Exchangeable Occupancy Models and Discrete Processes with the Generalized Uniform Order Statistics Property

0.36 MB·English

by  Francesca Collet

Exchangeable Occupancy Models and Discrete Processes with
the Generalized Uniform Order Statistics Property
Francesca Collet and Fabrizio Leisen
Universidad Carlos III de Madrid
and
Fabio Spizzichino
1
1 Sapienza Universit`a di Roma
0
and
2
c Florentina Suter
e
Universitatea din Bucuresti
D

5 December 6, 2011

]
R
P Abstract
.
h This work focuses on Exchangeable Occupancy Models (EOM) and their relations with
t
a theUniformOrderStatisticsProperty(UOSP)forpointprocessesindiscretetime. Asour
m main purpose, we show how definitions and results presented in Shaked, Spizzichino and
[ Suter (2004) can be unified and generalized in the frame of occupancy models. We first
showsomegeneralfactsaboutEOM’s. ThenweintroduceaclassofEOM’s,calledM(a)-
1
models,thatisusedforgeneralizingthenotionofUniformOrderStatisticsProperty. For
v
7 processes with this property, we prove a general characterization result. Our interest is
6 also focused on properties of closure w.r.t. some natural transformations of EOM’s.
8
0 Keywords: Transformations of Occupancy Models, M(a)-Models, Closure Properties,
n,r
.
2 Random Sampling of Arrivals
1
1 AMS 2010 Classification: 62G30, 60G09, 60G55
1
:
v
i 1 Introduction
X
r
a Thesocalledoccupancy distributionsgiverise,aswellknown,toaclassofmultivariatemodels
useful in the description of randomized phenomena. The name “occupancy” comes from the
interpretation in terms of particles that are randomly distributed among several cells. In
particular, three classical examples, related to as many well-known physical systems, belong
tothisclass: Maxwell-Boltzmann,Bose-EinsteinandFermi-Diracmodel,seeFeller (1968). In
theMaxwell-Boltzmann(MB)statisticsthecapacityofeachcellisunlimited,andtheparticles
are distinguishable. In the Bose-Einstein (BE) statistics, the capacity of each cell is unlimited
but the particles are indistinguishable. In the Fermi Dirac (FD) statistics, the particles are
indistinguishable, butcellscanonlyholdamaximumofoneparticle. Allthesethreestatistics
1
assumethatthecellsaredistinguishable. Thesemodelsareattractiveformanyreasons. First
of all they have wide range of applications in Sciences, Engineering and also in Statistics, as
pointed out from Charalambides (2005, Chapters 4 and 5) and Gadrich and Ravid (2011).
Moreover Mahmoud (2009) provided an interpretation of occupancy distributions in terms
of Polya Urns, which are very flexible and applicable to problems arising in various areas;
e.g. Clinical Trials (see Crimaldi and Leisen (2008) for some references), Economics (see
Aruka (2011)) and Computer Science (see Shah, Kothari, Jayadeva and Chandra (2011)).
From a probabilistic and combinatoric point of view, the three fundamental models (MB,
BE and BS) have many interesting properties. Indeed they are, in particular, exchangeable
and this is a basic remark for our aims. This paper is in fact concentrated on the theme of
ExchangeableOccupancyModels(EOM)andtheirrelationswiththeUniformOrderStatistics
Property (UOSP) of counting processes in discrete time. As one main purpose of ours, we
show that some notions and results given in Shaked, Spizzichino and Suter (2004, 2008)
admit completely natural generalizations in the frame of EOM’s.
We start proving some general properties of the EOM’s. In particular, we show that the ex-
changeabilitypropertyismaintainedunderremarkabletypesoftransformationsofoccupancy
models. Then, we introduce the notion of M(a)-models, a relevant sub-class of EOM’s that
turns out to have an important role in our derivations. In the final part of the paper, we
study discrete-time counting processes satisfying the generalized UOSP; in other words, for
which the joint distribution of the jump amounts is distributed according to a M(a)-model,
conditionally on a fixed number of arrivals up to time t. In particular, for these processes,
several characterizations are proved.
