J. Japan Statist. Soc.Vol. 31 No. 2 2001 193–205

POLYA URN MODELS UNDER GENERALREPLACEMENT SCHEMES

Kiyoshi Inoue* and Sigeo Aki*

In this paper, we consider a Polya urn model containing balls of m differentlabels under a general replacement scheme, which is characterized by an m×m ad-dition matrix of integers without constraints on the values of these m2 integers otherthan non-negativity. This urn model includes some important urn models treatedbefore. By a method based on the probability generating functions, we consider theexact joint distribution of the numbers of balls with particular labels which are drawnwithin n draws. As a special case, for m = 2, the univariate distribution, the prob-ability generating function and the expected value are derived exactly. We presentmethods for obtaining the probability generating functions and the expected valuesfor all n exactly, which are very simple and suitable for computation by computeralgebra systems. The results presented here develop a general workable frameworkfor the study of Polya urn models and attract our attention to the importance ofthe exact analysis. Our attempts are very useful for understanding non-classical urnmodels. Finally, numerical examples are also given in order to illustrate the feasibilityof our results.

Key words and phrases: Polya urn, replacement scheme, addition matrix, probabil-ity generating functions, double generating functions, expected value.

1. Introduction

Urn models have been among the most popular probabilistic schemes andhave received considerable attention in the literature (see Johnson et al. (1997),Feller (1968)). The Polya urn was originally applied to problems dealing withthe spread of a contagious disease (see Johnson and Kotz (1977), Marshall andOlkin (1993)).

We describe the Polya urn scheme briefly. From an urn containing α1 ballslabeled 1 and α2 balls labeled 2, a ball is drawn, its label is noted and the ballis returned to the urn along with additional balls depending on the label of theball drawn; If a ball labeled i (i = 1, 2) is drawn, aij balls labeled j (j = 1, 2)are added. This scheme is characterized by the following 2 × 2 addition matrixof integers,

(a11

a12

a21

a22

); whose rows are indexed by the label selected and whose

columns are indexed by the label of the ball added.Several Polya urn models have been studied by many authors in the various

addition matrices, which generate many fruitful results. The case of the classicalPolya urn model (a11 = a22, a12 = a21 = 0) was studied earlier and a detaileddiscussion can be found in Johnson and Kotz (1977). In the case of a11 = a22,a12 = a21 = 0, Aki and Hirano (1988) obtained the Polya distribution of order

Received March 14, 2001. Revised April 27, 2001. Accepted May 8, 2001.

*Department of Informatics and Mathematical Science, Graduate School of Engineering Science,

Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan.

194 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

k. In the case of aii = c, aij = 0 for i �= j (i, j = 0, 1, . . . ,m), Inoue andAki (2000) considered the waiting time problem for the first occurrence of apattern in the sequence obtained by an (m + 1) × (m + 1) Polya urn scheme.In the case of a11 = a22, a12 = a21, Friedman (1949) obtained the momentgenerating function of the total number of balls with a particular label remainingin the urn after n draws; Friedman’s urn can be used to model the growth ofleaves in recursive trees (see also Mahmoud and Smythe (1991)). In the caseof a11 + a12 = a21 + a22, Bagchi and Pal (1985) showed an interesting exampleof Polya urn scheme applied to data structures in computer. (Gouet (1989,1993) corrected some of the statements made by Bagchi and Pal (1985)). Ina p × p Polya urn scheme (constant row sums allowing negative entries on thediagonal, but having several constraints on the eigenvalue structure), Smythe(1996) considered a central limit theorem.

One interest has been focused on the exact distribution of the total numbersof balls with particular labels remaining in the urn after n draws, or the exactdistribution of the numbers of balls with particular labels which are drawn withinn draws from the urn. Their derivation involves a combinatorial method ofcounting paths representing a realization of the urn development.

For a long time, most investigations have been made under the special struc-ture of the constant addition matrix with constant row sums, which implies asteady linear growth of the urn size. The reason for the imposition of this con-straint is mathematical convenience; Urn schemes where the constraint is imposedare generally much simpler to analyze than those where it was not imposed.

Recently, Kotz et al. (2000) attempted to treat a Polya urn model containing2 different labels according to a general replacement scheme, and pointed out thatno constraint case is considerably more challenging even in 2 × 2 case. That is,the exact distribution of the number of balls with a particular label which aredrawn within n draws is rather convoluted and such an exact distribution israther unwieldy for large n for numerical computation.

