Occupancy statistics arising from weighted particle rearrangements

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Occupancy statistics arising from weighted particle rearrangements

Thierry Huillet

To cite this version:

Thierry Huillet. Occupancy statistics arising from weighted particle rearrangements. Journal of

Physics A: Mathematical and Theoretical, IOP Publishing, 2007, 40, pp.9179-9200. �hal-00120316�

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Occupancy statistics arising from weighted particle rearrangements

Thierry Huillet

Laboratoire de Physique Th´ eorique et Mod´ elisation, CNRS-UMR 8089 et Universit´ e de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302, Cergy-Pontoise, FRANCE

E-mail: Thierry.Huillet@u-cergy.fr December 13, 2006

Abstract

The box-occupancy distributions arising from weighted rearrangements of a particle system are investigated. In the grand-canonical ensemble, they are characterized by determinantal joint probability generating functions. For doubly non-negative weight matrices, fractional occupancy statistics, generalizing Fermi-Dirac and Bose-Einstein statistics, can be defined. A spatially extended version of these balls-in-boxes problems is investigated.

Keywords: occupancy problems, urn models, interaction, Boltzmann weight matrices, balls in boxes, MacMahon master theorem, permanents, correlation functions.

AMS Classification 2000: 60C05, 60E05, 05Axx, 82Bxx, 60-02.

PACS Numbers: 02.50.Ey, 02.50.-r, 05.30.Pr, 05.40.-a, 02.10.Yn

1 Introduction and outline

The purpose of this work is to study a class of combinatorial balls-in-boxes mod-

els as a random allocation scheme of particles. Although the urn model under

study is formally in the spirit of the ones found for example in Charalambides,

[6], Johnson and Kotz, [16], or Kolchin, [17], books, it is not covered by these

manuscripts. Another unrelated balls-in-boxes process of similar flavour was

recently revisited in [15], developing some equilibrium aspects of the zeta-urn

model first introduced in [4]. In some sense made precise later, the model we

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shall deal with here is concerned with random box filling of an interacting particle system derived from weighted permutations of its constitutive items.

We now describe our model in some detail. Suppose we are given a total amount of k ≥ 1 labeled particles (items) of n different types (or colors), with n ≥ 2. Let k

m

, m = 1, .., n be the number of type-m particles in some initial configuration, with |k

n

| := k

1

+ .. + k

n

= k and k

n

= (k

m

; m = 1, .., n). Place initially these k particles in boxes and more specifically, place the k

m

type-m particles in box number m, m = 1, .., n, respectively. Suppose the energy required to move a particle from box m to box m

⁰

is − log W

m,m⁰

where W

m,m⁰

is the m ×m

⁰

entry of some non-negative n×n weight matrix W . We shall address the problem of evaluating the configurational weight of all particle rearrangements which end up with k

_n

particles in the different boxes, regardless of their type.

Upon suitable normalization, for each Boltzmann weight matrix W , we shall use

this to define the Gibbs canonical probability generating function of the random

box-occupancies K

_n,k

= (K

_n,k

(m) ; m = 1, .., n), given a total population of k

particles. From this model, the event K

n,k

= k

n

will be realized either because

there are k

m

type-m particles in box number m but also for any rearrangement

of this peculiar configuration, the weight of each being needed to evaluate its

occurrence probability. After suitably randomizing the particle number k, we

shall rather work with the grand-canonical probability generating function of

the random occupancies K

n,z

where ‘fugacity’ parameter z is in one to one cor-

respondence with the average number κ of particles in the system. We shall

show that it has a determinantal form and that the associated probabilities are

rather permanantal, that is, can be expressed in terms of the permanent of some

enlarged matrix derived from W . Since W has non-negative entries, it turns out

that the joint distribution of K

_n,z

is infinitely divisible (that is in the compound

Poisson class). As a result, it makes sense to raise the probability generating

function of K

_n,z

to the power α, for all α ∈ (0, ∞). This leads to a fractional

occupancy statistics of order α, the special case α = 1 corresponding to the

standard Bose-Einstein occupancy model. Under the additional condition that

W is definite non-negative, it makes sense to raise the probability generating

function of K

n,z

to the power α, for all α ∈ {.., −2, −1}: for such distributions,

the maximal number of particles within each box cannot exceed −α. The special

case α = −1 corresponds to the usual Fermi-Dirac occupancy model involving

an exclusion principle: no more than one particle within each box. Therefore,

for doubly non-negative weight matrices (that is, both non-negative and definite

non-negative), fractional occupancies of all order α ∈ {.., −2, −1} ∪ (0, ∞) can

be defined. In the limit |α| % ∞, a Maxwell-Boltzmann occupancy model is

found: grand-canonical box-occupancies turn out to be independent and Pois-

son distributed. All these allegations can easily be derived from the version

of the MacMahon master theorem which is relevant to our balls-in-boxes con-

text, respecting transition weights. We now briefly describe the content of the

manuscript in more details.

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In Section 2, we start illustrating these ideas in the case where W is a weight matrix with {0, 1} −entries, starting with the flat matrix. Here the configurational weight of particle rearrangements simply counts the number of admissible rearrangements when transition from box m to box m

⁰

is either forbidden or allowed, depending on whether W

m,m⁰

is 0 or 1. We discuss some examples in Sub-sections 2.1 and 2.2; some are solvable, some are more involved.

Subsection 2.3 considers the full case of a real-valued weight matrix that is doubly non-negative. Here we compute for instance the factorial moments of K

n,z

, the 2−boxes joint law of the occupancies (K

n,z

(m

1

) ; K

n,z

(m

2

)) , the marginal law of K

n,z

(m). We prove that the limiting distribution of K

n,z

when α % ∞ is the Maxwell-Boltzmann distribution. We characterize the limit law of K

_n,z

/κ when the expected number κ of particles goes to infinity, in terms of a specific multivariate gamma distribution.

Section 3 is devoted to what is needed of MacMahon master theorem which is relevant to our balls-in-boxes context.

