Limit distributions for multitype branching processes of m-ary search trees

TL;DR

This paper studies the limit distributions of a continuous-time multitype branching process related to m-ary search trees, revealing a phase transition from Gaussian to complex distributions and analyzing their properties.

Contribution

It introduces a detailed analysis of the limit distributions for multitype branching processes of m-ary search trees, including existence, uniqueness, and properties of the solutions.

Findings

For m ≤ 26, the process is Gaussian.

For m ≥ 27, the limit distribution W is complex-valued and non-Gaussian.

W has an absolutely continuous distribution with support on the entire complex plane.

Abstract

A particular continuous-time multitype branching process is considered, it is the continuous-time embedding of a discrete-time process which is very popular in theoretical computer science: the m-ary search tree (m is an integer). There is a well-known phase transition: when m \leq 26, the asymptotic behavior of the process is Gaussian, but for m \geq 27 it is no more Gaussian and a limit W of a complex-valued martingale arises. Thanks to the branching property it appears as a solution of a smoothing equation of the type Z = e^{-{\lambda}T}(Z(1) + ... + Z(m)), where {\lambda} \in C, the Z(k) are independent copies of Z and T is a R_+-valued random variable, independent of the Z(k). This distributional equation is extensively studied by various approaches. The existence and unicity of solution of the equation are proved by contraction methods. The fact that the distribution of W is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of W is obtained by considering W as the limit of a complex Mandelbrot cascade.