The continuum P\'olya-like random walk

TL;DR

This paper introduces a generalized continuum version of the Pólya-like random walk, extending the classical urn model to include variable initial conditions, random transition entries, and higher-dimensional stochastic processes.

Contribution

It develops a broad framework for Pólya-like processes with variable and random parameters, expanding the classical urn model to a continuous, high-dimensional setting.

Findings

Analysis of parametric classes of generalized Pólya-like random walks

Extension of the urn model to include randomness in transition entries

Broader interpretation as high-dimensional random walks

Abstract

The P\'olya 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 P\'olya 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 P\'olya 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 P\'olya-like random walks.