Finally, we shall address the following problem: Assume particles can only be placed in boxes in the positions −∞ < x

1

≤ x

2

≤ .. ≤ x

n

< ∞ on the real line. The box number of a particle now is the index of its position on R and the model is spatially extended: a particle is attached to box number m if it stands at m−th position x

m

. The purpose of Section 4 is to construct natural doubly non-negative weight matrices W which are indexed by x := (x

1

, .., x

n

) ∈ R

ⁿ

, representing particle positions on the line. They are adapted to the spatially extended version of the occupancy problem. In such models, the m × m

⁰

entry W

m,m⁰

of W is of the form W (x

m

, x

m⁰

); using this, occupancy statistics can be considered when − log W (x

m

, x

m⁰

) is now the energy required to move a particle from position x

m

to position x

m⁰

. The construction of such matrices parallels the one of non-negative correlation functions whose spectral measure is positive. We shall also discuss the relevance of the recent notion of an infinitely divisible weight matrix, in the context of our investigations. Several examples are supplied.

2 Multi-type weighted particle rearrangements

As discussed in the Introduction, we start with the simplest rearrangement case before extending the construction to more general weight matrices.

2.1 The simple rearrangement case

Suppose we are given a total amount of k ≥ 1 labeled particles (items) of n

different types (or colors), with n ≥ 2. Let k

m

, m = 1, .., n be the number of

type-m particles in some initial configuration, with |k

n

| := k

1

+..+k

n

= k. Place

k

m

particles in box number m, m = 1, .., n, respectively. When considering the

problem of enumerating the number of ways to permute these k

_n

particles ending

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up with k

_n

particles in the different boxes (or urns), the following generating function proves necessary

1 1 − z (u

₁

+ .. + u

_n

) = X

k_n∈Nⁿ

z

^|kⁿ^|

|k

n

|!

Q

n m=1

k

_m

!

n

Y

m=1

u

^k_m^m

.

Here u : = (u

₁

, .., u

_n

) ∈ [0, 1]

ⁿ

‘marks’ the different types of particles and z ∈ [0, z

_c

:= 1/n) is a ‘marker’ of the total number of particles. From this, interpreting the above generating function as an ‘exponential’ generating function, extracting the Taylor coefficients in the variables Q

n

m=1

u

^k_m^m

of its series expansion, as conventional wisdom suggests, there are |k

_n

|! = k! ways to permute the k

n

labeled (distinguishable) particles. In other words, if S (k

n

) is the set of all such permutations, then |S (k

n

)| = |k

n

|!. This is poorly informative, so far.

Would the particles be unlabeled within each type class, there would clearly be

Qn^|kⁿ^|!

m=1km!

ways to permute the k

_n

unlabeled particles, looking the above generating function as an ‘ordinary’ generating function and simply extracting the coefficients in the variables Q

n

m=1

u

^k_m^m

of its series expansion. In this case, a permutation is called a rearrangement of the word 1

^k¹

...n

^kⁿ

. Clearly, when assigning u

1

= .. = u

n

= 1, the

z

^k

−coefficient of (1 − zn)

⁻¹

, which is n

^k

, will count the number of ways to permute (rearrange) k unlabeled particles of n different types, regardless of the number of particles within each class.

Assuming for example n = 2, k

₁

= 2, k

₂

= 1, k = 3, identifying the 2 particles of the first type, there are 3 different possible rearranged words out of 1

²

2

¹

, namely 11 2, 12 1 and 21 1 whereas there are 2

³

= 8 ways to permute 3 unlabeled particles of 2 different types: ∅ 222, (1 22, 2 12, 2 21), (11 2, 12 1, 21 1) and 111 ∅ corresponding respectively to the partitions (k

1

, k

2

) = (0, 3), (1, 2), (2, 1) and (3, 0) of k = 3. Here ∅ is the empty box ‘word’. For more details on rearrangements, see Cartier and Foata, [5].

Let W = J where J is the n×n flat ‘weight’ matrix whose entries are all equal to 1, for which Spect(W ) = {0, .. (n − 1) times.., 0; n} . With |W | := det (W ) and U :=diag(u), it turns out that

1 1 − z (u

1

+ .. + u

n

) = |I − zU W|

⁻¹

. Next, we note that

"

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

#

log |I − zU W |

⁻¹

= (|k

n

| − 1)!

counts the number of ways to permute cyclically the k

_n

labeled particles, when

restricting S (k

_n

) to S

c

(k

_n

) , the set of all connected (one cycle) permutations,

clearly with |S

_c

(k

_n

)| = (|k

_n

| − 1)!.

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With α > 0 now ‘marking’ the number of cycles, this suggests to also consider the generating function

|I − zU W|

^−α

= e

^−αlog|I−zU W|

.

With p ∈ N , the generating function [α

^p

] |I − zU W|

^−α

is the one counting the permutations of n types of particles, when restricting them to be a collection of p disconnected cycles. In particular, [α] |I − zU W|

^−α

= − log |I − zU W | counts the permutations of n types of particles, when restricting them to be simple cycles. We shall return to this point later.

Canonical and grand-canonical randomization.

Let K

_n,k

:= (K

_n,k

(m) ; m = 1, .., n) denote an integral-valued random vector which will stand for occupancies of the boxes m = 1, .., n given a total population of k particles.

Given |k

_n

| = k, define naturally the (conditional) probability that K

_n,k

(m) = k

_m

, m = 1, .., n by the ratio of their configurations numbers

P (K

n,k

= k

n

) =

k k1..kn

[z

^k

] (1 − zn)

⁻¹

· 1 (|k

n

| = k) .

The event K

n,k

= k

n

will be realized either because there are k

m

type-m particles in box number m but also for any rearrangement of this peculiar configuration. In other words,

E

n

Y

m=1

u

^Km^n,k^(m)

!

= z

^k

(1 − z (u

₁

+ .. + u

_n

))

⁻¹

[z

^k

] (1 − zn)

⁻¹

is its canonical (conditional) probability generating function.

To avoid considering the simplex |k

n

| = k, suppose one randomizes the total number k of particles as follows: let there be a random number K

n,z

of particles where

P (K

n,z

= k) = z

^k

z

^k

(1 − zn)

⁻¹

(1 − zn)

⁻¹

. In other words, with u ∈ [0, 1]

E u

^K^n,z

= |I − zW|

|I − zuW | = 1 − zn 1 − zun

is the (geometric) generating function of K

n,z

. Then, parameters z and 0 <

κ := E (K

n,z

) are related through

κ := κ (z) = zn

1 − zn ,

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and, since ‘fugacity’ z ∈ [0, z

_c

= 1/n), these are in one-to-one correspondence.

