ (a1 · · · ak ) = τ (a1 ) · · · τ (ak ),
provided that neighbor elements come from different algebras.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
13 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
Free: we first need the fact that A1 , A2 are freely independent iff
hA1 , 1A i , A2 are freely independent. If so, we can consider
ai := ai − τ (ai )1A without leaving the sub-algebra. Then, we require
τ (a1 · · · an ) = 0,
for a1 , . . . , ak coming from alternating algebras.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
14 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
Monotone: The ordered collection {Ai }i∈I is monotone independent
if and only if for every n, m ∈ N,
τ (bai aj ak c) = τ (aj )τ (bai ak c) if i < j > k
τ (aim · · · ai2 ai aj1 aj2 · · · ajm ) = τ (aim ) · · · τ (ai2 )τ (ai )τ (aj1 )τ (aj2 ) · · · τ (ajm ),
if im > · · · > i2 > i2 > i < j1 < j2 < · · · < jn , for all random variables
ai ∈ Ai , aj ∈ Aj , ak ∈ Ak , ail ∈ Ail , b, c ∈ A.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
15 / 25
The Five Natural Notions of Independence
The Five Natural Notions of Independence
Anti-monotone: Is the mirror image of the Monotone case: the
ordered collection {Ai }i∈I is anti-monotone independent if and only if
for every n, m ∈ N,
τ (bai aj ak c) = τ (aj )τ (bai ak c) if i > j < k
τ (aim · · · ai2 ai aj1 aj2 · · · ajm ) = τ (aim ) · · · τ (ai2 )τ (ai )τ (aj1 )τ (aj2 ) · · · τ (ajm ),
if im < · · · < i2 < i2 < i > j1 > j2 > · · · > jn , for all random variables
ai ∈ Ai , aj ∈ Aj , ak ∈ Ak , ail ∈ Ail , b, c ∈ A.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
16 / 25
The Five Natural Notions of Independence
Additive Convolutions
When considering a C ∗ -probability space (A, τ ) with self-adjoint elements
a, b ∈ A (compactly supported) having distributions µa , µb , then the
distribution of µa+b can be recovered from µa and µb , provided that a, b
satisfy any of the five independence relations.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
17 / 25
The Five Natural Notions of Independence
Additive Convolutions
When considering a C ∗ -probability space (A, τ ) with self-adjoint elements
a, b ∈ A (compactly supported) having distributions µa , µb , then the
distribution of µa+b can be recovered from µa and µb , provided that a, b
satisfy any of the five independence relations.
Hence, for a pair of compactly supported probability measures µ, ν, we can
denote by µ ∗ ν, µ ] ν, µ ν, and µ . ν, respectively, their tensor,
Boolean, free and monotone additive convolutions.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
17 / 25
Central Limit Theorems
Central Limit Theorems
Let {an }n≥1 be a sequence of centered, self-adjoint random variables in a
C ∗ -probability space (A, τ ), which are all independent in one of the five
senses, all having the same distribution.
Question: what is the asymptotic distribution of the sum
SN =
a1 + · · · + aN
√
.
N
Note: by asymptotic distribution, we mean, what happens with
limN→∞ τ ((SN )n ) for every n ≥ 1.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
18 / 25
Central Limit Theorems
Central Limit Theorems
Notice that
τ
a1 + · · · + aN
√
N
Arturo Jaramillo Gil (University of Kansas)
n =
1
N
n
2
X
τ (ar1 · · · arn ).
1≤r1 ,...,rn ≤N
The Five Natural Notions of Independence
September 30
19 / 25
Central Limit Theorems
Central Limit Theorems
Notice that
τ
a1 + · · · + aN
√
N
n =
1
N
X
n
2
τ (ar1 · · · arn ).
1≤r1 ,...,rn ≤N
Observe that for the cases of Boolean, classical and free independence,
since {ai }1≤i≤N are equi-distributed, we obtain that
r (i) = r (j)
⇔
p(i) = p(j) ∀i, j
holds. Then, τ (ar (1) . . . ar (n) ) depends only on which indices of the n-tuple
(r (1), . . . , r (n)) are the same.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
19 / 25
Central Limit Theorems
Central Limit Theorems
For such cases (Boolean, classical and free), all the information concerning
the different values of τ (ar (1) · · · ar (n) ) can be encoded by a partition of
the set of {1, . . . , n} in the following way. We associate to an n-tuple
(r (1), . . . , r (n)), the partition π = {V1 , . . . , Vs }, where 1 ≤ p, q ≤ n
belong to the same block if and only if r (p) = r (q).
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
20 / 25
Central Limit Theorems
Central Limit Theorems
For such cases (Boolean, classical and free), all the information concerning
the different values of τ (ar (1) · · · ar (n) ) can be encoded by a partition of
the set of {1, . . . , n} in the following way. We associate to an n-tuple
(r (1), . . . , r (n)), the partition π = {V1 , . . . , Vs }, where 1 ≤ p, q ≤ n
belong to the same block if and only if r (p) = r (q). Then, we can rewrite
our sum in the following way:
X
τ ((a1 + · · · aN )n ) =
kπ AN
π,
π partitions
of{1,...,n}
where AN
π := {(r (1), . . . , r (n)) ∼ π 1 ≤ r (i) ≤ N} and kπ is the value
(independent from N) of τ on the class represented by π.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
20 / 25
Central Limit Theorems
Central Limit Theorems
Observations:
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
21 / 25
Central Limit Theorems
Central Limit Theorems
Observations:
1 For any π containing a block with a single element r (i), we have
kπ = 0, for every type of independence (and hence, the number of
blocks of π can’t exceed n/2).