More in detail, the outline of the paper is as follows. Our central results will be presented in
the Section 5, where we show natural extensions of the results proved in Shaked, Spizzichino
andSuter (2004,2008). Section2isdevotedtosomepreliminaryargumentsaboutoccupancy
models and Section 3 presents some specific aspects of the EOM’s. In Section 4 we define the
class of occupancy M(a)-models and study some related properties.
2 Occupancy models and order statistics of discrete variables
For fixed n = 2,3,... and r = 1,2,..., let A be the set defined by
n,r
 
n
 (cid:88) 
A := x ≡ (x ,...,x ) : x = 0,1,...,r and x = r .
n,r 1 n j j
 
j=1
As it is well-known (see e.g. Feller (1968)) the cardinality of A is
n,r
(cid:18) (cid:19)
n+r−1
|A | = .
n,r
n−1
2
Starting from the classical scheme of r particles that are distributed stochastically into n
cells, we consider the random vector X ≡ (X ,X ,...,X ) where, for j = 1,2,...,n, X is
1 2 n j
the{0,1,...,r}-valuedrandomvariablethatcountsthenumberofparticlesfalleninj-thcell.
X ,X ,...,X are called occupancy numbers and the joint distribution of (X ,X ,...,X ),
1 2 n 1 2 n
describing the probabilistic mechanism of assignment of the particles to the cells, is called
an occupancy model. Since the vector (X ,X ,...,X ) takes its values in the set A , an
1 2 n n,r
occupancy model is then a probability distribution on A .
n,r
Let now B denote the set
r,n
B := {u ≡ (u ,u ,...,u ) : u = 1,2,...,n and u ≤ u ≤ ··· ≤ u }
r,n 1 2 r i 1 2 r
and consider the mapping ϕ : B −→ A defined as
r,n n,r
ϕ(u) = (ϕ (u),ϕ (u),...,ϕ (u)),
1 2 n
with
r
(cid:88)
ϕ (u) = 1 , for j = 1,2,...,n.
j {ui=j}
i=1
This is a one-to-one correspondence and then, for the cardinality of B , we have
r,n
(cid:18) (cid:19)
n+r−1
|B | = |A | = .
r,n n,r
n−1
As to ψ = ϕ−1 : A −→ B , we can write
n,r r,n
ψ(x) = (ψ (x),ψ (x),...,ψ (x)),
1 2 r
with
 (cid:12) 
(cid:12) s
 (cid:12)(cid:88) 
ψi(x) = min s(cid:12) xj ≥ i , for i = 1,2,...,r.
(cid:12)
 
(cid:12)j=1
We can thus consider the random vector U ≡ (U ,...,U ) defined by
1 r
U = ψ(X). (1)
For (u ,...,u ) ∈ B , one then has
1 r r,n
P{U = u ,...,U = u } = P{X = ϕ (u),...,X = ϕ (u)}.
1 1 r r 1 1 n n
Let P(A ) and P(B ) respectively denote the family of probability distributions on A
n,r r,n n,r
and the family of probability distributions on B . In view of the above one-to-one corre-
r,n
3
spondence between A and B , we can consider the induced (one-to-one) correspondence
n,r r,n
between P(A ) and P(B ). More precisely, we consider the mappings Φ and Ψ = Φ−1
n,r r,n
defined by
Ψ(Q)(x) = Q[ψ(x)] and Φ(P)(u) = P[ϕ(u)],
where P ∈ P(A ) and Q ∈ P(B ).
n,r r,n
Remark 2.1. Note that P is the uniform distribution on A (i.e., the Bose-Einstein model)
n,r
if and only if Q = Φ(P) is the uniform distribution on B .
r,n
ConsidernowrexchangeablerandomvariablesY ,Y ,...,Y ,whichtakevaluesin{1,2,...,n}.