Given κ := E (K

n,z

) =

_1−zn^zn

, define next the probability that K

n,z

(m) = k

m

, m = 1, .., n simply by

P (K

n,z

= k

n

) = z

^|kⁿ^| _k^|kⁿ^|

1..kn

(1 − zn)

⁻¹

, k

n

∈ N

ⁿ

. In other words, with W = J

E

n

Y

m=1

u

^K_m^n,z^(m)

!

= (1 − z (u

1

+ .. + u

n

))

⁻¹

(1 − zn)

⁻¹

= |I − zW|

|I − zU W|

stands for its (grand canonical) probability generating function, now conditioned on the expected number κ (z) of particles in the system.

Asymptotics for κ % ∞.

One suspects that, for fixed n, the following convergence in distribution will

hold K

_n,z

κ

→

dκ%∞

X

_n

,

where X

n

:= (X (1) , .., X (n)) is an n−dimensional random vector supported by R

ⁿ+

. Let us prove this and characterize the joint law of X

n

. Recalling κ := κ (z) =

_1−zn^zn

, let us assume z = (1 − ) /n where, being close to 0

⁺

, κ, which is of order

⁻¹

, tends to ∞. Then, with s := (s

1

, .., s

n

) and S :=diag(s)

E

n

Y

m=1

e

^−s^m^K^n,z^(m)/κ

!

∼ E

n

Y

m=1

e

^−s^m^K^n,n⁻^{1 (1−)}^(m)

!

=

1 − (1 − ) ( P

n

1

e

^−s^m

) /n ∼

1 − (1 − ) (n − P

n 1

s

m

) /n

∼ 1

1 + ( P

n

1

s

m

) /n = 1

|I + z

_c

SW | = E

n

Y

m=1

e

^−s^m^X(m)

! .

This is the Laplace Stieltjes transform of a symmetric exponentially distributed random vector X

_n

, with 1-dimensional marginals E e

^−s^m^X(m)

=

_1+s¹

m/n

, m = 1, .., n, the ones of exponentially distributed random variables on R

⁺

with mean n

⁻¹

.

Fractional statistics.

Let z

α

:= z/α. We shall also be interested in random variables with distributions P

α

defined by

E

α n

Y

m=1

u

^K_m^n,z^(m)

!

= 1 −

_α^z

(u

₁

+ .. + u

_n

)

−α

1 −

_α^z

n

−α

=

|I − z

α

W |

|I − z

_α

U W |

^α

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where α ∈ {.., −2, −1}∪(0, ∞) and z < α/n if α > 0 or z ≥ 0 if α ∈ {.., −2, −1} . Since W = J , with (α)

_k

:= α (α + 1) .. (α + k − 1), this is

P

α

(K

_n,z

= k

_n

) = (1 − z

_α

n)

^α

· z

^|k_αⁿ^|

(α)

_|k

n|

Q

n

m=1

k

m

! , k

_n

∈ N

ⁿ

.

When α = −i < 0, the range of K

n,z

is k

n

∈ {0, .., i}

ⁿ

. For instance, for k

_n

∈ {0, 1}

ⁿ

, P

−1

(K

_n,z

= k

_n

) is the (Fermi-) joint probability to find one (respectively, no) particle in box number m

_q

if k

_m_q

= 1 (respectively, if k

_m_q

= 0). Thus, when W = J, K

_n,z

is well-defined under P

α

for all α ∈ {.., −2, −1} ∪ (0, ∞) . In particular,

E

α

u

^K^n,z

=

1 − z

α

n 1 − z

_α

un

^α

,

showing that K

n,z

has negative binomial(α, z

α

n) distribution (when α > 0) or multinomial(−α, (−z

α

n) / (1 − nz

α

)) distribution (α ∈ {.., −2, −1}), with mean κ := E

α

(K

n,z

) =

_1−nz^zn

α

. Such α−distributions may be thought of as fractional occupancy statistics of order α. When α = −1, α = +1, |α| % ∞ we get a usual Fermi-Dirac, Bose-Einstein or Maxwell-Boltzmann distribution for K

_n,z

, respectively. When α = k/2 where k is any integer, we get a para-Boson statistics (see Aringazin et al and Greenberg et al, [1] and [14]). One of the questions raised next is: for which W is K

_n,z

well-defined for all α in the positivity spectrum {.., −2, −1} ∪ (0, ∞)?

2.2 Weighted restricted rearrangements: {0, 1} −weight matrix

These elementary considerations suggest first to introduce more generally the quantities |I − zU W |

⁻¹

where W is now a n × n weight matrix whose entries all belong to {0, 1} . By doing so, we address the problem of enumerating the number of ways to permute the |k

_n

| = k particles when the transition from type m to type m

⁰

is allowed at the only condition that entry W

_m,m⁰

= 1.

Define formally energy (cost) of transition m → m

⁰

as H

m,m⁰

:= − log W

m,m⁰

. Then, would W

m,m⁰

= 0, would the transition from type m to type m

⁰

be forbidden as the energy required to realize this transition is infinite. Would W

m,m⁰

= 1, the transition m → m

⁰

requires no particular energy. Proceeding in this way, the random occupancies K

n,z

are now dictated by the weighted restricted rearrangements encoded by W . For related problems, see Diaconis et al, [9].

