Interacting generalized Friedman's urn systems

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by  Giacomo Aletti

INTERACTING GENERALIZED FRIEDMAN’S URN SYSTEMS
GIACOMO ALETTI AND ANDREAGHIGLIETTI
6
Abstract. We consider systems of interacting Generalized Friedman’s Urns (GFUs) having irre-
1
0 duciblemean replacement matrices. The interaction is modeled through theprobability to sample
2 the colors from each urn, that is defined as convex combination of the urn proportions in the sys-
tem. From the weights of these combinations we individuate subsystems of urns evolving with
c
differentbehaviors. Weprovideacompletedescriptionoftheasymptoticpropertiesofurnpropor-
e
D tions in each subsystem by establishing limiting proportions, convergence rates and Central Limit
Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation

8 techniques.

Keywords. Interacting systems; Urn models; Strong Consistency; Central Limit Theorems; Sto-
]
R chastic approximation.
P
. 2010 MSC classification: 60K35; 62L20; 60F05; 60F15.
h
t
a 1. Introduction
m
The stochastic evolution of systems composed by elements which interact among each other has
[
always been of great interest in several areas of application, e.g. in medicine a tumor growth is
2
the evolution of a system of interacting cells [35], in socio-economics and life sciences a collective
v
0 phenomenon reflects the result of the interactions among the individuals [27], in physics the con-
5 centration of certain molecules within cells varies over time due to interactions between different
5
cells [31]. In thelastdecade several models have beenproposedinwhich theelements of the system
1
0 are represented by urns containing balls of different colors, in which the urn proportions reflect the
. status of the elements, and the evolution of the system is established by studying the dynamics at
1
0 discrete times of this collection of dependent urn processes. The main reason of this popularity is
6 concerned with the urn dynamics, which is (i) suitable to describe random phenomena in different
1
scientific fields (see e.g. [21]), (ii) flexible to cover a wide range of possible asymptotic behaviors,
:
v (iii) intuitive and easy to be implemented in several fields of application.
i
X The dynamics of a single urn typically consists in a sequential repetition of a sampling phase,
r when a ball is sampled from the urn, and a replacement phase, when a certain quantity of balls is
a
replaced in the urn. The basic model is the Po´lya’s urn proposed in [16]: from an urn containing
balls of two colors, balls are sequentially sampled and then replaced in the urn with a new ball of
the same color. This updating scheme is then iterated generating a sequence of urn proportions
whose almost sure limit is random and Beta distributed. Starting from this simple model, several
interesting variations have been suggested by considering different distributions in the sampling
phase, e.g. [19, 20], or in the replacement phase, e.g. [3, 18, 30]. In a general K-colors urn model,
the color sampled at time n is usually represented by a vector X such that X = 1 when the
n i,n
sampled color is i 1,...,K , X = 0 otherwise; the quantities of balls replaced in the urn at
i,n
∈ { }
time n are typically defined by a matrix D such that D indicates the number of balls of color
n ki,n
k replaced in the urn when the color i is sampled. Considering D ;n 1 as an i.i.d. sequence, a
n
{ ≥ }
Date: December 9, 2016.
1
2 G.ALETTIANDA.GHIGLIETTI
crucial element to characterize the asymptotic behavior of the urn is the mean replacement matrix
H := E[D ], typically called generating matrix.
n
Theclass of urnmodels consideredinthis paperis commonly denoted byGeneralized Friedman’s
Urn (GFU). The GFU model was introduced in [18] and its extensions and their asymptotic
behavior have beenstudiedinseveral works, seee.g. [4,5,6,33]. TheGFUconsidered inthis paper
is characterized by a non-negative irreducible generating matrix H with average constant balance,
i.e. the columns of H sum up at the same constant, K H = c > 0 for any i 1,...,K ,
k=1 ki ∈ { }
which implies that its maximum eigenvalues λ (H) = c has multiplicity one. The irreducibility
max
P
of H distinguishes the GFU from the Randomly Reinforced Urn (RRU) model, which includes the
classical Po`lya’s Urn, whose replacement matrix is diagonal: when the color i is sampled, the GFU
replaces in the urn more colors following the distribution of the ith column of D while the RRU
n
only adds balls of colors i; hence, the probability to sample color i at next step is reinforced in
the RRU, while it may increase or decrease according to the current urn composition in the GFU.
The asymptotic behavior is in general very different: in a GFU the urn proportion converges to a
deterministic equilibrium identified by H (see e.g. [4, 5, 6, 33]), while in a RRU the limit is random
and its distribution depends on the initial composition (see e.g. [1, 2, 15]).