Our purpose in the present paper is to develop a general workable frameworkfor the exact distribution theory for Polya urn models mentioned before and toemphasize the importance of the exact analysis. The approach is to solve a systemof equations of conditional probability generating functions (p.g.f.’s). Then, theprobability functions and moments are derived from an expansion of the solutionregardless of whether or not the constraint is imposed.

In this paper, a Polya urn model containing balls of m different labels andcharacterized by a general replacement scheme is considered, which include someimportant models treated before. We consider the exact joint distribution ofthe numbers of balls with particular labels which are drawn within n draws.As a special case, a univariate distribution is derived from a Polya urn modelcontaining balls of 2 different labels.

For the derivation of the main part of the results, we use the method basedon the conditional p.g.f.’s. This method was introduced by Ebneshahrashooband Sobel (1990), and was developed by Aki and Hirano (1993, 1999), Aki et al.(1996). The procedure is very simple and suitable for computation by computer

POLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES 195

algebra systems.Furthermore, we propose two methods for the Polya urn model. One is a

recurrence for obtaining the expected values for all n, which is derived fromthe system of equations of conditional p.g.f.’s. The other is a useful method forobtaining the p.g.f.’s. Here, a double variable generating function and a notion oftruncation parameter are introduced, where, for a sequence of p.g.f.’s {φn(tttttttt)}n≥0,we define the double variable generating function by Φ(tttttttt, z) =

∑∞n=0 φn(tttttttt)zn. The

p.g.f.’s are derived from the system of equations of the double variable generatingfunctions. The difference between the method of conditional p.g.f.’s and themethod based on the double variable generating function is that for a fixed n,the p.g.f. φn(tttttttt) is obtained by the former method, whereas, for a fixed truncationparameter u0, all the p.g.f.’s up to u0, φ0(tttttttt), φ1(tttttttt), . . . , φu0(tttttttt) are obtained bythe latter method. The procedures of two methods presented here are also verysimple and suitable for computation by computer algebra systems.

The rest of this paper is organized in the following ways. In Section 2, aPolya urn model containing balls of m different labels is introduced, which ischaracterized by the general replacement scheme. As a special case, a univariatedistribution is derived from a Polya urn model containing balls of 2 differentlabels. Section 3 gives two methods for the Polya urn models. One is a recurrencefor obtaining the expected values for all n. The other is a useful method forobtaining the p.g.f.’s. Here, double variable generating function and a notionof truncation parameter are introduced, which play an important role. Bothmethods are also very simple and suitable for computation by computer algebrasystems. In Section 4, numerical examples are given in order to illustrate thefeasibility of our main results.

2. The models

In this section, we consider a Polya urn model characterized by an m ×maddition matrix. As a special case, for m = 2, the univariate distribution, theprobability generating function and the expected value are derived exactly.

2.1. The Polya urn model containing m different labelsFrom an urn containing α1 balls labeled 1, α2 balls labeled 2, . . . , αm balls

labeled m, a ball is chosen at random, its label is noted and the ball is returnedto the urn along with additional balls according to the addition matrix of non-negative integers, A = (aij) i, j = 1, . . . ,m, whose rows are indexed by the labelof the ball chosen and whose columns are indexed by the label of the ball added.Always starting with the newly constituted urn, this experiment is continued ntimes. Let Z1, Z2, . . . , Zn be a sequence obtained by the above scheme, whichtake values in a finite set B = {1, 2, . . . ,m}. Let r be a positive integer suchthat 1 ≤ r ≤ 2m − 1 and let B1, B2, . . . , Br be subsets of B, where Bi �= ∅ andBi �= Bj for i �= j. Then, we define the numbers of balls whose labels belong to thesubsets Bi (i = 1, . . . , r) which are drawn within n draws by X(i)

n =∑n

j=1 IBi(Zj)(i = 1, . . . , r), where IBi(·) (i = 1, . . . , r) means the indicator function of thesubset Bi.

196 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

In the sequel, we will obtain the p.g.f. E[tX(1)n

1 tX(2)n

2 · · · tX(r)n

r ] of the jointdistribution of (X(1)

n , X(2)n , . . . , X

(r)n ). Hereafter, we denote the urn composi-

tion and the total of the balls in the urn by bbbbbbbb = (α1, α2, . . . , αm) and |bbbbbbbb| =α1 + α2 + · · · + αm, respectively. We denote the i-th row of the addition matrixA by aaaaaaaai = (ai1, ai2, . . . , aim). Needless to say, αi ≥ 0 (i = 1, . . . ,m) and |bbbbbbbb| �= 0are assumed throughout this paper.