The quantity |I − zU W |

⁻¹

is defined for u : = (u

₁

, .., u

_n

) ∈ [0, 1]

ⁿ

and z ∈

[0, z

_c

:= 1/ρ (W )) where, with λ

_m

; m = 1, .., n the spectrum of W , ρ (W ) =

kW k = max (|λ

m

| ; m = 1, .., n) is the spectral radius of W. The critical fugacity

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z

_c

is the reciprocal of the spectral radius of W and |I − zU W |

⁻¹

first becomes singular at z

c

. Since all such W are non-negative, by Perron-Foebenius theorem, the eigenvalue with largest modulus is real. Since W only has {0, 1} −entries, ρ (W ) ∈ [1, n] and so z

c

∈ [1/n, 1]. The weighted random occupancies K

n,z

(m), m = 1, .., n can be defined accordingly. As we shall indeed see later in the next section, such random variables are well-defined; in fact they are always well-defined as soon as W has non-negative entries. This is because the Taylor coefficients in the variables Q

n

m=1

u

^k_m^m

of the series expansion of

_{|I−zU W|}^|I−zW^|

are the permanents Per(W (k

n

)). Here W (k

n

) are the |k

n

| × |k

n

| matrix obtained by repeating (removing and deleting) each W

m,m⁰

into a size k

m

× k

m⁰

block if k

m

and k

m⁰

both positive (otherwise). Permanents of a matrix with non-negative entries are non-negative. Note that the basic quantity appearing in the law of K

n,z

reads |I − zW |

⁻¹

= Q

n

m=1

(1 − zλ

m

)

⁻¹

.

Similarly, random variables K

_n,z

with law P

α

parameterized by α > 0 are well-defined for all α > 0 when W has in particular non-negative entries. This is because the Taylor coefficients in the variables Q

n

m=1

u

^k_m^m

of the series expansion of

|I−zαW|

|I−zαU W|

α

now are proportional to the α−permanents Per

_α

(W (k

_n

)).

α−Permanents of a matrix with non-negative entries are non-negative (see the next Section). In all such cases, random variables K

n,z

with law P

α

(where α > 0) are infinitely divisible, that is in the compound Poisson class. When α ∈ {.., −2, −1}, a necessary and sufficient condition for K

_n,z

with law P

α

to be well-defined is that W has all its 2

ⁿ

principal minors non-negative (see Vere- Jones, [21], Prop. 6.1 for the necessity condition). If K

_n,z

with law P

α

is to be defined for all α ∈ {.., −2, −1} ∪ (0, ∞), a sufficient condition is that W it has all its principal minors non-negative and is non-negative. If W is in addition symmetric, then a doubly non-negative W would do (doubly non-negative matrices are those which are both definite non-negative and with non-negative entries).

Examples.

• Assume W = I. Then, with z < z

c

= 1

|I − zU W |

⁻¹

=

n

Y

m=1

(1 − zu

m

)

⁻¹

= 1 1 − ⊕

ⁿ_m=1

(zu

m

)

where x

1

⊕x

2

= x

1

+x

2

−x

1

x

2

is the commutative and associative probabilistic sum. Here,

"

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

_m

!

#

|I − zU W |

⁻¹

=

n

Y

m=1

k

m

!

counts the number of ways to permute the k

n

labeled particles when the image

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particle is forced to return back to its own type. Therefore, E u

^K^n,z

= |I − zW |

|I − zuW | =

1 − z 1 − zu

ⁿ

is the generating function of K

_n,z

, with mean κ = (nz) / (1 − z) . The z

^k

−coefficient of (1 − z)

⁻ⁿ

is

^n+k−1_k

.

Clearly, in this interaction-free case K

n,z

κ

→

dκ%∞

X

n

where the components of the random vector X

n

are independent and identically distributed (iid), mean 1, exponentially distributed random variables on R

+

.

• In the last two examples, the weight matrix is definite non-negative. It has all principal minors non-negative. The canonical form of a {0, 1} −valued definite non-negative weight matrix W is made of p flat Jordan blocks J

q

(of type J ), q = 1, .., p where the sizes of J

_q

are n

_q

× n

_q

, q = 1, ..p with P

p

q=1

n

_q

= n and 1 ≤ n

₁

≤ ... ≤ n

_p

. Since W is symmetric, outside the Jordan blocks, W has zero entries and so is block-diagonal. The number of such matrices therefore is the number of partitions of n. The spectrum of W is Spect(W ) =

∪

^p_q=1

{0, .. (n

_q

− 1) times.., 0; n

_q

} and with n

₀

:= 0 and n

_q

:= P

q

r=0

n

_q

, q = 0, .., p

|I − zU W|

⁻¹

=

p

Y

q=1

1 − z u

nq−1+1

+ .. + u

n_q

⁻¹

=

1 − ⊕

^p_q=1

z u

_n_q−1₊₁

+ .. + u

_n_q

−1

defined for z < 1/n

p

. Representing the above partition sequence 1 ≤ n

1

≤ ... ≤ n

p

by the sequence 0 =: m

0

< 1 ≤ m

1

< ... < m

r

where each m

q

, q = 1, .., r has multiplicity d

q

now with P

r

q=1

d

q

m

q

= n, E u

^K^n,z

= |I − zW |

|I − zuW | =

r

Y

q=1

1 − zm

q

1 − zum

q

^dq

is the generating function of K

n,z

, with mean κ =

r

X

q=1

(zd

q

m

q

) / (1 − zm

q

) .

The first introductory example (W = J ) corresponds to an irreducible case with r = 1, m

1

= n and d

1

= 1 whereas the second example (W = I) is completely reducible with r = 1, m

₁

= 1 and d

₁

= n.

For this class of W , K

_n,z

is well-defined under P

α

for all α ∈ {.., −2, −1} ∪

(0, ∞).

(11)

• (Derangements) Assume now that W = J − I which is symmetric but not definite-positive.

One can check that Spect(W ) = {−1, .. (n − 1) times.., −1; n − 1}. Then, with z < z

c

= 1/ (n − 1), it holds that

|I − zU W|

⁻¹

= 1 1 − P

n

p=2

z

^p

(p − 1) σ

p

(u) where

σ

_p

(u) = X

1≤m1<..<m_p≤n p

Y

q=1

u

_m_q

are the elementary symmetric functions. For this example,

"

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

#

|I − zU W |

⁻¹

counts the number of ways to permute the k

n

labeled particles, when no per- muted particle is allowed to return to its own type. A permutation in this class is called a derangement (see Gillis, [13]).

