Transient analysis of the subordinated chain of a state dependent pure birth process

TL;DR

This paper analyzes a pure birth process with specific intensities, showing its subordinated chain resembles a Bernoullian scheme with dependent success probabilities and links to a Polya urn model, with explicit transition probabilities and bounds.

Contribution

It introduces a novel analysis of the subordinated chain of a state-dependent pure birth process, connecting it to Bernoullian schemes and Polya urns, with explicit calculations.

Findings

Subordinated chain is similar to a Bernoullian scheme with dependent success probabilities.

Explicit transition probabilities are derived using generating functions.

Bounds are established using the centering sequence approach.

Abstract

Consider a pure birth process with intensities {\lambda}(k) =1/(1+k), with k=0,1,2,..., we show that the subordinated chain is assimilable to a Bernoullian scheme with dependent successes probabilities, also we show a direct link with a degenerate Polya urn replacement scheme \cite{Janson}. We compute explicitly transition probabilities, by generating function method, of the subordinated chain and give some interesting bounds using the "centering sequence" approach proposed by MacDiarmid \cite{McDiarmidcentering}.