The model proposed in this paper is a collection of N 1 GFUs that interact among each other
≥
duringthesamplingphase: theprobabilitytosampleacolorifromanurnj isaconvexcombination
of the urn proportions of the entire system. Hence, a crucial role to describe the system dynamics
is played by the interacting matrix W made by the weights of those combinations. Since the
asymptotic properties of the single GFUs are typically determined by the correspondinggenerating
matrices Hj;1 j N and since the interaction among them is ruled by W, the system
{ ≤ ≤ }
dynamics has been studied by defining a new object Q that merges the information contained in
Hj;1 j N andW. From theanalysis of theeigen-structure of Q,weare abletoestablish the
{ ≤ ≤ }
convergence and the second-order asymptotic behavior of the urn proportions in the entire system.
Hence, this paper extends the theory on GFU models in the sense that, in the special case of no
interaction, i.e. W = I, the results presented for the system reduce to the well-known results for a
single GFU.
Severalinteracting urnmodelshavebeenproposedinthelastdecade, especially forRRUmodels.
An early work is represented by [29] that considered a collection of two-colors RRU in which the
replacements depend on the colors sampled in the rest of the system and hence the sequence
D ;n 1 is not i.i.d. Consequently, the interaction in [29] is modeled through the definition of
n
{ ≥ }
D , instead of X as in our model. A completly different updating rule has been used in the two-
n n
color urn model proposed in [26], in which sampling color 1 in the urn j increases the composition
of color 1 in the urn j, while sampling color 2 increases the composition of color 2 in the neighbor
urns i = j and the urn j comes back to the initial composition. Asymptotic properties for this
6
system have been obtained in [26] where there is no convergence of the urn proportions. Other
models in which the interaction enters in the replacement matrices are for instance [8, 10, 11].
Recently therehave beenmoreworks concerningurnsystemsin whichtheinteraction is modeled
through the sampling probabilities as in our model. They differ from this paper since all of them
consider RRUs and the interaction is only modeled as mean-field interaction tuned by a parameter
α (0,1),i.e. theurnsinteractamongeachotheronlythroughtheaveragecompositionintheentire
∈
system. As a consequence, their asymptotic results lead to the synchronization property in which
all the urn proportions of the system converge to the same random limit. In particular, in [24, 25]
the asymptotic behavior of the urn system has been studied for a model that defines the sampling
probabilities throughtheexponentialoftheurncompositions. In[12,13]thesamplingprobabilities
are defined directly using the urn compositions and, in addition, the synchronization property has
INTERACTING GENERALIZED FRIEDMAN’S URN SYSTEMS 3
been proved; moreover, different convergence rates and second-order asymptotic distributions for
the urn proportion have been established for different values of the tuning parameter α. Since we
consider GFU models the asymptotic results established in this paper are totally different from
those proved in [12, 13], e.g. our limiting proportions are not random and they do not depend on
the initial compositions.
It is also significant to highlight that this work allows a general structure for the urn interaction,
which reduces to the mean-field interaction only for a particular choice of the interacting matrix
W. Moreover, from the analysis of the structure of W we are able to individuate subsystems of
urnsevolving withdifferentbehaviors(seeSubsection2.4): (i)theleading systems, whosedynamics
are independent of the rest of the system and (ii) the following systems, whose dynamics “follow”
the evolution of other urns of the system; in the special case of irreducible interacting matrix,
which includes the mean-field interaction considered in [12, 13], there is a unique leading system
and no following systems. These two classes of systems have been studied separately (leaders
in Section 4 and followers in Section 5), in order to provide an exhaustive description of the
asymptotic behavior in any part of the system. In fact, since different systems may converge at
different rates, a unique central limit theorem would not be able to characterize the convergence
of any urn proportion. Hence, through a careful analysis on the eigen-structure of Q realized in
Subsection 5.2, we individuate the components of the urn processes in the system that actually
“lead” or influence the following systems, so that we can establish the right convergence rate and
a non-degenerate asymptotic distribution for any subsystem.
A pivotal technique in the proofs consists in revisiting the dynamics of the urn proportions of
the system in the stochastic approximation (SA) framework, as suggested for the composition of
a single GFU in [23]. To this end, the dynamics of the urn compositions of the same subsystems
have been reformulated into a recursive stochastic algorithm (see Section 3). Then, the dynamics
of the urn proportions have been properly modified to embed the processes of the urn proportions
into the whole suitable space RK (see Subsection 4.1 and 5.1).
