Statistics and Probability Letters 83 (2013) 265–271

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Statistics and Probability Letters

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Phases in the two-color tenable zero-balanced Pólya processJoshua Sparks ∗, Hosam M. MahmoudDepartment of Statistics, The George Washington University, Washington, DC 20052, USA

a r t i c l e i n f o

Article history:Received 8 May 2012Received in revised form 23 August 2012Accepted 23 August 2012Available online 10 September 2012

MSC:60G2060F0535A99

Keywords:UrnPólya urnPólya processPhase transitionPartial differential equation

a b s t r a c t

The Pólya process is obtained by embedding the usual (discrete-time) Pólya urn schemein continuous time. We study the class of tenable Pólya processes of white and blue ballswith zero balance (no change in n, the total number of balls, over time). This class includesthe (continuous-time) Ehrenfest process and the (continuous-time) Coupon Collector’sprocess. We look at the composition of the urn at time tn (dependent on n). We identifya critical phase of tn at the edges of which phase transitions occur. In the subcriticalphase, under proper scaling the number of white balls is concentrated around a constant.In the critical phase, we have sufficient variability for an asymptotic normal distributionto be in effect. In this phase, the influence of the initial conditions is still somewhatpronounced. Beyond the critical phase, the urn is very well mixed with an asymptoticnormal distribution, in which all initial conditions wither away. The results are obtainedby an analytic approach utilizing partial differential equations.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Two-color urn schemes evolving in discrete time have drawn a lot of attention over the last century due to their potentialapplications inmany fields. These schemes include the Pólya–Eggenberger scheme (amodel for contagion) and the Ehrenfestscheme (a model for gas diffusion), among many other well known discrete-time urn schemes. For general background onurn models see Johnson and Kotz (1977) and Balakrishnan (1997) or Mahmoud (2008).

Associated with a Pólya urn scheme is a process obtained by embedding in continuous time. This embedded stochasticprocess has not been particularly named; we think it is befitting to call it the Pólya process. For clear readability, we adhereto calling growth in discrete time a ‘‘scheme’’, and growth in continuous time a ‘‘process’’. The process was introducedin Athreya and Karlin (1968) as a mathematical transform (Poissonization) to understand urns evolving in discrete-time.Athreya and Karlin (1968) reports difficulty in dePoissonization, to obtain the inverse transform.

A two-color Pólya urn is an urn containing balls of up to two colors, say white and blue, and following rules of evolutionover time. Balls are drawn at certain epochs in time. When a white ball is drawn, it is returned to the urn together withU white balls and V blue balls; when a blue ball is drawn, it is returned to the urn together with X white balls and Y blueballs (all four variables are random, with possible support including negative numbers). It is customary to represent thesedynamics by the replacement matrix

U VX Y

; (1)

∗ Corresponding author. Tel.: +1 606 316 7065.E-mail addresses: josparks@gwmail.gwu.edu (J. Sparks), hosam@gwu.edu (H.M. Mahmoud).

0167-7152/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2012.08.020

266 J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271

the rows of this matrix are indexed with the color of the ball picked at a stage (white and blue respectively from top tobottom), and the columns are indexed with the color of the balls added (white and blue respectively from left to right).

What distinguishes continuous-time Pólya processes from the usual discrete-time random urn schemes is the timing ofthe ball draws. In the discrete-time scheme, the balls are picked at equispaced time intervals (and the spacing is taken to bethe unit of time). In the continuous-time process, each ball is endowedwith a clock that rings in exponential time Exp(1) (anexponential random variable with mean 1). All the clocks are independent of each other and of any other random variablesrelated to the past. When a ball’s clock rings (a renewal point in the Pólya process or an epoch), we take it that the ball hasbeen picked from the urn, and execute the rules associatedwith its color instantaneously. All newballs added come endowedwith their own independent clocks. The collective process enjoys a memoryless property as it is induced by independentclocks with exponential ringing time. If a clock at an epoch is at a certain proportion of its ringing time, the associated balldoes not carry over that time to the next renewal. The clock’s remaining time to ring remains Exp(1), as if the process isreset to start afresh after each renewal. Thus, the Pólya process is Markovian, with no memory of the past, but the ratesmay change depending on how many balls are added. In general, the Markovian process may be an inhomogeneous jumpprocess, with the jumps occurring at the epochs.