1 2 r
To these random variables we can associate a vector of occupancy numbers by introducing
the random variables X ,X ,...,X defined by
1 2 n
r
(cid:88)
X = 1 , for j = 1,2,...,n.
j {Y =j}
h
h=1
The set of the possible values taken by Y ≡ (Y ,...,Y ) is then D := {1,2,...,n}r.
1 r r,n
We will also write X = ϕ˜(Y) or
X = ϕ˜ (Y ,Y ,...,Y ), for j = 1,...,n (2)
j j 1 2 r
where ϕ˜: D −→ A with
r,n n,r
r
(cid:88)
ϕ˜ (y ,y ,...,y ) = 1 , for y ∈ D and j = 1,2,...,n. (3)
j 1 2 r {y =j} r,n
h
h=1
It can be immediately seen that the probability distribution of (Y ,Y ,...,Y ) is uniquely
1 2 r
determined by the probability distribution of (X ,X ,...,X ) and vice versa.
1 2 n
In fact, the following relationships hold
P{X = ϕ˜ (y),X = ϕ˜ (y),...,X = ϕ˜ (y)}
1 1 2 2 n n
P{Y = y ,Y = y ,...,Y = y } = (4)
1 1 2 2 r r (cid:0) r (cid:1)
ϕ˜1(y)ϕ˜2(y)···ϕ˜n(y)
and
P{X = x ,X = x ,...,X = x } =
1 1 2 2 n n
(cid:18) (cid:19)
r
= P{Y = ψ (x),Y = ψ (x),...,Y = ψ (x)} (5)
1 1 2 2 r r
x ···x
1 n
We note that ϕ˜ : D −→ A is different from the previous transformation ϕ, which is
r,n n,r
defined on B ⊂ D and is bijective. However, ϕ˜(y) = ϕ(y) if y ∈ B and, for any
r,n r,n r,n
y ∈ D , we have
r,n
ϕ˜(y) = ϕ˜(y ,...,y ) = ϕ(y ,...,y ),
(1) (r) (1) (r)
4
where (y ,...,y ) is the vector of the coordinates of y rearranged in increasing order.
(1) (r)
(cid:0) (cid:1)
We consider now the vector Y ≡ Y ,...,Y of the order statistics of the exchangeable
(·) (1) (r)
vector (Y ,Y ,...,Y ). The set of values taken by Y is B . To a probability distribution
1 2 r (·) r,n
of (Y ,Y ,...,Y ) it corresponds one and only one probability distribution of Y and, for
1 2 r (·)
u ∈ B , we can write
r,n
P{Y = u ,...,Y = u }
(1) 1 (r) r
P{Y = u ,...,Y = u } = .
1 1 r r (cid:0) r (cid:1)
ϕ1(u)···ϕn(u)
Furthermore,
(cid:0) (cid:1) (cid:0) (cid:1)
ϕ Y = ϕ˜ Y = ϕ˜(Y) = X,
(·) (·)
and thus
Y = ϕ−1(X) = ψ(X).
(·)
We can then summarize the arguments above as follows.
Proposition 2.1. Let X ,X ,...,X be random variables such that the support of their
1 2 n
joint distribution is A , for some r ∈ N. Moreover, let Y ,Y ,...,Y be exchangeable,
n,r 1 2 r
{1,2,...,n}-valued random variables such that (2) holds.
Then, the random variables U ,U ,...,U , defined by equation (1), are the order statistics of
1 2 r
the vector (Y ,...,Y ).
1 r
In what follows we present an interpretation of the marginal distributions of the variables
Y ,Y ,...,Y ,intermsoftheassociatedoccupancynumbersX ,X ,...,X . Onthispurpose,
1 2 r 1 2 n
we now consider a special transformation, that map one occupancy model into another. Two
other natural transformations, also intersting for our next developments, will be introduced
later on.