Suppose that we have an urn composition bbbbbbbb = (α1, α2, . . . , αm) after � (� =0, 1, . . . , n) draws. Then, we denote by φn−(bbbbbbbb; tttttttt) the p.g.f. of the conditionaldistribution of the numbers of balls whose labels belong to the subsets Bi (i =1, . . . , r) which are drawn within (n− �) draws, where tttttttt = (t1, . . . , tr).

Theorem 2.1. From the definitions of φn−(bbbbbbbb; tttttttt) (� = 0, 1, . . . , n), we havethe following system of the equations;

φn(bbbbbbbb; tttttttt) =m∑i=1

αi|bbbbbbbb|tttttttt

IB(i)φn−1(bbbbbbbb+ aaaaaaaai; tttttttt),(2.1)

φn−(bbbbbbbb; tttttttt) =m∑i=1

αi|bbbbbbbb|tttttttt

IB(i)φn−−1(bbbbbbbb+ aaaaaaaai; tttttttt), � = 1, 2, . . . , n− 1,(2.2)

φ0(bbbbbbbb; tttttttt) = 1, where, tttttttt IB(i) = tIB1

(i)1 t

IB2(i)

2 · · · tIBr (i)r .(2.3)

Proof. It is easy to see that φ0(bbbbbbbb; tttttttt) = 1 by the definition of the p.g.f..Suppose that the urn composition is bbbbbbbb = (α1, α2, . . . , αm) after � (� = 0, 1, . . . , n−1) draws. Then, the p.g.f. of the conditional distribution of the numbers of ballswhose labels belong to the subsets Bj (j = 0, . . . , r) which are drawn within(n− �) draws is φn−(bbbbbbbb; tttttttt) (� = 0, 1, . . . , n− 1). We should consider the conditionof one-step ahead from every condition. Given the condition we observe the(� + 1)-th draw. For every i = 1, . . . ,m, the probability that we draw the balllabeled i is αi/|bbbbbbbb|. If we have the ball labeled i (i = 1, . . . ,m), then the p.g.f. of theconditional distribution of the numbers of balls whose labels belong to the subsetsBj (j = 0, . . . , r) which are drawn within (n − � − 1) draws is φn−−1(bbbbbbbb + aaaaaaaai; tttttttt)(� = 0, 1, . . . , n− 1). Therefore, we obtain the equations (2.1) and (2.2).

Example 2.1. Assume that B = {1, 2, 3, 4}, B1 = {2, 4}, B2 = {3, 4}, tttttttt =(t1, t2) and the addition matrix is equal to the 4×4 zero matrix. Suppose that wehave an urn composition bbbbbbbb = (α1, α2, α3, α4) after � (� = 0, 1, . . . , n) draws. Then,we denote by φn−(bbbbbbbb; tttttttt) the p.g.f. of the conditional distribution of the numbersof balls whose labels belong to the subsets B1, B2 which are drawn within (n− �)draws. Then, we have the following system of the equations;

φn−(bbbbbbbb; t1, t2) =(α1

|bbbbbbbb| +α2

|bbbbbbbb| t1 +α3

|bbbbbbbb| t2 +α4

|bbbbbbbb| t1t2)φn−−1(bbbbbbbb; t1, t2),(2.4)

� = 0, 1, . . . , n− 1,φ0(bbbbbbbb; t1, t2) = 1.(2.5)

POLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES 197

Under an initial urn composition bbbbbbbb0 = (α01, α02, α03, α04), we get

φn(bbbbbbbb0; t1, t2) =(α01

|bbbbbbbb0|+α02

|bbbbbbbb0|t1 +

α03

|bbbbbbbb0|t2 +

α04

|bbbbbbbb0|t1t2

)n

.(2.6)

In this example, if the labels 1, 2, 3, 4 are regarded as (0, 0), (1, 0), (0, 1),(1, 1), respectively, the equation (2.6) is the p.g.f. of joint distribution of thenumber of balls with the first label 1 and the number of balls with the secondlabel 1 which are drawn within n draws. The distribution is called the bivariatebinomial distribution (see Kocherlakota (1989), Marshall and Olkin (1985)).