The main results of the paper starts at Section 4. The first part of the paper is a necessary
formulation of the problem in its general form, together with all the assumptions and notations
that may appear tough at a first reading. We provide a guiding Example 3.1, that is recovered in
the Example 4.1 and Example 5.1, to help the reader to appreciate the main results, although not
in all their depth.
More precisely, the structure of the paper is the following. In Section 2 we present model
and main assumptions concerning the interacting GFU system. Specifically, in Subsection 2.1 we
describe how the composition of the colors in each urn of the system evolves at any time n 1.
≥
Then, in Subsection 2.2 the main assumptions required to establish the results of the paper are
presented. Subsection 2.3 contains a preliminary result. Subsection 2.4 is dedicated to analyze the
structure of the interacting matrix and hence to define the leading and the following subsystems
that compose the entire system.
Section 3 is concerned with the dynamics of the interacting GFU system expressed in the sto-
chastic approximation framework. In particular, in Subsection 3.1 we introduce the notation that
combines thecomposition oftheurnsinthesamesubsystem. Then,inSubsection3.2thedynamics
of the urn proportions in any subsystem is reformulated into a recursive stochastic algorithm.
Section4andSection5containthemainresultsofthepaper. Inparticular,Section4isconcerned
with the asymptotic behavior of the leading systems: the convergence of the urn proportions is
established in Subsection 4.2 and the corresponding CLT is presented in Subsection 4.3. Then,
Section 5 is focused on the asymptotic behavior of the following systems: in Subsection 5.3 we
4 G.ALETTIANDA.GHIGLIETTI
present the result on the convergence of the urn proportions, while in Subsection 5.4 we establish
the relative CLT.
Section 6contains abriefdiscussiononfurtherpossibleextensions oftheinteracting GFUmodel.
Theproofsof all the results presented in thepaper arecontained in Section 7. Finally, in Appendix
we report basic results of stochastic approximation that have been used in the main proofs.
2. Model Setting and main Assumptions
Considera collection of N 1 urnscontaining balls of K 1 different colors. At any time n 0
≥ j ≥ ≥
and for any urn j 1,...,N , let Y > 0 be the real number denoting the amount balls of color
∈ { } k,n
k 1,...,K , let Tj := K Yj be the total number of balls and let Zj := Yj /Tj be the
∈ { } n k=1 k,n k,n k,n n
proportion of color k.
P
2.1. Model. We now describe precisely how the system evolves at any time n 1. Denote by
≥
the σ-algebra generated by the urn compositions of the entire system up to time (n 1), i.e.
n−1
F −
j j
:= σ X ,Y , 1 j N, 1 k K, 1 t n 1 .
Fn−1 k,t k,t ≤ ≤ ≤ ≤ ≤ ≤ −
(cid:16) (cid:17)
The dynamics of the system is described by two main phases: sampling and replacement.
Sampling phase: for each urn j 1,...,N , a ball is virtually sampled and its color is repre-
∈ { }
j j
sented as follows: X = 1 indicates that the sampled ball is of color i, X = 0 otherwise. We
i,n i,n
denote by Z˜j the probability to sample a ball of color i in the urn j at time n, i.e.
i,n−1
Z˜j := E Xj .
i,n−1 i,n | Fn−1
h i
Given the sampling probabilities Z˜j ,1 j N,1 i K , the colors are sampled in-
{ i,n−1 ≤ ≤ ≤ ≤ }
dependently in all the urns of the system and hence, for any i 1,...,K , X1 ,...,XN are
∈ { } i,n i,n
independent conditionally on . We define the sampling probabilities as convex combinations
n−1
F
of theurnproportionsof thesystem. Formally, for any urnj 1,...,N we introducetheweights
∈ { }
w ;1 h N such that 0 w 1 and N w = 1. Thus, the probability to sample the
{ jh ≤ ≤ } ≤ jh ≤ h=1 jh
color i in the urn j is defined as follows
P
N
(1) Z˜j := w Zh .
i,n−1 jh i,n−1
h=1
X
Replacement phase: after that a ball of color i has been sampled from the urn j, we replace
j j
D balls of color k 1,...,K in the urn j. For any urn j we assume that D ;n 1 is a
ki,n ∈ { } { n ≥ }
j j j
sequence of i.i.d. non-negative random matrices, where D := [D ] . We will refer to D as
n ki,n ki n
replacement matrix andtoHj := E[Dj]asgenerating matrix. NoticethatHj aretime-independent
n
j
since D ;n 1 are identically distributed (see Subsection 6 for possible extensions). Moreover,
n
{ ≥ }
j
we assume that at any time n the replacement matrix for the urn j, i.e. D , is independent of the
n
j
sampled colors, i.e. X ;1 j N,1 i K , and independent of the replacement matrices of
{ i,n ≤ ≤ ≤ ≤ }
the other urns of the system, i.e. Dj0 with j = j.