A Pólya process is tenable if it is always possible to draw balls and apply the rules on every stochastic path. To illustratethe notion of tenability, suppose we have an urn of white and blue balls with replacement matrix

−2 22 −2

.

This matrix, together with an even initial number of white balls and an even initial number of blue balls, form a tenable urnscheme or process. By contrast, if we start with three white balls and four blue balls, the process gets stuck upon drawing awhite ball twice.

In this note, we deal with the class of two-color tenable zero-balanced Pólya processes—these have the replacementmatrix (1)with row sumequal to 0 on each row. In tenable zero-balanced Pólya processes the number of balls never changes,it is only the proportion of balls of each color that may. We shall call the number of balls in a zero-balanced Pólya process n.It is shown in Kholfi and Mahmoud (2012b) that the only possible replacement matrices for tenable zero-balanced urns arein the form

c

−Ber(p) Ber(p)Ber(r) −Ber(r)

, (2)

for a positive integer c that divides n, and Ber(s) is a Bernoulli randomvariablewith success probability s ∈ [0, 1].1 When thenumber of balls does not change, the Pólya process becomes a Markov chain with a well understood stationary distribution.However, in many applications it is even more important to understand the state of the urn after a short period of time.Take for instance the Ehernfest urn, a classic model for the exchange of gas particles or heat between two chambers. Inapplications like cooking and air-conditioning, one would be quite interested in the amount of gas or heat in each chamberafter a period of time of the order of hours, rather than be interested in what happens at infinite time.

The study of the distribution of balls in phases of drawing in discrete Pólya urn schemes has recently received attention(see Balaji et al. (2010),Mahmoud (2010), Smythe (2011) and Kholfi andMahmoud (2012a,b)). This has not yet been done forcontinuous-timemodels. The study of phases is also of interest in related areas, such as the evolution of randomhypergraphs(see Bender et al. (1997)) and occupancy problems; see Vatutin and Mikhailov (1982). The book Kolchin et al. (1978) givesa broad coverage of occupancy problems.

It is our goal in this note to study the phases in the two-color tenable zero-balanced Pólya process along the way to itsstationary distribution and find approximations for the probability distribution in the various phases.

2. Technical setup

Let W (t) = Wn(t), respectively B(t) = Bn(t), be the number of white, respectively blue, balls at time t in a tenablezero-balanced Pólya process on a total of nwhite and blue balls. Formally speaking, the two-color Pólya process is the two-component vector

W (t)B(t)

. Let

φn(t, u, v) := EeuW (t)+vB(t)

be the joint moment generating function of

W (t)B(t)

. Mahmoud (2008, p. 75) gives the partial differential equation (PDE)

∂φn(t, u, v)∂t

+1 − ηU,V (u, v)

∂φn(t, u, v)∂u

+1 − ηX,Y (u, v)

∂φn(t, u, v)∂v

= 0, (3)

1 In Kholfi and Mahmoud (2012b) the authors restrict r and p to be positive, to avoid dealing with reducible urn schemes, where one of the two colorsmay never appear, if the starting urn is void of it. Also, the authors develop their result for the urn scheme evolving in discrete time. The principle does notchange in continuous time, as tenability is a property of the replacement matrix and the initial conditions; in particular it is independent of the timing ofthe draws.