In this case, with 0 ≤ z < 1/ρ (W ) = 1/ (n − 1), we get

|I − zW |

⁻¹

= 1 1 − P

n

p=2

z

^p

(p − 1)

ⁿ_p

= 1

(1 + z)

ⁿ⁻¹

[1 − (n − 1) z]

with z

^k

|I − zW |

⁻¹

= P

k

l=0

(−1)

^l ^n+l−2_l

(n − 1)

^k−l

≥ 0. So, for instance E u

^K^n,z

= |I − zW |

|I − zuW | =

1 + z 1 + zu

n−1

· 1 − (n − 1) z 1 − (n − 1) zu is the generating function of K

n,z

, with mean

κ = (n − 1) z

1 1 − (n − 1) z − z 1 + z

.

For this class of W , K

n,z

is well-defined under P

α

for all α ∈ (0, ∞) (see Foata and Zeilberger, [12]) but, since the principal minors can be negative, not for α ∈ {.., −2, −1}.

• If W is any of the (n − 1)! orthogonal matrices of a cyclic (connected) permutation

|I − zU W|

⁻¹

= 1 1 − Q

n

m=1

(zu

m

) , z < z

_c

= 1.

For this example, there is no way to permute the k

_n

labeled particles, unless all k

_m

= k

₁

, m = 1, .., n in which case this number is (k

₁

!)

ⁿ

. Here,

E u

^K^n,z

= |I − zW |

|I − zuW | = 1 − z

ⁿ

1 − (zu)

ⁿ

(12)

is the generating function of K

_n,z

, with mean κ = (nz

ⁿ

) / (1 − z

ⁿ

) . Note that the spectrum of W consists in the n roots of unity. With λ

m

= e

^{2iπ(m−1)/n}

, m = 1, .., n note the identity P

n

m=1 zλ_m

1−zλm

=

_1−z^nzⁿn

= κ.

Clearly, in this case

K

n,z

κ

→

d_κ%∞

X

n

where, with E Q

n

m=1

e

^−s^m^X(m)

= (1 + ( P

n

1

s

m

) /n)

⁻¹

, X

n

has symmetric multivariate exponential distribution.

For this class of W , K

_n,z

is well-defined under P

α

for all α ∈ (0, ∞) but, since the principal minors can be negative, not for α ∈ {.., −2, −1}.

The next two examples are more involved and would require some additional knowledge of the weight matrix spectrum:

• (periodic nearest neighbors interactions):

Assume W

1,2

= W

1,n

= 1; W

m,m+1

= W

_m,m−1

= 1, m = 2, .., n − 1 and W

n,1

= W

_n,n−1

= 1, the other entries being all equal to 0. In this case, ρ (W ) = 2 and z

c

= 1/2. Matrix W is symmetric with real eigenvalues. In a standard (non-periodic) nearest neighbors interactions model, W

1,n

= W

n,1

= 0 and W is strictly tridiagonal.

• tournaments: assume W

m,m

= 0, m = 1, .., n. For all m

⁰

> m, fix W

m,m⁰

∈ {0, 1} and force W

m⁰,m

= 1 − W

m,m⁰

so that P

m,m⁰

W

m,m⁰

=

ⁿ₂

. Each W fulfills W + W

⁰

= J − I. There are 2

^n(n−1)/2

tournaments as issues of pair matching games with n players. 2

Remark.

The distribution of K

n,z

is exchangeable if and only if E Q

n

m=1

u

^Km^n,z^(m)

= E

Q

n

m=1

u

^Kσ_m^n,z^(m)

for all permutation σ of {1, .., n} ; in other words, if and only if |I − zU W | is a symmetric function of u := (u

₁

, .., u

_n

) . This will be the case under the following special conditions: W = aI +b (J − I) where a and b belong to {0, 1} . Thus, when W = I, W = J − I or W = J. And also when W is the weight matrix of a cyclic (connected) permutation. In all these cases, for each order, all principal minors of W coincide, which is a necessary and sufficient condition for the distribution of K

_n,z

to be exchangeable. This is clear from the development of |I − zU W | in terms of all principal minors of W . 3

2.3 Rearrangements from doubly non-negative weight matrices

Let W be an arbitrary weight matrix with real non-negative entries and with

non-negative principal minors. With z

_α

:= z/α, we are interested in random

(13)

variables K

_n,z

with probability generating functions E

α

n

Y

m=1

u

^K_m^n,z^(m)

!

=

|I − z

_α

W |

|I − z

α

U W |

α

where α ∈ {.., −2, −1}∪(0, ∞) and z < α/n if α > 0 or z ≥ 0 if α ∈ {.., −2, −1} . These random variables are well-defined for all α in the prescribed range and infinitely divisible when α > 0. They correspond to occupancy α−distributions arising from rearrangements with weight W. If W is in addition symmetric, then W belongs to the class of doubly non-negative matrices.

For general doubly non-negative real-valued matrices, K

n,z

is well-defined under P

α

where α varies in the announced range.

Factorial moments of K

n,z

.

Let k

m

≥ 0, m = 1, .., n and {n}

_k

:= n (n − 1) .. (n − k + 1) , {n}

₀

:= 1.

Define the factorial moments of K

n,z

to be µ

α

(k

n

) := E

α

n

Y

m=1

{K

n,z

(m)}

_k

m

! . We have

E

α n

Y

m=1

u

^K_m^n,z^(m)

!

=

|I − z

α

W |

|I − z

_α

U W |

^α

=

|I − z

α

W |

|I − z

_α

W − z

_α

(U − I) W |

^α

=

I − (U − I) z

α

W (I − z

α

W )

⁻¹

−α

=: (|I − (U − I) W

z_α

|)

^−α

. Therefore, the factorial moments µ

_α

(k

_n

) of K

_n,z

exist; they are given by the Taylor coefficients in the variables Q

n

m=1

(u

m

− 1)

^k^m

of the series expansion of (|I − (U − I) W

_z_α

|)

^−α

, namely Per

_α

(W

_z_α

(k

_n

)), where W

_z_α

= z

_α

W (I − z

_α

W )

⁻¹

is the resolvent matrix of W. If W has non-negative entries, so does W

_z_α

and therefore the factorial moments of K

n,z

, namely Per

α

(W

z_α

(k

n

)) are non- negative. If W is definite-positive with positive eigenvalues λ

m

, so does W

z_α

with eigenvalues (z

α

λ

m

) / (1 − z

α

λ

m

). For doubly non-negative weight matrices, µ

α

(k

n

) are well defined for all α in the full range {.., −2, −1} ∪ (0, ∞).

