has been a popular model in the research of applied probability and stochastic analysis due to its simplicity and versatility. In general, the dynamics of Pólya urns are considered on a discrete-time scale. Two-color Pólya urn schemes have found applications in many fields. Two of the most classical Pólya urn models are thePólya-Eggenberger urn (see ) and the Ehrenfest urn (see ), which were respectively used to model disease contagion and gas diffusion. In modern times, different types of Pólya urns are utilized to study scientific models from more diverse areas, such as the play-the-winner scheme in clinical trials (e.g., [21, 23]), and recursive tree models and searching algorithms in computer science (e.g., ). In addition, mathematicians, probabilists and theorists are committed to developing rigorous methods for characterizing the dynamics of Pólya urns analytically. Some exactly solvable Pólya urn models are summarized and listed in , and the extensions to solvable randomized Pólya urns appear in . We refer the interested readers to  for a general background, and to [14, 19] for textbook style expositions.
Embedding discrete-time processes into continuous time has been around for a long time; see , for example. The idea was first employed in the context of urns in . A Poisson-like transformation was introduced to create a bridge between discrete-time urn schemes and the associated continuous-time urn processes obtained by embedding. In some articles or texts, this type of transformation is called “poissonization.” In the literature, sometimes the poissonized Pólya urns are called Pólya processes, first so named in . An inverse transformation (called “depoissonization”) was also reported in . Problems arising in depoissonization are highly nontrivial, and the inverse transformation itself seems elusive till today.
2 The Pólya process
To begin with, we give a few words about Pólya urns. A two-color Pólya urn scheme is an urn containing balls of up to two colors, say white and blue, and evolving over time according to some predetermined rules, which govern the dynamics of urn evolution. At each time point, a ball is drawn at random (all balls in the urn being equally likely), its color is observed, and it is placed back in the urn along with some new balls according to the pertinent addition rule. This is a sampling process with replacement and possible growth (or shrinkage) of the urn population. The addition rules are as follows. When a white ball is drawn, it is returned to the urn together with white balls and blue balls; when a blue ball is drawn, it is returned to the urn together with white balls and blue balls. In general, these numbers can be positive, zero or negative, where a negative value is interpreted as removing that many balls out of the urn, or even random. The class of urn processes we investigate in this manuscript calls only for fixed replacements.
An urn is called tenable if the rule can be executed forever on all possible stochastic paths, and it never gets “stuck,” due to being empty (in which case the next ball cannot be drawn) or not having enough balls to be removed for at least one color, while executing the ball replacement rules after a draw.
It is customary to represent these dynamics by the (ball addition) replacement matrix
the rows of this matrix are indexed with white and blue, respectively from top to bottom, and the columns are indexed with white and blue, respectively from left to right.
An associated Pólya urn process is obtained by embedding a discrete-time Pólya urn scheme into continuous time. It can be thought of as a renewal process, where the predetermined rule is executed at each renewal point (called epoch). It was suggested in  to use exponential random variables with mean (which we call ) for modeling interarrival time
, as the exponential distribution has tractable properties. In particular, the exponential distribution appears to be a useful device for the embedding owing to its appealingmemoryless property and scalability.
More precisely, consider that each ball in the urn is endowed with a clock that rings in time, and all these clocks are independent of each other and of any other random variables related to the past. At an epoch, the pertinent rules associated with its replacement matrix are executed. All newly-added balls are endowed with their own independent clocks. The execution of the rules takes place instantaneously after a draw, without any time lapse.
Most investigations of urn schemes or processes with a constant ball replacement matrix assume a condition called balance, in view of which the row sum of the ball replacement matrix is constant, and thus the total number of balls added at each step remains unchanged; see [5, 6] for instance. The balance condition leads to a mathematically-convenient model, whereby conditional proportions have a deterministic denominator.
In this short communication, we deviate from the balance condition, a situation in which the total number of balls is a nondegenerate random variable. We are interested in a class of two-color Pólya urns with the replacement matrix
for integer . We characterize the asymptotic behavior of the possonized urn. In this particular class of urn models, we always add or subtract balls of the two colors in the same amount. The difference between the number of blue and number of white balls remains the same at all times. We call the differential index of the urn, as it appears as a governing parameter in the underlying asymptotic distributions.