n 0
6
INTERACTING GENERALIZED FRIEDMAN’S URN SYSTEMS 5
In conclusion, the composition of the color k 1,...,K in the urn j 1,...,N evolves at
∈ { } ∈ { }
time n 1 as follows:
≥
K
j j j j
(2) Y = Y + D X .
k,n k,n−1 ki,n i,n
i=1
X
2.2. Main Assumptions. We now present the main conditions required to establish the results
of the paper. The first assumption is concerned with bounds for the moments of the replacement
distributions. Specifically, we require the following condition:
(A1) there exists δ > 0 and a constant 0 < C < such that, for any j 1,...,N and any
δ
∞ ∈ { }
k,i 1,...,K , E[(Dj )2+δ] < C .
∈ { } ki,n δ
j
Note that C does not depend on n since D ;n 1 are identically distributed.
δ n
{ ≥ }
Thesecondassumptionistheaverageconstantbalanceoftheurnsinthesystemanditisimposed
by the following condition on the generating matrices H1,...,HN:
(A2) for any j 1,...,N and i 1,...,K , there exists a constant 0 < cj < such that
∈ { } ∈ { } ∞
K Hj = cj.
k=1 ki
Note that (A2) guarantees that the average number of balls replaced in any urn is constant, re-
P
gardless its composition. Assumption (A2) is essential to obtain the asymptotic configuration of
the system, i.e. the limiting urn proportions. The second-order asymptotic properties of the inter-
acting urn system, namely the rate of convergence and the limiting distributions, are obtained by
assuming a stricter assumption than (A2). This condition is expressed as follows:
(A’2) foranyj 1,...,N ,i 1,...,K ,P K Dj = cj = 1,i.e. eachurnisupdated
∈{ } ∈ { } k=1 ki,n
with a constant total amount of balls. (cid:16) (cid:17)
P
Naturally, (A’2) implies (A1) with C = (max cj )2+δ.
δ j
{ }
Noticethat,bydefiningYj = (cj)−1Yj andDj = (cj)−1Dj foralln 1,theurndynamics
k,n k,n ki,n ki,n ≥
in (2) can be expressed in the following equivalent form:
b b
K Yj Yj
j j j j j k,n−1 k,n−1 j
Y = Y + D X , Z = = = Z .
k,n k,n−1 ki,n· i,n k,n−1 K Yj K Yj k,n−1
Xi=1 kb=1 k,n−1 k=1 k,n−1
Thebrefore, frbom now on wbe will denote bybYj and DPj the normalizePd quantities Yj and Dj
k,n ki,n b k,n ki,n
and hence (A2) and (A’2) are replaced by the following conditions:
(A2) for any j 1,...,N and i 1,...,K , K Hj = 1. b b
∈{ } ∈ { } k=1 ki
(A’2) for any j 1,...,N and i 1,...,K , PP K Dj = 1 = 1.
∈{ } ∈ { } k=1 ki,n
In this case, (A’2) implies (A1) with Cδ = 1. (cid:16) P (cid:17)
Finally, we consider Generalized Friedman’s Urns (GFUs) with irreducible generating matrices,
as expressed in the following condition:
(A3) for any j 1,...,N , Hj is irreducible.
∈{ }
This assumption will guarantee deterministic asymptotic configurations for the urn proportions in
the system.
Remark 2.1. Extensions to non-homogeneous generating matrices H ;n 0 are possible, as
n
{ ≥ }
discussed in Section 6. In that case, assumption (A2) should be referred to the limiting matrix
Hj := a.s. lim Hj.
n→∞ n
−
6 G.ALETTIANDA.GHIGLIETTI
2.3. A preliminary result. Theassumptions(A2)and(A’2)ontheconstantbalanceareessential
to obtain the following result on the total number of balls in the urns of the system:
Theorem 2.1. Let Tj = K Yj be the total number of balls contained in the urn j at time
n k=1 k,n
n. Then, under assumptions (A1) and (A2), Tj n;n 1 is an L2 martingale and, for any
P n
{ − ≥ }
α < 1/2,
j
(3) nα Tn 1 a.s./L2 0.