2.2. The Polya urn model containing 2 different labelsAs a special case, for m = 2, we study the Polya urn model containing 2

different labels. Assume that B = {1, 2}, B1 = {2} and A = (aij) i, j = 1, 2.Let Yn =

∑ni=1 IB1(Zi). Suppose that we have an urn composition bbbbbbbb = (α1, α2)

after � (� = 0, 1, . . . , n) draws. Then, we denote by ψn−(bbbbbbbb; t1) the p.g.f. of theconditional distribution of the number of balls labeled 2 which are drawn within(n− �) draws. From Theorem 2.1, we have the following Corollary 2.1.

Corollary 2.1. From the definitions of ψn−(bbbbbbbb; t1) (� = 0, 1, . . . , n), wehave the following system of the equations;

ψn(bbbbbbbb; t1) =α1

|bbbbbbbb|ψn−1(bbbbbbbb+ aaaaaaaa1; t1) +α2

|bbbbbbbb| t1ψn−1(bbbbbbbb+ aaaaaaaa2; t1),(2.7)

ψn−(bbbbbbbb; t1) =α1

|bbbbbbbb|ψn−−1(bbbbbbbb+ aaaaaaaa1; t1) +α2

|bbbbbbbb| t1ψn−−1(bbbbbbbb+ aaaaaaaa2; t1),(2.8)

� = 1, 2, . . . , n− 1,ψ0(bbbbbbbb; t1) = 1.(2.9)

We will solve the system of the equations (2.7), (2.8) and (2.9) under aninitial urn composition bbbbbbbb0 = (α01, α02). First, we note that the above equation(2.7) can be written in matrix form as

ψn(bbbbbbbb0; t1) =α01

α01 + α02ψn−1(bbbbbbbb0 + aaaaaaaa1; t1) +

α02

α01 + α02t1ψn−1(bbbbbbbb0 + aaaaaaaa2; t1),

=(

α01

α01 + α02

α02

α01 + α02t1

) (ψn−1(bbbbbbbb0 + aaaaaaaa1; t1)ψn−1(bbbbbbbb0 + aaaaaaaa2; t1)

),

= C1(t1)ψψψψψψψψn−1(t1), (say).

Next, for � = 1, we write the equation (2.8) as

ψn−1(bbbbbbbb0 + aaaaaaaa1; t1) =α01 + a11

α01 + α02 + a11 + a12ψn−2(bbbbbbbb0 + 2aaaaaaaa1; t1)

+α02 + a12

α01 + α02 + a11 + a12t1ψn−2(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2; t1),

ψn−1(bbbbbbbb0 + aaaaaaaa2; t1) =α01 + a21

α01 + α02 + a21 + a22ψn−2(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2; t1)

+α02 + a22

α01 + α02 + a21 + a22t1ψn−2(bbbbbbbb0 + 2aaaaaaaa2; t1),

198 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

or, equivalently,(ψn−1(bbbbbbbb0 + aaaaaaaa1; t1)ψn−1(bbbbbbbb0 + aaaaaaaa2; t1)

)

=

α01 + a11

α01 + α02 + a11 + a12

α02 + a12

α01 + α02 + a11 + a12t1 0

0α01 + a21

α01 + α02 + a21 + a22

α02 + a22

α01 + α02 + a21 + a22t1

·

ψn−2(bbbbbbbb0 + 2aaaaaaaa1; t1)ψn−2(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2; t1)ψn−2(bbbbbbbb0 + 2aaaaaaaa2; t1)

.

We write ψψψψψψψψn−1(t1) = C2(t1)ψψψψψψψψn−2(t1). For non-negative integers �1, �2 such that�1 + �2 = �, let

ψψψψψψψψn−(t1) =

ψn−(bbbbbbbb0 + �aaaaaaaa1; t1)ψn−(bbbbbbbb0 + (�− 1)aaaaaaaa1 + aaaaaaaa2; t1)ψn−(bbbbbbbb0 + (�− 2)aaaaaaaa1 + 2aaaaaaaa2; t1)

...ψn−(bbbbbbbb0 + �1aaaaaaaa1 + �2aaaaaaaa2; t1)

...ψn−(bbbbbbbb0 + �aaaaaaaa2; t1)

.

Then, the system of the equations (2.7), (2.8) and (2.9) can be written in ma-trix form as ψψψψψψψψn−+1(t1) = C(t1)ψψψψψψψψn−(t1) (� = 1, . . . , n), and ψψψψψψψψ0(t1) = 1(n+1) =(1, 1, . . . , 1)′, where, 1(n+1) denotes the (n + 1) × 1 column vector whose com-ponents are all unity and C(t1) denotes the � × (� + 1) matrix whose (i, j)-thcomponent is given by,

cij(�; t1) =

α01 + (�− i)a11 + (i− 1)a21

α01 + α02 + (�− i)(a11 + a12) + (i− 1)(a21 + a22),

j = i, i = 1, . . . , �,α02 + (�− i)a12 + (i− 1)a22

α01 + α02 + (�− i)(a11 + a12) + (i− 1)(a21 + a22)t1,

j = i+ 1, i = 1, . . . , �,0,

otherwise.