2.1 Transformation K
1
The first transformation we consider can be described as follows.
Consider r objects distributed on n cells according to a given occupancy model and let
X ,X ,...,X be the related occupancy numbers. We drop one of the particles from this
1 2 n
population randomly (i.e. so that each of the r particles has the same probability to be
dropped, independently of its cell). We denote by X(cid:48),X(cid:48),...,X(cid:48) the occupancy numbers
1 2 n
associated with the new population of r−1 particles.
This transformation will be denoted by K . Notice that K : P(A ) −→ P(A ).
1 1 n,r n,r−1
Lemma 2.1. Consider the event E = {X(cid:48) = x(cid:48),...,X(cid:48) = x(cid:48) }. Then,
x(cid:48) 1 1 n n
(cid:88)n x(cid:48) +1
P {E } = h P{X = x(cid:48),...,X = x(cid:48) +1,...,X = x(cid:48) }. (6)
x(cid:48) r 1 1 h h n n
h=1
5
Proof. The proof consists of simple manipulations. In fact, we can write
P {E } = P{X(cid:48) = x(cid:48),...,X(cid:48) = x(cid:48) }
x(cid:48) 1 1 n n
n
(cid:88)
= P{E ∩(the eliminated particle is from the cell h)}
x(cid:48)
h=1
n
= (cid:88)P{E ∩(cid:0)X = x(cid:48),...,X = x(cid:48) +1,...,X = x(cid:48) (cid:1)}
x(cid:48) 1 1 h h n n
h=1
n
(cid:88)
= P{E |X = x(cid:48),...,X = x(cid:48) +1,...,X = x(cid:48) }×
x(cid:48) 1 1 h h n n
h=1
×P{X = x(cid:48),...,X = x(cid:48) +1,...,X = x(cid:48) }
1 1 h h n n
(cid:88)n x(cid:48) +1
= h P{X = x(cid:48),...,X = x(cid:48) +1,...,X = x(cid:48) }.
r 1 1 h h n n
h=1
Let us now consider the exchangeable vectors (Y ,...,Y ) and (cid:0)Y(cid:48),...,Y(cid:48) (cid:1), corresponding
1 r 1 r−1
to the occupancy numbers X ,...,X and X(cid:48),...,X(cid:48), respectively.
1 n 1 n
Proposition 2.2. The joint distribution of (cid:0)Y(cid:48),...,Y(cid:48) (cid:1) coincides with the (r−1)-dimen-
1 r−1
sional marginal distribution of (Y ,...,Y ).
1 r
Proof. In view of (4) and (6), the joint distribution of (cid:0)Y(cid:48),...,Y(cid:48) (cid:1) is given by
1 r−1
P{Y(cid:48) = y(cid:48),Y(cid:48) = y(cid:48),...,Y(cid:48) = y(cid:48) } =
1 1 2 2 r−1 r−1
P{X(cid:48) = ϕ˜ (y(cid:48)),X(cid:48) = ϕ˜ (y(cid:48)),...,X(cid:48) = ϕ˜ (y(cid:48))}
= 1 1 2 2 n n
(cid:0) r−1 (cid:1)
ϕ˜1(y(cid:48))ϕ˜2(y(cid:48))···ϕ˜n(y(cid:48))
ϕ˜1(y(cid:48))ϕ˜2(y(cid:48))···ϕ˜n(y(cid:48)) (cid:88)n ϕ˜h(y(cid:48))+1
= ×
(r−1)! r
h=1
×P{X = ϕ˜ (y(cid:48)),...,X = ϕ˜ (y(cid:48))+1,...,X = ϕ˜ (y(cid:48))}.