n − −→
!
j j
Moreover, under assumption (A’2), T = T +n a.s. and hence (3) holds for any α < 1.
n 0
2.4. The interacting matrix. The interaction among the urns of the system is modeled through
the sampling probabilities Z˜j , that are defined in (1) as convex combinations of the urn pro-
i,n−1
portions of the system. Formally, we denote by W the N N matrix composed by the weights
×
w ,1 j,h N of such linear combinations and we refer to it as interacting matrix. We
jh
{ ≤ ≤ }
now consider a particular decomposition of W that individuates subsystems of urns evolving with
different behaviors. The same decomposition is typically applied to the transition matrices in the
context of discrete time-homogeneous Markov chains (see [28]) to characterize the state space. For
this reason, we first present the decomposition of W in this framework, and then we identify the
subsystems of urns as the communicating classes of the state space.
Consider a discrete time-homogeneous Markov chain with state space 1,...,N and transition
{ }
matrix W, i.e. the element w now represents the probability of a Markov chain to move from
jh
state j to state h in one step. It is well-known (see [28]) that the communication relationship
(i j if there exist m,n 0 such that [Wm] > 0 and [Wn] > 0) induces a partition of the
ij ji
∼ ≥
state space into communicating classes (some of them are necessarily closed and recurrent, with
possibly some transient classes). The maximum eigenvalue is λ = 1 and its multiplicity reflects
the number of recurrent classes. Accordingly, let us denote by the set of labels that identify the
L
communicating classes, n 1 the multiplicity of λ (W) = 1, and define the integers n 0
L max F
≥ ≥
and 1 rL1 < ... < rLnL < rF1 < ... < rFnF = N such that W can be decomposed as follows (see
≤
[28, Example 1.2.2] for the analogous upper triangular case):
WL 0
W := ,
WFL WF
(cid:18) (cid:19)
WL1 0 ... 0
(4) WL := 0 WL2 ... ...
... ... ... 0
0 0 ...WLnL !
WFL := WF..1.L1 ...... WF.1.L.nL , WF := WWFF2F11 W0F2 ...... 00 .
... ... ... ...
(cid:18)WFnFL1 ...WFnFLnL (cid:19) WFnFF1 WFnFF2 ...WFnF !
where:
(1) for any l , Wl is an sl sl irreduciblematrix, wherewe let sl := rl rl− and l− indicates
the eleme∈ntLin that pre×cedes l (by convention L− and F− L−);
L 1 ≡ ∅ 1 ≡ nL
(2) := , := L1,...,LnL and := F1,...,FnF aresets of labels that identify,
L F L F
L L ∪L L { } L { }
respectively, recurrentandtransientcommunicatingclassesinthestatespace( = when
F
L ∅
n = 0);
F
(3) foranyl ,thereisatleastanl ,l = l ,suchthatWl1l2 = 0;hence,λ (Wl)= 1
1 F 2 1 2 max
∈ L ∈ L 6 6
if l and λ (Wl) < 1 if l .
L max F
∈ L ∈ L
INTERACTING GENERALIZED FRIEDMAN’S URN SYSTEMS 7
Naturally, whennF = 0theelementsinWFL andWF donotexistandweconsiderrLnL = N. This
occurs when all the classes are closed and recurrent and hence the state space can be partitioned
into irreducible and disjoint subspaces. In the case of W irreducible, there is only one closed and
recurrent class and hence n = 1 and r1 = N.
L
In the framework of urn systems, W indicates the interacting matrix and hence the element
w represents how the color sampled from the urn j is influenced by the composition of the urn
jh
h. Hence, the probability of the Markov process to move from j to h in the state space can be
interpreted as the influence that h has on j in the urn system. As a consequence, recurrent classes
may be seen as subsystems of urns which are not influenced by the rest of the system; analogously,
transientclassesmayrepresentsubsystemsofurnswhichareinfluencedbyotherurnsofthesystem.
Hence, from an interacting matrix W expressed as in (4), we can decompose the urn system in:
(i) leading systems Sl,l , Sl := rl− + 1 < j rl , that evolve independently with
L
{ ∈ L } { ≤ }
respect to the rest of the system;
(ii) if n 0, following systems Sl,l , Sl := rl− +1 < j rl , that evolve depend-
F F
≥ { ∈ L } { ≤ }
ing on the proportions of the urns in the leaders SL1,...,SLnL and their upper followers
SF1,...,Sl−.