J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271 267

where ηU,V (u, v) is the jointmoment generating function ofU and V , and ηX,Y (u, v) is the jointmoment generating functionofX and Y . This functional equation has only been solved in a very limited number of cases, as for instance the case of forwardand backward diagonal processes (Balaji and Mahmoud, 2006) and the Ehrenfest process (Balaji et al., 2006). However, wewould like to note that the PDE is valid for general tenable urns and not just those which are zero-balanced. Earlier formsof this PDE appear in Balaji and Mahmoud (2006), where the functional equation is derived for a deterministic replacementmatrix.

In a tenable zero-balanced urn on a total of nwhite and blue balls, we must have

W (t)+ B(t) = n, (4)

at all times t ≥ 0. For this class of urns, the PDE (3) assumes a simpler form. In fact, in view of (4), it suffices to find onemarginal distribution at time t to completely determine the urn status at time t . So, we shall only study the distribution ofW (t). Let

ψn(t, u) = φn(t, u, 0) = E[euW (t)]

be the moment generating function ofW (t). We can now manipulate (3) to find a PDE for ψn(t, u). Observe that

∂

∂tφn(t, u, v)

v=0

=∂

∂tψn(t, u);

∂

∂uφn(t, u, v)

v=0

=∂

∂uψn(t, u);

∂

∂vφn(t, u, v)

v=0

= EB(t)euW (t)

= E

(n − W (t)) euW (t)

= nψn(t, u)−

∂

∂uψn(t, u).

A two-color tenable zero-balanced Pólya process has the replacement matrix (2) and so, upon setting v = 0, the functionalEq. (3) becomes

∂ψn(t, u)∂t

+1 − e−ucBer(p) ∂ψn(t, u)

∂u+

1 − eucBer(r)

nψn(t, u)−

∂

∂uψn(t, u)

= 0.

After rearrangement, this becomes

∂ψn(t, u)∂t

+p − r + recu − pe−cu ∂ψn(t, u)

∂u+ rn(1 − ecu)ψn(t, u) = 0.

The latter PDE has the solution

ψn(t, u) =e−nt(p+r)

(p + r)nc

rect(p+r)+cu

+ pect(p+r)− p + pecu

rect(p+r)+cu + pect(p+r) + r − recu

W (0)c

rect(p+r)+cu+ pect(p+r)

+ r − recu n

c . (5)

To facilitate subsequent asymptotic analysis, we write this as

ψn(t, u) =e−nt(p+r)

(p + r)nc

rect(p+r)+cu

+ pect(p+r) nc A1A2

=

recu + pp + r

nc

A1A2, (6)

with

A1 := A1(n, t, u) =

1 +

(p + r)(ecu − 1)rect(p+r)+cu + pect(p+r) + r − recu

W (0)c

, (7)

and

A2 := A2(n, t, u) =

1 +

r − recu

rect(p+r)+cu + pect(p+r)

nc

. (8)

268 J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271

3. Scope of main result and organization

We study the Pólya process governed by the replacement matrix (2). The case p = r = 0 is trivial, as the urn neverchanges. So, we shall assume that p + r > 0, assuring that at least one of the two probabilities is positive. As mentioned,for tenable zero-balanced Pólya processes, Wn(t) is a Markov chain. For n fixed and t approaching infinity, it is straightfor-ward to show that Wn(t) has the binomial stationary distribution Bin( nc ,

rp+r ), which counts the number of successes in n

cindependent trials with rate of success r

p+r .Our attention is on the case with very large n, which arises naturally in many applications. For example, according to

Avogadro’s number, amole of gas contains about 6.022×1023 particles. So, in a real application of gas diffusion between twochambers, we are dealing with billions of gas particles. The assumption that n grows to ∞ gives a realistic approximation.For large n, the asymptotic behavior ofψn(t, u) depends on how t relates to n. From this point on, we let t = tn be a functionof n. It will turn out that we have three phases:(i) The subcritical, when tn → ∞, but tn = o(ln n), or when tn ∼ K ln n, for some K < 1

2c(p+r) ;(ii) The critical, when tn ∼

12c(p+r) ln(bnn), for some positive bn that is bounded away from 0 and infinity, i.e.M1 ≤ bn ≤ M2,

for some positiveM1 and M2.(iii) The supercritical, when tn ∼ K ln n, for some K > 1

2c(p+r) , or when ln n = o(tn). One will notice that this phase has theform of the critical phase, except in this phase bn → ∞.