Marginal distribution of K

n,z

(m).

With U

m

a diagonal matrix whose m × m entry is u

m

, all other diagonal entries being 1, with z

α

:= z/α, we have

E

α

u

^K_m^n,z^(m)

=

|I − z

α

W |

|I − z

α

U

m

W |

α

=

|I − z

α

W |

|I − z

α

W − z

α

(U

m

− I) W |

α

=

I − (U

m

− I) z

α

W (I − z

α

W )

⁻¹

−α

= (|I − (U

m

− I) W

z_α

|)

^−α

(14)

=

1 − (u

m

− 1) (W

z_α

)

_m,m

−α

=:

1 − p

m

1 − p

m

u

m

α

.

Since (W

_z_α

)

_m,m

≥ 0, when α > 0, this is the distribution of a negative binomial random variable with parameters

α, p

m

:=

_1+(W^(W^zα⁾^m,m

zα)_m,m

. Here p

m

is the success probability depending on z and α. When α ∈ {.., −2, −1}, this generating function is the one of a multinomial distribution with parameters −α, π

m

:= − (W

_z_α

)

_m,m

where π

_m

now is the probability that the underlying Bernoulli random variable takes the value 1.

Note from this analysis that in any case for α κ := E

α

(K

n,z

) = α

n

X

m=1

(W

z_α

)

_m,m

= α · tr (W

z_α

) .

Joint distribution of (K

_n,z

(m

₁

) , K

_n,z

(m

₂

)).

Let 1 ≤ m

₁

< m

₂

≤ n. With U

_m₁_,m₂

a diagonal matrix whose entries m

₁

× m

₁

and m

₂

× m

₂

are u

_m₁

and u

_m₂

, all other diagonal entries being 1, we have

E

α

u

^K_m^n,z₁ ^(m¹⁾

u

^K_m^n,z₂ ^(m²⁾

=

|I − z

α

W |

|I − z

_α

U

_m₁_,m₂

W |

^α

= (|I − (U

m₁,m₂

− I) W

z_α

|)

^−α

=



1 − X

i∈{1,2}

(u

_m_i

− 1) (W

_z_α

)

_m

i,m_i

+ (u

_m₁

− 1) (u

_m₂

− 1) |W

zα

(m

₁

, m

₂

)|





−α

where |W

z_α

(m

1

, m

2

)| is the corresponding minor of W

z_α

. From this, we get in particular

Cov

_α

(K

_n,z

(m

₁

) , K

_n,z

(m

₂

)) = α h (W

_z_α

)

_m

1,m₁

(W

_z_α

)

_m

2,m₂

− |W

zα

(m

₁

, m

₂

)| i

= α (W

z_α

)

_m

1,m₂

(W

z_α

)

_m

2,m₁

which is non-negative when W

_z_α

(or W ) is non-negative or when W

_z_α

(or W ) is symmetric (in particular definite positive). When α > 0, (K

_n,z

(m

₁

) , K

_n,z

(m

₂

)) are positively correlated. Note that the above covariance decreases with α and that Cov

α

(K

n,z

(m

1

) , K

n,z

(m

2

)) →

α%∞

0. In sharp contrast, when α ∈ {.., −2, −1}, these random variables are negatively correlated.

Limiting distribution of K

n,z

when α % ∞.

To the first order in α, the exponential-trace expression of the determinant gives

E

α n

Y

m=1

u

^K_m^n,z^(m)

!

=

|I − z

_α

W |

|I − z

α

U W |

^α

α%∞

→

(15)

e

^−tr(zW^{)+tr(zU W)}

=

n

Y

m=1

e

^−z(1−u^m^)W^m,m

. This shows that asymptotically

K

n,z

→

d α%∞

P

n,z

where P

n,z

:= (P

n,z

(1) , .., P

n,z

(n)) are mutually independent Poisson distributed random variables with respective intensities zW

m,m

, proportional to the diagonal terms of W. This is the limiting Maxwell-Boltzmann distribution for K

n,z

.

Limit law of K

n,z

/κ when κ % ∞.

Assume first α = 1 and let z

c

:= 1/ρ (W ) . Assume W differs from the identity. Letting z := z

_c

(1 − ) , κ = tr(W

_z

) grows like

⁻¹

, to the first order in . This is because κ = P

n

1

(zλ

_m

) / (1 − zλ

_m

) where (λ

_m

; m = 1, .., n) are the eigenvalues of W with ρ (W ) the largest (real) of these: this model does not show up phase transition phenomena because κ := κ (z) is always divergent at critical fugacity z

_c

.

Now, with s := (s

₁

, .., s

_n

)

⁰

and S :=diag(s) E

n

Y

m=1

e

^−s^m^K^n,z^(m)

!

&0

∼

|I − z

c

(1 − ) W |

|I − z

c

(1 − ) (1 − S) W |

&0

∼

|I − z

_c

W + z

_c

W |

|I − z

c

W + z

c

(I + S) W | .

Observing |I − z

c

W | = 0, the exponential-trace expansion of the determinant gives in this singular case

|I − z

c

W + z

c

W | ∼

&0

· tr (z

c

W · adj (I − z

c

W )) where adj(A) is the adjugate matrix of A. Thus

E

n

Y

m=1

e

^−s^m^K^n,z^(m)

!

&0

→

tr (z

c

W · adj (I − z

c

W ))

tr ((I + S) z

_c

W · adj (I − z

_c

W )) =: E

n

Y

m=1

e

^−s^m^Xⁿ^(m)

! . This shows that asymptotically

K

n,z

κ

→

d κ%∞

X

n

where the law of X

n

:= (X

n

(1) , .., X

n

(n)) is characterized by the above joint Laplace-Stieltjes transform. Developing the traces, with hs, X

n

i the scalar product of s ≥ 0 and X

n

, we get more specifically

E

e

^−hs,Xⁿⁱ

= 1 +

n

X

m=1

s

_m

µ

_m

!

