A note on the asymptotic behavior of the height for a birth-and-death process

TL;DR

This paper investigates the asymptotic properties of the height in a birth-and-death process related to mean-field models, providing variance analysis, a weak law of large numbers, and showing the failure of a central limit theorem.

Contribution

It derives the asymptotic variance of the height and demonstrates the convergence in distribution to a degenerate distribution, highlighting the non-applicability of the CLT.

Findings

Asymptotic variance of the height is obtained.

A weak Law of Large Numbers for the height is established.

The Central Limit Theorem does not hold for the normalized height.

Abstract

This paper focuses on the asymptotic behaviors of the {\it height} for a birth-and-death process which related to a mean-field model \cite{FFS}(or the Anick-Mitra-Sondhi model \cite{DDM}). Recently, the asymptotic mean value of the height for the model is given in \cite{LAV}. In this paper, first, the asymptotic variance of the height is given, and as a consequence, a weak Law of Large Number for the height is obtained. Second, the centered and normalized height is proved to converge in distribution to a degenerate distribution, this indicates that the desired Central Limit Theorem fails.