Note that we require for tenability; if , the initial number of white balls exceeds the number of blue balls, and it is possible for the process to get locked in a path that depletes the blue balls and ultimately the process halts. Besides, this class of Pólya processes does not belong to generalized urn processes investigated in the seminal article 
, as several assumptions therein are not satisfied, rendering our research novel. The study of this class in discrete time presents many algebraic difficulties owing to the lack of balance condition. It was pointed out by Basil Morcrette during his presentation in the 24th International Meeting on the Analysis of Algorithms (AofA 2013) that the problems of unbalanced Pólya urn are challenging even by using the very powerful analytic combinatoric methods, since the ordinary differential equations generated by the so-called “analytic urn” technique are significantly harder to solve than those that appear upon applying the method to a balanced urn.
3 Continuous-time constant-differentials Pólya processes
In this section, we investigate the constant-differentials Pólya processes that are generated by the Pólya urns associated with the replacement matrix in (1) and a positive differential index. Our strategy is to exploit the methodology developed in , and our goal is to characterize the asymptotic distribution of and , the number of white balls and blue balls, respectively, in the urn at time , after proper scaling. The principal idea is to establish a partial differential equation (PDE) that governs the dynamics of the Pólya process as time goes by.
be the joint moment generating function ofand . The PDE in  is valid for all kinds of tenable urn processes (not necessarily balanced).
Lemma 1 ([3, Lemma 2.1]).
The general solution to PDE (2) is known, but not in a closed form for most cases. In essence, the solution is an integration along characteristic curves. These integrals are often too difficult to obtain.
There are only a few cases for which the solutions are developed: the forward and backward diagonal processes in , the Ehrenfest processes in , a class of zero-balanced processes with replacement matrix of Bernoulli entries in , the Apollonian processes in , the triangular urn processes in  and the Bagchi-Pal processes in .
In particular, the PDE for constant-differentials Pólya processes is
We are able to simplify PDE (3) by observing that the (marginal) process of is fully specified by the (marginal) process of since the difference remains constant for all . We thus only need to focus on the evolutionary behavior of , i.e., the moment generating function of only. Consider .
we are able to rewrite PDE (3) in terms of ; that is,
which is equivalent to the stated PDE. ∎
PDE (4) is an initial-value problem, to be solved under the boundary condition . The problem is amenable to the method of characteristics in . In fact, the PDE is of a known type called the transport equation. The presence of the term renders it inhomogeneous.
For time , let and be respectively the number of white and blue balls in a constant-differentials Pólya process on white and blue balls with the replacement matrix
and of differential index . As , we have
The characteristic curves for PDE (4) are the solutions to the ordinary differential equations (ODEs):
The characteristic curves are
We need the intersection point with the -axis to construct the solution; it is
Unlike the case of homogenious transport equation, where the PDE solution is constant over the characteristic curves, in an inhomogenious case like the one at hand, the PDE solution varies on the characteristic curves. In our case, the PDE solution must satisfy the ODE
An initial condition on the PDE is . It is required then to solve the ODE for under the initial condition . The ODE in question is first-order and linear, with a standard solution. As a solution to the PDE, we obtain
Next, we derive the asymptotic distribution of by considering scale , i.e., , as . Recall the moment generating function
. We replace the dummy variableby , where is the new dummy variable, and obtain
Consider the local expansion of exponential function,
We come up with
In the last convergence relation, the right-hand side is the moment generating function of a gamma random variable with shape parameter and scale parameter . We thus conclude the stated convergence in distribution according to Lévy’s continuity theorem; see [16, p. 172]. ∎
The mean and variance
The mean and variance ofare given by
The mean, at time , is readily obtained by taking the derivative of at . Likewise, the second moment of is obtained by taking the derivative of twice at , i.e.,
The variance then follows after simplifying . ∎
In view of the constant differential condition, we have
With convergent in distribution and , by Slutsky’s theorem (see [16, p. 146]) we can assert the corollary. ∎
4 An alternative approach via martingale
We use the superscript on a matrix to denote its transpose, and consider . It is folklore that , where
is a random vector containing the number of different kinds of balls in a Pólya process at time, is a continuous-time martingale, which has been observed in many classes of Pólya processes. This is shown in 
for a broad class of Pólya processes that have a particular eigenvalue structure, with a positive real principal eigenvalue. However, this condition is not satisfied in the case of constant-differentials Pólya processes, where the principal eigenvalue is a repeated. We prove next that indeed, is a martingale, too, for the class of constant-differentials Pólya processes.