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
21 / 25
Central Limit Theorems
Central Limit Theorems
Observations:
1 For any π containing a block with a single element r (i), we have
kπ = 0, for every type of independence (and hence, the number of
blocks of π can’t exceed n/2).
2 AN is the number of ways in which we can associate different indices
π
to each block of π. This can be done in N(N − 1) · · · (N − |π| + 1)
|π| asymptotically. Then,
ways. Hence, AN
π ∼N
X
X
n
(a1 + · · · aN )n
− n2
√
) = lim
kπ AN
= lim
kπ N |π|− 2 .
lim τ (
πN
N→∞
N→∞
N→∞
N
π
π
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
21 / 25
Central Limit Theorems
Central Limit Theorems
Observations:
1 For any π containing a block with a single element r (i), we have
kπ = 0, for every type of independence (and hence, the number of
blocks of π can’t exceed n/2).
2 AN is the number of ways in which we can associate different indices
π
to each block of π. This can be done in N(N − 1) · · · (N − |π| + 1)
|π| asymptotically. Then,
ways. Hence, AN
π ∼N
X
X
n
(a1 + · · · aN )n
− n2
√
) = lim
kπ AN
= lim
kπ N |π|− 2 .
lim τ (
πN
N→∞
N→∞
N→∞
N
π
π
Then, the only partitions that have non-vanishing contributions are the
blocks of size 2, and
X
(a1 + · · · aN )n
√
lim τ
=
kπ .
N→∞
N
π pairing
of {1,...,n}
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
21 / 25
Central Limit Theorems
Central Limit Theorems
Then, the odd moments are zero, and when n = 2k, depending on the type
of independence, we obtain the following limit behavior for the moments.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
22 / 25
Central Limit Theorems
Central Limit Theorems
Then, the odd moments are zero, and when n = 2k, depending on the type
of independence, we obtain the following limit behavior for the moments.
1
Classical case
(a1 + · · · aN )n
√
lim τ
=
N→∞
N
π
Arturo Jaramillo Gil (University of Kansas)
X
pair partition
of {1,...,2k}
The Five Natural Notions of Independence
σ 2k kπ = σ 2k
(2k)!
.
2k k!
September 30
22 / 25
Central Limit Theorems
Central Limit Theorems
Then, the odd moments are zero, and when n = 2k, depending on the type
of independence, we obtain the following limit behavior for the moments.
1
2
Classical case
(a1 + · · · aN )n
√
lim τ
=
N→∞
N
π
Boolean case
(a1 + · · · aN )n
√
lim τ
=
N→∞
N
Arturo Jaramillo Gil (University of Kansas)
X
σ 2k kπ = σ 2k
pair partition
of {1,...,2k}
X
(2k)!
.
2k k!
σ 2k kπ = σ 2k .
π only on the partition
π={{1,2},...,{2k−1,2k}}
The Five Natural Notions of Independence
September 30
22 / 25
Central Limit Theorems
Central Limit Theorems
Free case:
(a1 + · · · aN )n
√
lim τ
=
N→∞
N
π
Arturo Jaramillo Gil (University of Kansas)
X
σ 2k =
non-crossing
The Five Natural Notions of Independence
σ 2k
k +1
2k
k
September 30
.
23 / 25
Central Limit Theorems
Central Limit Theorems
Free case:
(a1 + · · · aN )n
√
lim τ
=
N→∞
N
π
X
σ 2k =
non-crossing
σ 2k
k +1
2k
k
.
For the monotone and anti-monotone cases, the partitions must
take order into consideration, so is a little different. Nevertheless, we
can follow the same methodology to prove that
(a1 + · · · aN )n
σ 2k
2k
√
lim τ
= k
.
k
N→∞
2
N
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
23 / 25
Central Limit Theorems
Central Limit Theorems
We have hence proved
that depending on the type of independence, the
(a1 +···aN )n
√
converges in law to a law with density f (x) given by
sequence,
N
x2
1
2
1
Classical case f (x) = σ√
e − 2σ2 ,
π
√
1
Free case f (x) = 2π
4 − x 2 1[−2,2] (x),
3
Boolean case f (x) = 21 (δ−1 + δ1 ),
4
Monotone and anti-monotone case f (x) =
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
√1
π 4−x 2
1[−2,2] (x).
September 30
24 / 25
Central Limit Theorems
Bibliography
Naofumi Murak. The five independences as Natural Products.
Victor Perez Abreu. Part of a book still not published.
Arturo Jaramillo Gil (University of Kansas)
The Five Natural Notions of Independence
September 30
25 / 25