As we will see in the following sections, the asymptotic behaviors of the leading systems and the
following systems are quite different. For completeness of the paper, we will present the results for
both the types of systems, assuming that n 1.
F
≥
Remark 2.2. Extensions to random and time-dependent interacting matrices W ;n 0 are
n
{ ≥ }
possible, as discussed in Section 6. In that case, the structure presented in (4) is concerned with
the limiting matrix W := a.s. lim W .
n→∞ n
−
3. The interacting urn system in the stochastic approximation framework
Acrucialtechniquetocharacterizetheasymptoticbehavioroftheinteractingurnsystemconsists
in revisiting its dynamics into the stochastic approximation (SA) framework. A similar approach
has been adopted in [23] to establish the asymptotic behavior of a single urn. However, since here
we deal with systems of urns, we need to extend the dynamics (2) to jointly study the urns that
interact among each other. To this end, we first introduce in Subsection 3.1 a compact notation
thatcombines thecompositionoftheurnsinthesamesubsystemSl,l . Then,inSubsection3.2
∈ L
we embed each subsystem dynamics into the classical SA form: given a filtered probability space
(Ω, ,( ) ,P), we consider the following recursive procedure
n n≥0
A F
1 1
(5) n 1, θ = θ f(θ )+ (∆M +R ),
n n−1 n−1 n n
∀ ≥ − n n
where f : Rd Rd is a locally Lipschitz continuous function, θ an -measurable finite random
n n
→ F
vector and, for every n 1, ∆M is an -martingale increment and R is an -adapted
n n−1 n n
≥ F F
remainder term. In our framework, the process θ satisfying (5) will represent the proportions
n
of the colors of the urns in the same subsystem. In the next sections we apply the “ODE” and
the “SDE” methods for SA reported in Theorem A.1 and in Theorem A.2 (see Appendix), that
establish first and second-order asymptotic results for θ . Specifically, Theorem A.1 states that,
n
under suitable hypotheses on ∆M and R , the set Θ∞ of the limiting values of θ as n +
n n n
is a.s. a compact connected set, stable by the flow of ODE θ˙ = f(θ); moreover, if θ∗→ Θ∞∞
f
≡ − ∈
is a uniformly stable equilibrium on Θ∞ of ODE , then θ a.s. θ∗. In addition, under further
f n
−→
assumptions on ∆M and R , Theorem A.2 establishes the CLT for θ in which the convergence
n n n
8 G.ALETTIANDA.GHIGLIETTI
rate and the asymptotic distribution depend on the eigen-structure of the Jacobian matrix of f(θ)
evaluated at the equilibrium point θ∗.
3.1. Notation. The quantities related to the urn j 1,...,N at time n are random variables
∈ { }
denoted by:
(1) Yj = (Yj ,...,Yj )′ RK,
n 1,n K,n ∈ +
(2) Zj = (Zj ,...,Zj )′ K, where K indicates the K-simplex,
n 1,n K,n ∈ S S
(3) Z˜j = (Z˜j ,...,Z˜j )′ K,
n 1,n K,n ∈ S
(4) Xj = (Xj ,...,Xj )′ K 0,1 K,
n 1,n K,n ∈ S ∩{ }
while the corresponding terms of the system Sl, l , given by the sl urns labeled by rl− +
∈ L {
1,...,rl , are denoted by:
}
−
(1) Yl := (Yrl +1,...,Yrl)′ RslK,
n n − n ∈ +
(2) Zl := (Zrl +1,...,Zrl)′ slK, where slK indicates the Cartesian product of sl K-
n n − n ∈ S S
simpleces where Zrl +1,...,Zrl are defined,
n n
−
(3) Z˜l := (Z˜rl +1,...,Z˜rl)′ slK,
n n − n ∈ S
(4) Xl := (Xrl +1,...,Xrl)′ slK 0,1 slK,
n n− n ∈ S ∩{ }
(5) Tl := (Trl +11 ,...,Trl1 )′ RslK, where 1 indicates the K-vector of all ones.
n n K n K ∈ + K
The replacement matrix for the system Sl is defined by a non-negative block diagonal matrix
Dl of dimensions slK slK, where the sl blocks are the replacement matrices of the urns rl− +
n ×− {
1,...,rl in Sl, i.e. Drl +1,...,Drl. Analogously, the generating matrix for Sl is defined by a
} n n −
block diagonal matrix Hl of the same dimensions, where the sl blocks are Hrl +1,...,Hrl. The
n
interaction within the system Sl is modeled by the slK slK matrix Wl with values in [0,1]
×
defined as follows: starting from Wl in (4), each weight w is replaced by the corresponding
jh
diagonal matrix w I , where here I indicates the K K-identity matrix. Analogously, the
jh K K
×
interaction between a following system Sl1, l , and another system Sl2, l L ,...,l− ,
1 ∈ LF 2 ∈ { 1 1 }
is modeled by the matrix Wl1l2, obtained by replacing each weight w of Wl1l2 in (4) with the
jh
corresponding diagonal matrix w I . Finally, we will denote by I the identity matrix composed
jh K
by more matrices I .