1 1 h h n n
For brevity sake, let us set
ϕ˜(h)(y(cid:48)) := (cid:0)ϕ˜ (y(cid:48)),...,ϕ˜ (y(cid:48))+1,...,ϕ˜ (y(cid:48))(cid:1) ,
1 h n
6
so that we rewrite the previous equation as
P{Y(cid:48) = y(cid:48),Y(cid:48) = y(cid:48),...,Y(cid:48) = y(cid:48) } =
1 1 2 2 r−1 r−1
= ϕ˜1(y(cid:48))ϕ˜2(y(cid:48))···ϕ˜n(y(cid:48)) (cid:88)n ϕ˜h(y(cid:48))+1P{X = ϕ˜(h)(y(cid:48))}.
(r−1)! r
h=1
We can now use (5) and get
P{Y(cid:48) = y(cid:48),Y(cid:48) = y(cid:48),...,Y(cid:48) = y(cid:48) } =
1 1 2 2 r−1 r−1
ϕ˜1(y(cid:48))!ϕ˜2(y(cid:48))!···ϕ˜n(y(cid:48))! (cid:88)n ϕ˜h(y(cid:48))+1 r!
= ×
(r−1)! r ϕ˜ (y(cid:48))!···(ϕ˜ (y(cid:48))+1)!···ϕ˜ (y(cid:48))!
1 h n
h=1
(cid:110) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:111)
×P Y = ψ ϕ˜(h)(y(cid:48)) ,...,Y = ψ ϕ˜(h)(y(cid:48)) ,Y = ψ ϕ˜(h)(y(cid:48))
1 1 r−1 r−1 r r
n
(cid:88) (cid:110) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:111)
= P Y = ψ ϕ˜(h)(y(cid:48)) ,...,Y = ψ ϕ˜(h)(y(cid:48)) ,Y = ψ ϕ˜(h)(y(cid:48)) .
1 1 r−1 r−1 r r
h=1
Let us focus on the vector
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
ψ ϕ˜(h)(y(cid:48)) ,...,ψ ϕ˜(h)(y(cid:48)) ,ψ ϕ˜(h)(y(cid:48)) .
1 r−1 r
The first r − 1 components are defined by the composition (ψ ◦ ϕ˜)(y(cid:48)) and the result is a
(r−1)-tuple
(cid:16) (cid:17)
y(cid:48) ,...,y(cid:48) ∈ B
σ(1) σ(r−1) r−1,n
for every permutation σ of {1,2,...,r−1}. The last component ψ (cid:0)ϕ˜(h)(y(cid:48))(cid:1) is not deter-
r
mined by the previous composition. It is a “free” element indicating the cell in which the
additional r-th particle fell. Thus, it is a value randomly chosen in {1,...,n}.
Finally, by recalling the exchangeability of (Y ,...,Y ), we get the equality
1 r
n
(cid:88)
P{Y(cid:48) = y(cid:48),Y(cid:48) = y(cid:48),...,Y(cid:48) = y(cid:48) } = P{Y = y(cid:48),...,Y = y(cid:48) ,Y = h}.
1 1 2 2 r−1 r−1 1 1 r−1 r−1 r
h=1
and then the conclusion follows.
2.2 Transformation K
2
Here we consider the transformation K : P(A ) −→ P(A ) simply obtained by “eras-
2 n,r n−1,r
ing” one of the n cells. More precisely, we start once again from an occupancy model in
P(A ) and let X ,...,X denote occupancy numbers jointly distributed according to such
n,r 1 n
7
a model. We now consider the case where the n-th cell is eliminated and any of the X
n
particles that had fallen within it, is put at random, and independently, within the remaining
cells (see also details about the MB model later on).