(2.10)

Proposition 2.1. The probability generating function ψn(bbbbbbbb0; t1), the exactdistribution of Yn and its expected value are given by

ψn(bbbbbbbb0; t1) = C1(t1)C2(t1) · · ·Cn(t1)1(n+1) =n∏

i=1

Ci(t1)1(n+1),

POLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES 199

P (Yn = y) =∑

1≤n1<···<ny≤n

C1(0) · · · Cn1(0) · · · Cny(0) · · ·Cn(0)1(n+1),

E[Yn; bbbbbbbb0] =n∑

i=1

C1(1) · · · Ci(1) · · ·Cn(1)1(n+1), where,

Ck(t1) =dCk(t1)dt1

=(dcij(k; t1)dt1

).

In a similar way, under an initial urn composition bbbbbbbb0 = (α01, α02, . . . , α0m), wecan solve the system of the equations in Theorem 2.1 by virtue of their linearityand obtain the p.g.f.. However, we do not write it due to lack of space.

Remark 1. In this Polya urn model, Kotz et al. (2000) derived the exactdistribution of Yn by another approach, and derived the recurrence relation forthe expected value. They also reported that the expected value can be derivedfrom the recurrence relation in a case that the constraint is imposed, whereasthe expected value can not be derived from it in a case that the constraint is notimposed. Then, we present a useful recurrence for the expected values, as willbe shown later.

3. Methods for computation

In this section, we present two methods for the exact analysis, which arevery simple and suitable for computation by computer algebra systems. One isa recurrence for obtaining the expected values for all n. The other is a methodfor obtaining p.g.f.’s.

3.1. The recurrences for the expected values

Theorem 3.1. (The Polya urn model containing m different labels)The expected values of X(i)

n (i = 0, 1, . . . , r), E[X(i)n ; bbbbbbbb] say , satisfy the recur-

rences;

E[X(i)n ; bbbbbbbb] =

m∑j=1

αj|bbbbbbbb| (IBi(j) + E[X(i)

n−1; bbbbbbbb+ aaaaaaaaj ]), n ≥ 1, i = 1, . . . , r,(3.1)

E[X(i)0 ; bbbbbbbb] = 0, i = 1, . . . , r.(3.2)

Proof. It is easy to check the equation (3.2). The equation (3.1) is ob-tained by differentiating both sides of the equation (2.1) with respect to ti(i = 1, . . . , r) and then setting t1 = · · · = tr = 1. The proof is completed.

As a special case, for m = 2, we consider the Polya urn model containing2 different labels treated in Section 2.2. Then, from Theorem 3.1, we have thefollowing Corollary 3.1.

200 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

Corollary 3.1. (The Polya urn model containing 2 different labels)The expected value of Yn, E[Yn; bbbbbbbb] say , satisfies the recurrence;

E[Yn; bbbbbbbb] =α1

|bbbbbbbb|E[Yn−1; bbbbbbbb+ aaaaaaaa1] +α2

|bbbbbbbb|E[Yn−1; bbbbbbbb+ aaaaaaaa2] +α2

|bbbbbbbb| , n ≥ 1,(3.3)

E[Y0; bbbbbbbb] = 0.(3.4)

3.2. The double generating functionsUntil now, our results are derived from the p.g.f. directly. The most of this

section will be devoted to the double variable generating functions. First, we willbegin by considering the Polya urn model containing 2 different labels treated inSection 2.2. We define

Ψ(bbbbbbbb) =∞∑n=0

ψn(bbbbbbbb; t1)zn.