⁻¹

= (1 + hs, µ

_n

i)

⁻¹

(16)

where µ

_n

:= (µ

₁

, .., µ

_n

) , µ

_m

= b

_m

/ P

n

1

b

_k

and b

_m

= (z

_c

W · adj (I − z

_c

W ))

_m,m

, m = 1, .., n are the diagonal terms of the matrix z

c

W ·adj(I − z

c

W ). In any case, it holds that P

n

1

µ

m

= 1 so that P

n

1

X

n

(m) has mean 1 exponential law.

The marginals of X

n

are exponentially distributed with mean µ

m

. Note that, unless some special conditions hold on W , the distributions of K

n,z

and X

n

are not exchangeable. The telling feature of X

n

−law is that any linear non-negative combination of its components remains exponentially distributed.

When α > 0, considering the model K

n,z

now under P

α

, similar arguments would show that, with κ = tr(W

z_α

)

K

n,z

κ

→

d κ%∞

X

_n

where the law of X

_n

is now a multivariate-gamma distribution given by

E

e

^−hs,Xⁿⁱ

= 1 +

n

X

m=1

s

m

µ

m

!

−α

= (1 + hs, µ

_n

i)

^−α

where µ

_n

is now given in terms of the following b

_m

:

b

_m

= (z

_c,α

W · adj (I − z

_c,α

W ))

_m,m

, m = 1, .., n

with z

c,α

:= z

c

/α. The latter Laplace-Stieltjes transform of X

n

is well-defined because for all s > 0, all λ > 0, E e

^−λhs,Xⁿⁱ

, which is the Laplace-Stieltjes transform of the positive scalar random variable hs, X

n

i, is a Bernstein (completely monotone) function, of gamma type. The general shape of the moment generating function of a multivariate gamma random variable X

n

is determinantal such as: E e

^−hs,Xⁿⁱ

= |I + SM|

^−α

for some admissible matrix M . In our case, M takes the particular form: M = [µ

1

1, ..,µ

n

1] where 1 : = (1, .., 1)

⁰

. The spectrum of such matrices is {0, .. (n − 1) times.., 0; P

n

1

µ

_m

= 1} with non- negative real eigenvalues only; by Proposition 4.6 of Vere-Jones, [21], it gives a well-defined Laplace-Stieltjes transform.

3 Combinatorics related to MacMahon Master Theorem

In this Section, we shall develop some combinatorial aspects of the MacMahon master theorem which are useful to our purposes. We shall let N = {0, 1, 2, ..}, N

∗

= {1, 2, ..} .

We shall assume that W is an n ×n non-negative weight matrix, with |W | =

det (W ). We shall let PerW stand for the permanent of W. We recall that

k

n

= (k

1

, .., k

n

) ∈ N

ⁿ

, |k

n

| = k

1

+ .. + k

n

. u = (u

1

, .., u

n

) ∈ [0, 1]

ⁿ

and that

U =diag(u), I =Identity. Define W (k

n

) to be the |k

n

| × |k

n

| matrix obtained

(17)

by repeating (removing and deleting) each W

_m,m⁰

into a size k

_m

× k

_m⁰

block if k

m

and k

m⁰

both positive (otherwise). Assuming that z ∈ [0, 1/ρ (W )), we have

|I − zU W|

⁻¹

= X

kn∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

"

_n

Y

m=1

u

^k_m^m

#

_n

Y

m=1

(W u)

^k_m^m

= X

k_n∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

_m

! PerW (k

n

)

= X

kn∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈S(kn)

Y

m,lm

W

m,σ₁(m,l_m)

where, if σ (m, l

_m

) = (m

⁰

, l

⁰

), l

_m

= 1, .., k

_m

, m = 1, .., n, σ

₁

(m, l

_m

) = m

⁰

gives the type of the image of particle l

m

in the class m. In the formulae displayed above, the first identity is due to Mac-Mahon (see Comtet, [8], for a short proof).

It took some time to realize that the Taylor coefficients in u of the above expression can in fact be identified to the permanent of W (k

n

) which is the content of the subsequent expressions (see [21] and [22] for historical remarks and back- ground).

Remark. If W is diagonal, with positive diagonal entries (λ

m

, m = 1, .., n), one can check that PerW (k

n

) = Q

n

m=1

k

m

!λ

^k_m^m

so that, as required

|I − zU W |

⁻¹

= X

k_n∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

λ

^k_m^m

=

n

Y

m=1

(1 − zu

_m

λ

_m

)

⁻¹

. 3

Let α > 0 or α ∈ {−1, −2, ...}. Let cyc (σ) be the number of cycles in σ ∈ S (k

n

). Then

|I − zU W |

^−α

= X

kn∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

! Per

α

W (k

n

)

= X

k_n∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

_m

!

X

σ∈S(kn)

α

^cyc(σ)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

= X

k_n∈Nⁿ

(αz)

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈S(k_n)

1 α

|kn|−cyc(σ)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

. Thus,

I − z

α U W

−α

= X

kn∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈S(kn)

1 α

|kn|−cyc(σ)

Y

m,lm

W

m,σ₁(m,l_m)

(18)

and

I − z

α U W

−α

= 1

1 − 1 −

I −

_α^z

U W

α

with 1−

I − z

α U W

α

= X

k_n∈Nⁿ

(−z)

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈S(kn)

−1 α

^|kn|−cyc(σ)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

.

Further, with ‘tr’ standing for trace, the weight function σ → w

_α

(σ) = α

^cyc(σ)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

being multiplicative (equal to the product of the weights over the connected components of σ)

−α log I − z

α U W = X

k≥1

1 α

^k−1

z

^k

k tr

{U W }

^k

=

X

kn∈Nⁿ∗

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈Sc(k_n)

1 α

|kn|−1

Y

m,lm

W

m,σ₁(m,l_m)

where S

c

(k

n

) is the subset of connected permutations σ from S (k

n

), satisfying cyc (σ) = 1 and σ (n, k

n

) = (1, 1) . Thus

"

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

_m

!

#

n −α log I − z

α U W o

= X

σ∈Sc(kn)

1 α

|kn|−1

Y

m,l_m

W

_m,σ₁_(m,l_m₎

involves cyclic weight products Q

n m=1

Q

km

lm=1

W

_m,σ₁_(m,l_m₎

, since σ ∈ S

c

(k

n

) with σ

1

(n, k

n

) = 1. In particular,

"

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

#

{− log |I − zU W |} = X

σ∈S_c(k_n)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

= w

₁

(S

_c

(k

_n

)) .