By denoting X(cid:48)(cid:48),...,X(cid:48)(cid:48) the occupancy numbers obtained by this procedure, and for
1 n−1
(cid:0)x(cid:48)(cid:48),...,x(cid:48)(cid:48) (cid:1) ∈ A , we can write
1 n−1 n−1,r
P{X(cid:48)(cid:48) = x(cid:48)(cid:48),...,X(cid:48)(cid:48) = x(cid:48)(cid:48) } =
1 1 n−1 n−1
r (cid:0) x (cid:1)
= (cid:88) (cid:88) ξ1···ξn−1 P{X = x(cid:48)(cid:48)−ξ ,...,X = x(cid:48)(cid:48) −ξ ,X = x}, (7)
(n−1)x 1 1 1 n−1 n−1 n−1 n
x=0 ξ∈An−1,x
where we obviously mean P{X = x ,...,X = x } = 0, whenever some of the coordinates
1 1 n n
x ,...,x turns out to be smaller than 0.
1 n−1
2.3 Transformation K(N,r)
n,s
Here and in the rest of the paper we will use the notation S = (cid:80)N X . We consider the
N i=1 i
transformation that maps P(A ) onto P(A ) (with 1 ≤ n ≤ N − 1,1 ≤ s ≤ r) that
N,r n,s
is simply obtained by conditioning on a fixed value s for the partial sum S . For variables
n
X ,...,X distributed according to a given occupancy model in P(A ) we consider the
1 N N,r
model in A defined by
n,s
P{X = x ,...,X = x }
1 1 n n
P{X = x ,...,X = x |S = s} =
1 1 n n n
P{S = s}
n
for (x ,...,x ) ∈ A and with
1 n n,s
P{X = x ,...,X = x } =
1 1 n n
(cid:88)
= P{X = x ,...,X = x ,X = η ,...,X = η },
1 1 n n n+1 1 N N−n
η∈AN−n,r−s
(cid:88) (cid:88)
P{S = s} = P{X = x ,...,X = x ,X = η ,...,X = η }.
n 1 1 n n n+1 1 n N−n
x∈An,sη∈AN−n,r−s
3 Exchangeable Occupancy Models
Often relevant occupancy models are such that the random variables X ,...,X are ex-
1 n
changeable. The class of the Exchangeable Occupancy Models (EOM) is actually a wide and
interesting one. We then devote this Section to analyze several special aspects related with
such a condition. We notice, in particular, that the most well-known occupancy models, i.e.
Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac, are exchangeable. They are defined as
follows.
8
• Maxwell-Boltzmann:
1 r!
P{X = x ,X = x ,...,X = x } = ,
1 1 2 2 n n nr x !x !···x !
1 2 n
for x ≡ (x ,x ,...,x ) ∈ A .
1 2 n n,r
• Bose-Einstein:
1
P{X = x ,X = x ,...,X = x } =
1 1 2 2 n n (cid:0)n+r−1(cid:1)
n−1
for x ≡ (x ,x ,...,x ) ∈ A .
1 2 n n,r
• Fermi-Dirac:
1
P{X = x ,X = x ,...,X = x } =
1 1 2 2 n n (cid:0)n(cid:1)
r
for x ≡ (x1,x2,...,xn) ∈ A(cid:98)n,r, where A(cid:98)n,r := {x ∈ An,r : xj ∈ {0,1}}.
As mentioned above, we can easily see that any vector (X ,...,X ) distributed according to
1 n
one of these models is an exchangeable random vector.
In the next Section we will introduce a wide class of EOM’s that contains these fundamental
models. For an account about MB, BE, FD see Feller (1968) and Charalambides (2005). In
the following Propositions we state some general properties of EOM’s.
Proposition 3.1. If X ,X ,...,X are exchangeable, then the univariate marginals of Y ,
1 2 n 1
Y ,..., Y are uniform on {1,2,...,n}.
2 r
Proof. We prove the statement by showing that for any a,b ∈ {1,2,...,n}, a (cid:54)= b, we have
P{Y = a} = P{Y = b}. This would mean that P{Y = a} = 1 and hence the univariate
1 1 1 n
marginal distributions of (Y ,Y ,...,Y ) are uniform on {1,2,...,n}.