Then, the equations (2.7) and (2.9) in Corollary 2.1 lead to

Ψ(bbbbbbbb) = 1 +α1

|bbbbbbbb| zΨ(bbbbbbbb+ aaaaaaaa1) +α2

|bbbbbbbb| zt1Ψ(bbbbbbbb+ aaaaaaaa2).(3.5)

Given an initial urn composition bbbbbbbb0 = (α01, α02), we have

Ψ(bbbbbbbb0) = 1 +α01

|bbbbbbbb0|zΨ(bbbbbbbb0 + aaaaaaaa1) +

α02

|bbbbbbbb0|zt1Ψ(bbbbbbbb0 + aaaaaaaa2).(3.6)

We will show that the truncated generating function of Ψ(bbbbbbbb0), say Ψ(bbbbbbbb0), canbe obtained in a polynomial form of z up to the arbitrary order by using theequation (3.5) for the right-hand side of the equation (3.6) recursively. The ideaof truncation is also illustrated. From the equation (3.5), we have

Ψ(bbbbbbbb0 + aaaaaaaa1) = 1 +α01 + a11

|bbbbbbbb0 + aaaaaaaa1|zΨ(bbbbbbbb0 + 2aaaaaaaa1) +

α02 + a12

|bbbbbbbb0 + aaaaaaaa1|zt1Ψ(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2),(3.7)

Ψ(bbbbbbbb0 + aaaaaaaa2) = 1 +α01 + a21

|bbbbbbbb0 + aaaaaaaa2|zΨ(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2) +

α02 + a22

|bbbbbbbb0 + aaaaaaaa2|zt1Ψ(bbbbbbbb0 + 2aaaaaaaa2).(3.8)

Substituting (3.7) and (3.8) into the right-hand side of (3.6), we have

Ψ(bbbbbbbb0)(3.9)

= 1 +α01

|bbbbbbbb0|z

(1 +

α01 + a11

|bbbbbbbb0 + aaaaaaaa1|zΨ(bbbbbbbb0 + 2aaaaaaaa1) +

α02 + a12

|bbbbbbbb0 + aaaaaaaa1|zt1Ψ(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2)

)

+α02

|bbbbbbbb0|zt1

(1 +

α01 + a21

|bbbbbbbb0 + aaaaaaaa2|zΨ(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2) +

α02 + a22

|bbbbbbbb0 + aaaaaaaa2|zt1Ψ(bbbbbbbb0 + 2aaaaaaaa2)

).

Setting Ψ(bbbbbbbb0 + 2aaaaaaaa1) = Ψ(bbbbbbbb0 + aaaaaaaa1 + aaaaaaaa2) = Ψ(bbbbbbbb0 + 2aaaaaaaa2) = 0 in the right-hand sideof (3.9) (if we need Ψ(bbbbbbbb0) up to the first order), we obtain

Ψ(bbbbbbbb0) = 1 +(α01

|bbbbbbbb0|+α02

|bbbbbbbb0|t1

)z.

POLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES 201

Thus, the above substitution enables us to obtain Ψ(bbbbbbbb0) up to the first order.Hence, ψ0(bbbbbbbb0; t1) and ψ1(bbbbbbbb0; t1) are obtained. If we need Ψ(bbbbbbbb0) up to the secondorder, we should use the equation (3.5) again. Similarly, by substituting theequation (3.5) into the terms Ψ(bbbbbbbb0 +2aaaaaaaa1),Ψ(bbbbbbbb0 +aaaaaaaa1 +aaaaaaaa2),Ψ(bbbbbbbb0 +2aaaaaaaa2) in the right-hand side of (3.9), respectively, and then setting Ψ(bbbbbbbb0 +3aaaaaaaa1) = Ψ(bbbbbbbb0 +2aaaaaaaa1 +aaaaaaaa2) =Ψ(bbbbbbbb0 + aaaaaaaa1 + 2aaaaaaaa2) = Ψ(bbbbbbbb0 + 3aaaaaaaa2) = 0, we have Ψ(bbbbbbbb0) up to the second order.Therefore, ψ0(bbbbbbbb0; t1), ψ1(bbbbbbbb0; t1) and ψ2(bbbbbbbb0; t1) are obtained. Since we can repeatthe above substitution a sufficient number of times, we can obtain the truncatedgenerating function Ψ(bbbbbbbb0) up to the arbitrary order and obtain all the p.g.f.’sψn(bbbbbbbb0; t1) (n = 0, 1, . . .). A parameter of a non-negative integer u is introducedinto the generating function Ψ(bbbbbbbb), denoted by Ψ(bbbbbbbb;u). From the equations (3.5)and (3.6), we have

Ψ(bbbbbbbb0; 0) = 1 +α01

|bbbbbbbb0|zΨ(bbbbbbbb0 + aaaaaaaa1; 1) +

α02

|bbbbbbbb0|zt1Ψ(bbbbbbbb0 + aaaaaaaa2; 1),(3.10)

Ψ(bbbbbbbb;u) = 1 +α1

|bbbbbbbb| zΨ(bbbbbbbb+ aaaaaaaa1;u+ 1) +α2

|bbbbbbbb| zt1Ψ(bbbbbbbb+ aaaaaaaa2;u+ 1).(3.11)

By using the equation (3.11) for the right-hand side of the equation (3.10) recur-sively, we can obtain the truncated generating function of Ψ(bbbbbbbb0; 0), say Ψ(bbbbbbbb0; 0),in a polynomial form up to the arbitrary order. For the truncation parameterof a non-negative integer u0, the following Proposition 3.1 gives the method forobtaining the p.g.f.’s ψi(bbbbbbbb0; t1) (i = 0, . . . , u0).

