Distributions in the constant-differentials P\'olya process

TL;DR

This paper analyzes a class of Pólya processes with constant differentials, showing that scaled counts of balls converge to gamma distributions, with results derived through an analytic PDE approach.

Contribution

It introduces a new class of unbalanced Pólya processes and characterizes their asymptotic distributions using PDE methods, independent of initial conditions.

Findings

Counts of balls converge to gamma distributions

Results depend on differential index and ball addition

Initial conditions do not affect the limiting distribution

Abstract

We study a class of unbalanced constant-differentials P\'olya processes on white and blue balls. We show that the number of white balls, the number of blue balls, and the total number of balls, when appropriately scaled, all converge in distribution to a gamma random variables with parameters depending on the differential index and the amount of ball addition at the epochs, but not on the initial conditions. The result is obtained by an analytic approach utilizing partial differential equations.