1 2 r
Let (y ,...,y ) be a (r−1)-tuple in D and consider a ∈ {1,2,...,n}. Then
2 r r−1,n
(cid:88)
P{Y = a} = P{Y = a,Y = y ,...,Y = y }.
1 1 2 2 r r
(y2,...,yr)∈Dr−1,n
9
By using (4) we obtain that
(cid:88) P{X1 = ϕ˜1(a,y2,...,yr),...,Xn = ϕ˜n(a,y2,...,yr)}
P{Y = a} =
1 (cid:0) r (cid:1)
(y2,...,yr)∈Dr−1,n ϕ˜1(a,y2,...,yr)ϕ˜2(a,y2,...,yr)···ϕ˜n(a,y2,...,yr)
(cid:88) ϕ˜1(a,y2,...,yr)!···ϕ˜n(a,y2,...,yr)!
= ×
r!
(y2,...,yr)∈Dr−1,n
×P{X = ϕ˜ (a,y ,...,y ),...,X = ϕ˜ (a,y ,...,y )}
1 1 2 r n n 2 r
Taking into account that for any j ∈ {1,2,...,n}, the function ϕ˜ is given by (3), we obtain
j
(cid:88) ϕ˜1(y2,...,yr)!···(ϕ˜a(y2,...,yr)+1)!···ϕ˜n(y2,...,yr)!
P{Y = a} = ×
1
r!
(y2,...,yr)∈Dr−1,n
×P{X = ϕ˜ (y ,...,y ),...,X = ϕ˜ (y ,...,y )+1,...,X = ϕ˜ (y ,...,y )}
1 1 2 r a a 2 r n n 2 r
(cid:88) ϕ˜1(y2,...,yr)!···ϕ˜n(y2,...,yr)!
= (ϕ˜ (y ,...,y )+1)
a 2 r
r!
(y2,...,yr)∈Dr−1,n
×P{X = ϕ˜ (y ,...,y ),...,X = ϕ˜ (y ,...,y )+1,...,X = ϕ˜ (y ,...,y )}.
1 1 2 r a a 2 r n n 2 r
(8)
Now, we take b ∈ {1,2,...,n}, b (cid:54)= a. Without loss of generality we can suppose that a < b.
Then, the univariate marginal of Y computed in b is
1
(cid:88) ϕ˜1(y2,...,yr)!···ϕ˜n(y2,...,yr)!
P{Y = b} = (ϕ˜ (y ,...,y )+1)
1 b 2 r
r!
(y2,...,yr)∈Dr−1,n
×P{X = ϕ˜ (y ,...,y ),...,X = ϕ˜ (y ,...,y )+1,...,X = ϕ˜ (y ,...,y )}.
1 1 2 r b b 2 r n n 2 r
(9)
The sums (8) and (9) have nr−1 terms and are computed using all possible (r − 1)-tuples
(y ,...,y ) ∈ D . Therefore for any j ∈ {1,2,...,n} and i ∈ {0,1,2,...,r−1} there are
2 r r−1,n
summation terms in (8) and in (9), such that ϕ˜ (y ,...,y ) = i. Let α,β ∈ {0,1,...,r−1}
j 2 r
two fixed values. Then there are summation terms in (8) such that ϕ˜ (y ,...,y ) = α and
a 2 r
ϕ˜ (y ,...,y ) = β. Also there are summation terms in (9) such that ϕ˜ (y ,...,y ) = β and
b 2 r a 2 r
ϕ˜ (y ,...,y ) = α.
b 2 r
Let us now consider (x ,...,x ,...,x ,...,x ) ∈ A such that x = α and x = β. Then,
1 a b n n,r−1 a b
there are (cid:0) r−1 (cid:1) summation terms in (8), all equal to
x1x2···xn
x !···x !
1 n
(α+1)P{X = x ,...,X = α+1,...,X = β,...,X = x }
1 1 a b n n
r!
10