Proposition 3.1. (The Polya urn model containing 2 different labels)For any non-negative integer u0, the following system of the equations leads

to the truncated generating function of Ψ(bbbbbbbb0; 0), Ψ(bbbbbbbb0; 0) say , which is in a poly-nomial form of z up to the u0-th order , so that ψi(bbbbbbbb0; t1) (i = 0, . . . , u0) areobtained.

Ψ(bbbbbbbb0; 0) = 1 +α01

|bbbbbbbb0|zΨ(bbbbbbbb0 + aaaaaaaa1; 1) +

α02

|bbbbbbbb0|zt1Ψ(bbbbbbbb0 + aaaaaaaa2; 1),(3.12)

Ψ(bbbbbbbb;u) = 1 +α1

|bbbbbbbb| zΨ(bbbbbbbb+ aaaaaaaa1;u+ 1) +α2

|bbbbbbbb| zt1Ψ(bbbbbbbb+ aaaaaaaa2;u+ 1),(3.13)

for 0 ≤ u ≤ u0,

Ψ(bbbbbbbb;u) = 0, for u > u0.(3.14)

Proof. We continue to substitute the equation (3.11) into the right-handside of the equation (3.10) until all the parameters in the generating functionsin the right-hand side of the equation (3.10) are equal to u0 + 1. Then, Ψ(bbbbbbbb0; 0)takes the following form;

Ψ(bbbbbbbb0; 0) =u0+1∑v=0

gv(t1)zu0+1Ψ(bbbbbbbb0 + vaaaaaaaa1 + (u0 + 1− v)aaaaaaaa2;u0 + 1) + fu0(z),(3.15)

where, gv(t1) is a polynomial of t1, and fu0(z) is a polynomial of z up to theu0-th order. Setting Ψ(bbbbbbbb0 + vaaaaaaaa1 + (u0 + 1 − v)aaaaaaaa2;u0 + 1) = 0 for all v such that

202 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

0 ≤ v ≤ u0 + 1 in the right-hand side of (3.15), we can obtain the truncatedgenerating function Ψ(bbbbbbbb0; 0) in a polynomial form up to the u0-th order. Noticethat the order of the vanished terms is greater than u0. The proof is completed.

It is easy to extend this result to the general case. Similarly, the parameterof the non-negative integer u is introduced into the generating function Φ(bbbbbbbb), sayΦ(bbbbbbbb;u), where, Φ(bbbbbbbb) = Φ(bbbbbbbb;u) =

∑∞n=0 φn(bbbbbbbb; tttttttt)zn. From the following equations,

Φ(bbbbbbbb0; 0) = 1 + zm∑i=1

α0i

|bbbbbbbb0|tttttttt IB(i)Φ(bbbbbbbb0 + aaaaaaaai; 1),(3.16)

Φ(bbbbbbbb;u) = 1 + zm∑i=1

αi|bbbbbbbb|tttttttt

IB(i)Φ(bbbbbbbb+ aaaaaaaai;u+ 1),(3.17)

we can obtain the truncated generating function of Φ(bbbbbbbb0; 0), say Φ(bbbbbbbb0; 0).

Theorem 3.2. (The Polya urn model containing m different labels)For any non-negative integer u0, the following system of the equations leads

to the truncated generating function of Φ(bbbbbbbb0; 0), Φ(bbbbbbbb0; 0) say , which is in a polyno-mial form of z up to the u0-th order , so that φi(bbbbbbbb0; tttttttt) (i = 0, . . . , u0) are obtained .