If in particular α = −1, then

|I + zU W | = X

kn∈Nⁿ

z

^|kⁿ^|

n

Y

m=1

u

^k_m^m

k

m

!

X

σ∈S(k_n)

(−1)

^|kⁿ^|−cyc(σ)

Y

m,lm

W

_m,σ₁_(m,l_m₎

where

X

σ∈S(kn)

(−1)

^|kⁿ^|−cyc(σ)

Y

m,l_m

W

_m,σ₁_(m,l_m₎

= |W (k

_n

)|

(19)

which is null unless k

_n

∈ {0, 1}

ⁿ

. Thus, with |W ({m

₁

, .., m

_p

})| a principal minor of W

|I + zU W | = 1 +

n

X

p=1

z

^p

X

1≤m1<..<mp≤n p

Y

q=1

u

m_q

|W ({m

1

, .., m

p

})| .

Note that

1 − |I − zU W| =

n

X

p=1

(−1)

^p−1

z

^p

X

1≤m1<..<mp≤n p

Y

q=1

u

m_q

|W ({m

1

, .., m

p

})|

so that

|I − zU W|

⁻¹

= 1

1 − (1 − |I − zU W|) .

From these combinatorial developments, it is clear (from positivity of in- duced probabilities) that if W is definite non-negative, the occupancies K

_n,z

are well-defined under P

^α

for all α > 0 and infinitely divisible. If in addition W is definite non-negative, all principal minors of W are non-negative and so K

n,z

are well-defined under P

α

for all α ∈ {.., −2, −1}. For more on permanents, determinants, MacMahon master theorem and the like, see [19], [2], [7], [21], [22] and [5].

4 Doubly non-negative and infinitely divisible weight matrices. Spatially extended boxes.

Recall a weight matrix which is both non-negative and non-negative definite is doubly non-negative matrix. As underlined before, doubly non-negative weight matrices play a special role in our occupancy problems.

A probabilitic construction of doubly non-negative weight matrices.

Let us first give a systematic construction which generates doubly non- negative matrices. In the process, we shall obtain the spatially extended system on the real line discussed in the introduction.

Let (X

₁

, X

₂

) be a pair of iid random variables on R with a density, say φ (x).

Consider the random variable X = X

1

− X

2

. Its density, say f (x), exists and is given by f (x) = φ∗

^∨

φ (x), x ∈ R , where

∨

φ (x) := φ (−x) and ∗ stands for

convolution. Clearly f (x) = f (−x) and f is symmetric. Let Φ (iλ) = E e

^iλX¹

be the common characteristic function of both X

1

and X

2

. Then |Φ (iλ)|

²

is

the (real) Fourier transform of f (x) or the characteristic function of X. Clearly

g (λ) :=

_2π¹

|Φ (iλ)|

²

≥ 0 for almost all λ and since f is continuous and integrable,

(20)

by Bochner theorem, f is definite non-negative in that, for all integer n, all real numbers x := (x

m

, m = 1, .., n) and all z ∈ C

ⁿ

: z

⁰

[f (x

m

− x

m⁰

)] z ≥ 0. In this interpretation, the function g (λ) should be regarded as the integrable spectral density associated to the correlation function f . Here, [f (x

m

− x

m⁰

)] is the n×n square matrix whose (m, m

⁰

) entry is f (x

m

− x

m⁰

). Assuming W = [W

m,m⁰

] and W

m,m⁰

= f (x

m

− x

m⁰

), then the weight matrix is doubly non-negative. It should be emphasized that strictly speaking, W = W (x), now is a function of x. Note also that this construction yields a correlation function which is also a probability density function so that total mass of f is 1, which is purely arbitrary.

Let ψ (x) ≥ 0 be any non-negative function on R . Then, with W

m,m⁰

= ψ (x

_m

) f (x

_m

− x

_m⁰

) ψ (x

_m⁰

), W := [W

_m,m⁰

] is diagonally congruent to the latter and so is also doubly non-negative. Clearly, its entries are indeed non negative and definite positiveness is preserved under congruence. If ψ (x) = C > 0, this constant can be used to readjust the mass of f if needed.

Remarks.

(i) Let P be the permutation matrix which maps the indices of (x

m

, m = 1, .., n) into the ones of −∞ < x

1

≤ .. ≤ x

n

< ∞ where now the x

m

are ordered on the real line. By considering instead the novel congruent weight matrix P

⁰

W P , we can assume, without loss of generality, that W

m,m⁰

= ψ (x

m

) f (x

m

− x

m⁰

) ψ (x

m⁰

) where the x

m

are ordered. Note that in this construction, a particle is attached to box number m if and only if it stands at m−th position x

m

on the real line.

In this context, H

m,m⁰

:= − log W

m,m⁰

is now the energy required to move a particle from site x

m

to site x

m⁰

.

(ii) let w

x

be a (stationary) white noise (δ−correlated) process, indexed by x ∈ R . Let z

_x

:= (φ ∗ w)

_x

where φ is as above. Then z

_x

, x ∈ R , is a stationary process whose correlation function is Cov [z

x⁰

z

x⁰+x

] = f (x) = φ∗ φ

^∨

(x).

(iii) Assume both (X

1

, X

2

) have common law supported by (0, ∞) in the above construction. Then, it can be checked that f (x) = h (|x|) where h (z) = R

∞

0

φ (y + z) φ (y) dy, z > 0.

(iv) Assume both (X

1

, X

2

) have symmetric common law supported by R . Then f (x) = φ

^∗2

and E e

^iλX¹

is real. A given f belongs to this class if it is 2−divisible on R . Note that f (x) = h (|x|) where h (z) = R

R

φ (z − y) φ (y) dy, z > 0.

(v) Exploiting this Fourier isomorphism on integrable positive functions, reversing the role played by ‘space’ x and wave number λ, we conclude that f (x) =

_2π¹

|Φ (ix)|

²

is a correlation function whose associated spectral measure is g (λ) = φ∗ φ