The continuum Pólya-like random walk
arXiv:1608.01233
The paper extends the classic Pólya urn scheme to a continuous‑time setting where initial values and transition matrix entries can be non‑integer or random, interpreting the model as a high‑dimensional random walk.
Abstract
The Pólya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the Pólya process. We deal with a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard Pólya process. In one of the scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn," but a broader interpretation as a random walk in a possibly high number of dimensions. In this paper, we study several parametric classes of these generalized continuum Pólya-like random walks.