K
Example 3.1. Consider a system of N = 2 urns containing balls of K = 2 colors. Let the
generating matrices H1, H2 and the interacting matrix W be as follows:
3/4 1/2 7/8 7/8 α 1 α
(6) H1 := , H2 := , W := − ,
1/4 1/2 1/8 1/8 1 β β
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
where α and β are given constants in [0,1].
In the case of no interaction α = β = 1, from the classical theory on single GFUs (see [4, 5, 6,
33]), we have that
(1) Z1 = (Z1 ,Z1 )′ converges a.s. to (2/3,1/3)′, i.e. the right eigenvector of H1 associated
n 1,n 2,n
to λ = 1; moreover the convergence rate is √n, since the second eigenvalue of H1 is 0.25.
(2) Z2 = (Z2 ,Z2 )′ converges a.s. to (1/2,1/2)′, i.e. the right eigenvector of H2 associated
n 1,n 2,n
to λ = 1; moreover the convergence rate is n0.25, since the second eigenvalue of H2 is 0.75.
When both α < 1 and β < 1, W is irreducible. Using the notation introduced in Subsec-
tion 2.4 and Subsection 3.1, in this case the two urns belong to the same leading subsystem
INTERACTING GENERALIZED FRIEDMAN’S URN SYSTEMS 9
SL1 = 1,2 . We have sL1 = 2, W = WL = WL1, and the joint quantities read as follows:
{ }
Z1 := (Z1 ,Z1 ,Z2 ,Z2 )′ 2,2,
n 1,n 2,n 1,n 2,n ∈S
3 1 0 0 α 0 1 α 0
4 2 −
1 1 0 0 0 α 0 1 α
H := 4 2 , W :=  − .
0 0 7 1 1 β 0 β 0
8 8 −
0 0 1 7  0 1 β 0 β 
 8 8  − 
   
We will discuss the asymptotic properties of this system in Example 4.1.
When α = 1 and β < 1, the first urn forms a leading system, while the second one exhibits the
behavior of a following system (see Example 5.1).
3.2. The system dynamics in the SA form. For any system Sl, l , the dynamics in (2)
∈ L
can be written, using the notation of Subsection 3.1, as follows:
(7) Yl = Yl + Dl Xl .
n n−1 n n
We now express (7) in the SA form (5), where the process θ ;n 1 is represented by the urn
n
{ ≥ }
proportions of the system Sl, i.e. Zl ;n 1 . Since Yl = diag(Tl )Zl for any n 1, from (7) we
{ n ≥ } n n n ≥
have
diag(Tl )Zl = diag(Tl )Zl + Dl Xl ,
n n n−1 n−1 n n
that is equivalent to
(8) diag(Tl )(Zl Zl ) = diag(Tl Tl )Zl + Dl Xl .
n n − n−1 − n − n−1 n−1 n n
Now, notice that, for any n 1,
≥
(1) E[diag(Tl Tl ) ] = I by Theorem 2.1;
n − n−1 |Fn−1
(2) E[Dl Xl ] = E[Dl ]E[Xl ] = HlZ˜l , since Dl and Xl are independent
n n|Fn−1 n|Fn−1 n|Fn−1 n−1 n n
conditionally on .
n−1
F
Hence, defining the martingale increment
(9) ∆Ml := Dl Xl HlZ˜l (diag(Tl Tl ) I)Zl ,
n n n− n−1 − n − n−1 − n−1
we can express (8) as follows:
(10) diag(Tl )(Zl Zl ) = Zl + HlZ˜l + ∆Ml .
n n − n−1 − n−1 n−1 n
Now, multiplying by diag(Tl )−1 and defining the remainder term
n
(11) Rl := n diag(Tl )−1 I Zl + HlZ˜l + ∆Ml ,
n · n − − n−1 n−1 n
(cid:16) (cid:17)(cid:16) (cid:17)
we can write (10) as follows:
1 1
(12) Zl Zl = (Zl HlZ˜l ) + ∆Ml + Rl .
n− n−1 −n n−1 − n−1 n n n
(cid:16) (cid:17)
The term (Zl HlZ˜l ) in (12) should represent the function f in (5) in the SA form. However,
n−1− n−1
although in a leader Sl, l , we have that Z˜l only depends on Zl , in a follower Sl,
∈ LL n−1 n−1
l , the term Z˜l is in general a function of the composition of all the urns of the system, i.e.