For a given n, the chosen phases conform with the forward sense of time. We should note, however, that these are notthe only time growths that can be studied. One can study, for example, the evolution across various values of n of thePólya process for the sequence bn = {1, 1

2 , 2,13 , 3,

14 , 4, . . .}, such that lim infn→∞ bn = 0 and lim supn→∞ bn = ∞, which

does not fall into any of the three phases above. However, certain subsequences can be approached using a combination ofthese three phases. We do not cover these or other types of growths in this work, and recommend such research for theinterested reader.

We shall find the exact and asymptotic mean and variance in Section 4; they guide us to capture the structure of thephases. The main result of the paper is the following.

Theorem 3.1. Suppose we have a tenable zero-balanced Pólya process on n white and blue balls in total, governed by (2) andstarting with W (0) = αnn + o(n) (with 0 ≤ αn ≤ 1) white balls. Let W (t) be the number of white balls at time t. Then(a) In the subcritical phase, when tn = o(ln n), or tn ∼ K ln n, for K < 1

2c(p+r) , we have

W (tn)−r

p+r n

ne−c(p+r)tn− αn

P−→ −

rp + r

.

(b) Let bn be a positive function of nwith lim infn→∞ bn > 0. In the critical (bn bounded from above) and supercritical (bn → ∞)phases, when tn =

12c(p+r) ln(bnn), for r > 0 we have

W (tn)−r

p+r n −

αn −

rp+r

nbn

√n

D−→ N

0,

prc(p + r)2

,

and if r = 0 we have

W (tn)− αn

nbn

√n

P−→ 0.

The proof of this result appears in Section 5. We conclude the paper in Section 6 with some remarks and interpretations.

4. Exact moments

From the exact expression (5), we can find moments in exact form. The kth moment is obtained by taking the kthderivative with respect to u and setting u to 0. For k = 1, the mean is

E [Wn(t)] =r

p + rn +

Wn(0)−

rnr + p

e−ct(p+r)

which is valid for all t ≥ 0. Note that asymptotically, as n → ∞, the mean function experiences phase transitions:

E [W (tn)] =

r

p + r+

αn −

rp + r

e−c(p+r)tn

n + o

ne−c(p+r)tn

, in the subcritical phase;

rp + r

n −

αn −

rp + r

nbn

+ o

nbn

, in the critical phase;

rp + r

n + o(√n), in the supercritical phase.

J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271 269

By taking the second derivative of (5) with respect to u and setting u to 0, we get the second moment, and the variancefollows:

Var [Wn(t)] =c

(p + r)2prn +

(p2 − r2)Wn(0)+ rn(r − p)

e−ct(p+r)

−(p2 − r2)Wn(0)+ r2n

e−2ct(p+r) ,

which is valid for all t ≥ 0. Note that, like the mean, as n → ∞, the variance asymptotically has phase transitions:

Var [W (tn)] =

cpr

(p + r)2n + o(n e−c(p+r)tn), in the subcritical phase;

cpr(p + r)2

n + O(√n), in the critical and supercritical phases.

5. Proof of the main result

We take up the proofs in this section, starting with the subcritical phase in Section 5.1. The proofs for the critical andsupercritical phases are combined in Section 5.2.

5.1. The subcritical phase

In this phase tn = o(ln n), or tn ∼ K ln n, for some positive K < 1(2c(r+p)) . Let sn = ne−c(p+r)tn . In the solution (6), the part

A1 (cf. (7)) gives

A1

n, t,

usn

=

1 +

(p + r)e

cusn − 1

ec(p+r)tn

re

cusn + p

− r

e

cusn − 1

W0(n)c

=

1 +

(p + r)

cusn

+ O

1s2n

ec(p+r)tn

p + r + O

1sn

+ O

1sn

αnc n+o(n)

∼ exp

(p + r)

cusn

+ O

1s2n

αnc n + o(n)

e−c(p+r)tn

p + r + O

1sn

∼ e(αnu+o(1)).

