Strong laws for balanced triangular urns

TL;DR

This paper establishes strong laws of large numbers for balanced triangular urn models, revealing how the counts of different colored balls behave asymptotically based on the replacement matrix's structure.

Contribution

It derives strong laws for balanced triangular urns and applies these results to analyze three-color urn models, highlighting the influence of the replacement matrix's diagonal elements.

Findings

Strong laws depend on the diagonal entries of the replacement matrix.

Asymptotic behavior of ball counts is characterized by specific scalings.

Results are used to analyze more complex three-color urn models.

Abstract

Consider an urn model whose replacement matrix is triangular, has all entries nonnegative and the row sums are all equal to one. We obtain the strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use the strong laws obtained to study further behavior of certain three color urn models.