∈ LF n−1
ZL1 ,...,Zl . Hence, the dynamics of a leading system can be expressed as in (12), while the
n−1 n−1
dynamics of a following system needs to be incorporated with other systems to be fully described.
For this reason, the asymptotic behavior of these two types of systems are studied separately: the
leading systems in Section 4 and the following systems in Section 5.
10 G.ALETTIANDA.GHIGLIETTI
4. Leading Systems
Inthis section wepresentthe main asymptotic results concerning theleading systems Sl, l .
L
∈ L
We recall that these systems are characterized by irreducible interacting matrices Wl such that
λ (Wl) = 1 (see (4) in Subsection 2.4). For this reason, their dynamics is independent of the
max
rest of the system and hence, by using Z˜l = WlZl in (12), we have
n−1 n−1
1 1
Zl Zl = hl(Zl ) + ∆Ml + Rl ,
(13) n− n−1 − n n−1 n n n
hl(x) := (I Ql)x, Ql := (cid:16)HlWl (cid:17)
−
4.1. Extension of the urn dynamics to RslK. Since hl is defined on RslK, while the process
Zl ;n 0 takes values in the subset slK, then applying theorems based on the SA directly
{ n ≥ } S
to (13) may lead to improper results for the process Zl . To address this issue, we appropriately
n
modify the dynamics (13) by replacing hl with a suitable function fl := hl+mgl, where m > 0 is
m
an arbitrary constant and gl is a function defined in RslK that satisfies the following properties:
(i) the derivative gl is positive semi-definite and its kernel is Span (x y) : x,y slK :
D { − ∈ S }
hence, gl does not modify the eigen-structure of hl(x) on the subspace slK, where the
D S
process Zl is defined, while it changes the eigen-structure outside slK, where it can be
n S
arbitrary redefined;
(ii) gl(z) = 0 for any z slK: hence, since fl (z) = hl(z) for any z slK, the modified
∈ S m ∈ S
dynamics restricted to the subset slK represents the same dynamics as in (13).
S
Let us now provide an analytic expression of gl. First note that, since by definition of convex
combination we always have Wl1 = 1 , the left eigenvectors of Wl (possibly generalized) are
sl sl
such that U′1 = 1 and U′1 = 0 for all i = 1. Denote by Sp(A) the set of the eigenvalues of
1 sl i sl 6
a matrix A and note that, since by (A2) we always have 1′ Hj = 1′ , then Sp(Wl) Sp(Ql)
K K ⊂
and the sl left eigenvectors of Ql associated to any λ Sp(Wl) Sp(Ql), i 1,...,sl , present
i
∈ ⊂ ∈ { }
the following structure: U := (U 1 ,...,U 1 )′. As a consequence, for any z slK, we have
U′z = U′1 = 1 and U′zi= U′1i1 =K 0 for aislll iK 2,...,sl . Hence, denoting by∈VS and U the
1 1 sl i i sl ∈ { } 2 2
matrices whose columns are V ,...,V and U ,...,U , respectively, we define the function gl as
2 sl 2 sl
follows:
(14) gl(x) := V U′x 1 + V U′x,
1 1 − 2 2
and the dynamics of the process Zl in (13) c(cid:0)an be rep(cid:1)laced by the following:
n
1 1
Zl Zl = fl (Zl ) + ∆Ml + Rl ,
(15) n − n−1 − n m n−1 n n n
fl (x) := (I Ql)x + mV U(cid:16)′x 1 + m(cid:17)V U′x.
m − 1 1 − 2 2
4.2. First-order asymptotic results. We now prese(cid:0)nt the ma(cid:1)in convergence result concerning
the limiting proportion of the urns in the leading systems.
Theorem 4.1. Assume (A1), (A2) and (A3). Thus, for any leading system Sl, l , we have
L
∈ L
that
(16) Zl a.s. Zl := V ,
n −→ ∞ 1
where V indicates the right eigenvector associated to the simple eigenvalue λ = 1 of the matrix
1
Ql, with V = 1.
i 1i
P