Similarly, for the part A2 (cf. (8)) we have

A2

n, t,

usn

=

1 +r − re

cusn

ec(p+r)tnre

cusn + p

n

c

=

1 −

cusnr + O

1s2n

ec(p+r)tn

p + r + O

1sn

nc

∼ e−r

p+r u.

Therefore, as n → ∞, we have

ψn

tn,

usn

∼

p

p + r+

rp + r

1 +

cusn

+ O

1s2n

nc

eαn−

rp+r

u

∼ ern

(p+r)snue

αn−

rp+r

u.

And so,

EeW (tn)− r

p+r n

usn

∼ e

αn−

rp+r

u.

It follows thatW (tn)−

rp+r n

sn− αn

P−→ −

rp + r

. �

270 J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271

5.2. The critical phase and beyond

In the critical phase

tn =1

2c(p + r)ln(bnn),

for some positive bn that is bounded away from 0 and ∞, that is, there are two positive numbers M1 and M2, such thatM1 ≤ bn ≤ M2, for all n ≥ 1. In the supercritical phase tn ∼ K ln(n), for K > 1

(2c(p+r)) , or tn is superlogarithmic all together,with ln n = o(tn). With either of the two supercritical forms, tn can be written as 1

2c(p+r) ln(bnn), with bn growing to infinity.In the solution (6), the part A1 (cf. (7)) gives

A1

n, t,

u√n

=

1 +

(p + r)e

cu√n − 1

e

12 ln(bnn)

re

cu√n + p

− r

e

cu√n − 1

W0(n)

∼

1 +

(p + r)

cu√n + O

1n

e

12 ln(bnn)

re

cu√n + p

+ O

1

√n

αnc n+o(n)

∼ eαnu√bn .

Similarly, for the part A2 (cf. (8)) we have

A2

n, t,

u√n

=

1 +r − re

cu√n

e12 ln(bnn)

re

cu√n + p

n

c

∼ e−

r(p+r)

√bn

u.

Therefore, as n → ∞, we have

ψn

tn,

u√n

∼

p

p + r+

rp + r

ecu√n

nc

e1√bn

αn−

rp+r

u.

In the right-hand side of the latter asymptotic relation, the parenthesized expression raised to the power nc is the generating

function of c × Bin( nc ,r

p+r ), evaluated at u√n . According to the standard approximation of the binomial distribution, we can

write

ψn

tn,

u√n

e−

r

p+r

√n ue

−1√bn

αn−

rp+r

u

∼ e−

r

p+r

√n u

p

p + r+

rp + r

ecu√n

nc

→ eprcu2

2(p+r)2 .

The right-hand side is the moment generating function of the normal random variable N (0, prc(p+r)2

), with mean 0 andvariance prc

(p+r)2. In other words, in the critical and supercritical phases we have

W (tn)−r

p+r n −

αn −

rp+r

nbn

√n

D−→ N

0,

prc(p + r)2

.

Furthermore, if r = 0 (and p > 0), as in the case of coupon collection, the limiting normal distribution degenerates into amass probability at 0, and we can thus upgrade the convergence to be in probability at 0. �

Observe that the starting condition (the initial proportion of white balls) has a somewhat pronounced influence in theasymptotic distribution in the critical phase, but not in the supercritical phase, where this influence is attenuated as we getdeeper in that phase till it eventually dies out.

6. Concluding remarks

To put the result in perspective, let us see how it applies to a specific Pólya urn. Consider the replacement matrix−1 11 −1

of the Ehrenfest urn. Here p = r = 1, and c = 1. Suppose also that W0(n) = ⌊

23n + 6

√n + 3 ln n⌋ ∼

23n,

so that αn =23 . After time tn = ln ln n, according to Theorem 3.1(a), we have

W (ln ln n)−12n

n/ ln2 nP

−→16.