The vector is a bivariate martingale with respect to the natural filtration generated by the evolution of a constant-differentials Pólya process associated with Matrix (1).
Let be the vector . Note that is a nilpotent matrix, as . In view of this nilpotency, the matrix is easily found (from the series expansion) to be . We thus have
Let us interpret the top row, which reads
We consider as the status of white balls, in an urn process starts at time with an initial number of white balls and evolves over a period of . By Corollary 1, we have . Consequently, we arrive at
so is a martingale. Similarly, we have
At the vectorial level we get
and we conclude that is a two-dimensional martingale. ∎
The mean in Corollary 1 can be obtained from the martingale formulation, as we have
for any . We thus have , or in other words,
However, it is harder to obtain higher moments or the distribution by this technique.
is bounded as .
By the triangle inequality, we have
where the second line is followed by Corollary 1. By Cauchy-Schwarz inequality, we thus get
which proves that is bounded for large value of . ∎
The mean, , is not uniformly bounded for all ; more precisely, it blows up when is close to . We thus conjecture that may not converge to gamma almost surely.
-  Athreya, K. and Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching process and related limit theorems. The Annals of Mathematical Statistics 39, 1801–1817.
-  Balaji, S. and 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. and 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. Birkhäuser Boston Inc., Boston, MA.
-  Chauvin, B., Pouyanne, N. and Sahnoun, R. (2011). Limit distributions for large Pólya urns. The Annals of Applied Probability 21, 1–32.
-  Chauvin, B., Mailler, C. and Pouyanne, N. (2015). Smoothing equations for large Pólya urns. Journal of Theoretical Probability 28, 923–957.
-  Chen, C. and Mahmoud, H. (2018). The continuous-time triangular Pólya process. Annals of the Institute of Statistical Mathematics 70, 302–321.
-  Chen, C. and Zhang, P. (in press). Characterizations of asymptotic distributions of continuous-time Pólya processes. To appear Communications in Statistics—Theory and Methods.
-  DasGupta, A. (2010). Fundamentals of Probability: A First Course. Springer, New York, NY.
-  Eggenberger, F. and Pólya, G. (1923). Über die statistik verketteter vorgänge Zeitschrift für Angewandte Mathematik und Mechanik 3, 279–289.
-  Ehrenfest, P. and Ehrenfest, T. (1907). Über zwei bekannte Einwände gegen das Boltzmannsche H-theorem. Physikalische Zeitschrift 8, 311–314.
-  Janson, S. (2006). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Processes and their Applications 110, 177–245.
-  Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In the DMTCS proceedings AG, 59–118.
Johnson, N. and Kotz, S. (1977).
Urn Models and their Application: An Approach to Modern Discrete Probability Theory. John Wiely & Sons Inc., New York, NY.
-  Kac, M. (1949). On deviations between theoretical and empirical distributions. Proceedings of the National Academy of Sciences of the United States of America 35, 252–257.
-  Karr, A. (1993). Probability. Springer-Verlag, New York, NY.
-  Levine, H. (1997). Partial Differential Equations. American Mathematical Society, Providence, RI.
-  Mahmoud, H. (2003). Pólya urn models and connections to random trees: A review. Journal of Iranian Statistical Society 2, 53–114.
-  Mahmoud, H. (2009). Pólya Urn Models. CRC Press, Boca Raton, FL.
-  Morcrette B. and Mahmoud, H. (2012). Exactly solvable balanced tenable urns with random entries via the analytic methodology. DMTCS proceedings AQ, 219–232.
-  Rosenberger, W. (1999). Randomized play-the-winner clinical trials: Review and recommendations. Controlled Clinical Trials 20, 328–342.
-  Sparks, J. and Mahmoud, H. (2013). Phases in the two-color tenable zero-balanced Pólya process. Statistics & Probability Letters 83, 265–271.
-  Wei, L.-J. (1979). The generalized Pólya urn design for sequential medical trials. The Annals of Statistics 7, 291–296.
-  Zhang, P. and Mahmoud, H. (206). Distributions in a class of Poissonized urns with an application to Apollonian networks Statistics & Probability Letters 115, 1–7.