Φ(bbbbbbbb0; 0) = 1 + zm∑i=1

α0i

|bbbbbbbb0|tttttttt IB(i)Φ(bbbbbbbb0 + aaaaaaaai; 1),(3.18)

Φ(bbbbbbbb;u) = 1 + zm∑i=1

αi|bbbbbbbb|tttttttt

IB(i)Φ(bbbbbbbb+ aaaaaaaai;u+ 1), for 0 ≤ u ≤ u0,(3.19)

Φ(bbbbbbbb;u) = 0, for u > u0, where, tttttttt IB(i) = tIB1

(i)1 t

IB2(i)

2 · · · tIBr (i)r .(3.20)

4. Numerical examplesIn this section, we illustrate how to obtain the distributions and the expected

values by using computer algebra systems.

Example 4.1. The Polya urn model containing 4 different labelsAssume that bbbbbbbb0 = (1, 2, 1, 2), B = {1, 2, 3, 4}, B1 = {2, 4}, B2 = {3, 4} and

A =

1 0 1 1

0 1 0 1

1 0 1 0

1 1 0 1

. Let X(i)

n =∑n

j=1 IBi(Zj) (i = 1, 2). For n = 3, the p.g.f. is

φ3(bbbbbbbb0; t1, t2) =1

108+

733564

t1 +73

2376t2 +

2697920

t12 +

320935640

t1t2 +2697920

t22

+120t1

3 +336123760

t12t2 +

319135640

t1t22 +

180t2

3 +2691980

t13t2

+495135640

t12t2

2 +269

11880t1t2

3 +73594

t13t2

2 +73

2376t1

2t23

+127t1

3t23.

POLYA URN MODELS UNDER GENERAL REPLACEMENT SCHEMES 203

Figure 1. The exact joint probability function of (X(1)10 , X

(2)10 ) in the Example 4.1, given

bbbbbbbb0 = (1, 2, 1, 2) and the addition matrix A.

Table 1. The exact joint probability function of (X(1)3 , X

(2)3 ), given bbbbbbbb0 = (1, 2, 1, 2).

X(1)3 = 0 X

(1)3 = 1 X

(1)3 = 2 X

(1)3 = 3

X(2)3 = 0 0.009259 0.020482 0.033965 0.05

X(2)3 = 1 0.030724 0.090039 0.141456 0.135859

X(2)3 = 2 0.033965 0.089534 0.138917 0.122896

X(2)3 = 3 0.0125 0.022643 0.030724 0.037037

For n = 10, we give Fig. 1, which is the three-dimensional plot of the exact jointprobability function of (X(1)

10 , X(2)10 ), given bbbbbbbb0 = (1, 2, 1, 2) and the addition matrix

A.Marshall and Olkin (1990) discussed this model in case that the addition

matrix is the identity matrix. So far as we know, it was first proposed by Kaiserand Stefansky (1972).

Example 4.2. The Polya urn model containing 2 different labelsAssume that bbbbbbbb0 = (2, 3), B = {1, 2}, B1 = {2} and A =

(10

11

). Let Yn =∑n

j=1 IB1(Zj). For n = 10, the p.g.f. and the expected value are, respectively,

ψ10(bbbbbbbb0; t1) =191

+1252913153150

t1 +440455750450400

t12 +

52734593367567200

t13

+865985887346313467200

t14 +

9851047075028319296

t15 +

1956313731197218880

t16

+891357184651840

t17 +

16000323323

t18 +

10240676039

t19 +

1536676039

t110,

E[Y10; bbbbbbbb0] = ψ10(bbbbbbbb0; 1) =117504597558292529873145800

= 4.644683381.

We give Fig. 2, which is the two-dimensional plot of the exact expected values of

204 J. JAPAN STATIST. SOC. Vol.31 No.2 2001

Figure 2. The exact expected values of Yn, given three initial urn compositions

bbbbbbbb0 = (1, 1), (2, 3), (5, 1) and the addition matrix A in Example 4.2. Value I, II, III are, re-

spectively, the values given initial urn compositions bbbbbbbb0 = (1, 1), (2, 3), (5, 1).

Yn, given three initial urn compositions bbbbbbbb0 = (1, 1), (2, 3), (5, 1) and the additionmatrix A.

Remark 2. In this example, Kotz et al. (2000) suggested that the fixedvalues of the initial condition will be asymptotically negligible with regard to Ynfor large n and E[Yn] ∼ n/ lnn, as n → ∞. By calculating the exact expectedvalues of Yn given three initial conditions, we observe that their values dependon the initial conditions when n is small.

However, when n comes to 250, it seems that the exact expected values ofYn still heavily depend on their initial conditions. Therefore, we think that theexact analysis is important.

AcknowledgementsWe wish to thank the editor and the referees for careful reading of our paper

and helpful suggestions which led to improved results.

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