J. Sparks, H.M. Mahmoud / Statistics and Probability Letters 83 (2013) 265–271 271

This is a law of sharp concentration. However, after a longer period of time reaching the critical phase, more variabilityappears, and we get a central limit theorem.

For instance, the time tn =14 ln n + 5 + 3 cos n =

14 ln(ne20+12 cos n) falls inside the critical phase, with bn = e20+12 cos n.

According to Theorem 3.1(b), we have

W 14 ln n + 5 + 3 cos n

−

12 n −

16

n

e20+12 cos n√n

D−→ N

0,

14

.

As the Pólya process carries on for a very long period, such as a superlogarithmic amount of time, it enters in the supercriticalphase. For example, suppose tn =

611n

2. We can write this amount of time in the form 14 ln(n × ( 1n e

2411 n

2)), i.e., bn =

1n e

2411 n

2,

which grows to infinity, as n → ∞. According to Theorem 3.1(b), we get the central limit result

W 611n

2−

12 n

√n

D−→ N

0,

14

.

In general, when bn is a fast growing function, it annihilates the role of αn. As we expect, after a very long period of gasmixing, the influence of the initial split is obliterated.

Acknowledgments

The first author would like to acknowledge the Department of Statistics at The GeorgeWashington University andwouldlike to express his gratitude for their support and financial assistance through the Jerome Cornfield Award.

Some of this research was carried out while the second author was on sabbatical leave at Purdue University. Heacknowledges the support of the Department of Statistics at Purdue University, with special thanks to Dr. Mark DanielWardfor his care and hospitality.

The authors would also like to thank the reviewer for several suggestions that have improved the overall presentationand accuracy of this paper.

References

Athreya, K., Karlin, S., 1968. Embedding of urn schemes into continuous time Markov branching processes and related limit theorems. The Annals ofMathematical Statistics 39, 1801–1817.

Balaji, S., Mahmoud, H., 2006. Exact and limiting distributions in diagonal Pólya processes. Annals of the Institute of Statistical Mathematics 58, 171–185.Balaji, S., Mahmoud, H., Tong, Z., 2010. Phases in the diffusion of gases via the Ehrenfest urn model. Journal of Applied Probability 47, 841–855.Balaji, S., Mahmoud, H., Watanabe, O., 2006. Distributions in the Ehrenfest process. Statistics & Probability Letters 76, 666–674.Balakrishnan, N., 1997. Advances in Combinatorial Methods and Applications to Probability and Statistics. Birkhüser.Bender, E., Canfield, R., McKay, B., 1997. The asymptotic number of labeled graphswith n vertices, q edges, and no isolated vertices. Journal of Combinatorial

Theory, Series A 80, 124–150.Johnson, N., Kotz, S., 1977. Urn Models and their Application: An Approach to Modern Discrete Probability Theory. Wiley.Kholfi, S., Mahmoud, H., 2012. The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors. Statistics & Probability Letters 82,

49–57.Kholfi, S., Mahmoud, H., 2012. The class of tenable zero-balanced Pólya urn schemes: characterization andGaussian phases. Advances in Applied Probability

44, 702–728.Kolchin, V., Sevastianov, B., Chistiakov, V., 1978. Random allocations; translation. In: Balakrishnan, A.V. (Ed.), Scripta Series in Mathematics. V.H.Winston,

distributed solely by Halsted Press, Washington, New York.Mahmoud, H., 2008. Pólya Urn Models. CRC Press.Mahmoud, M., 2010. Gaussian phases in generalized coupon collection. Advances in Applied Probability 42, 994–1012.Smythe, R., 2011. Generalized coupon collection: the superlinear case. Journal of Applied Probability 48, 189–199.Vatutin, V., Mikhailov, V., 1982. Limit theorems for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory of

Probability and its Applications 27, 684–692.