Support and density of the limit $m$-ary search trees distribution

TL;DR

This paper investigates the distributional properties of the limit random variable in $m$-ary search trees, revealing its density, support, and moments, especially for $m \\geq 27$ where non-Gaussian behavior occurs.

Contribution

It proves that the limit distribution has a square integrable density, full support on the complex plane, and finite exponential moments, advancing understanding of the phase transition in $m$-ary search trees.

Findings

The distribution of the limit W has a density on the complex plane.

The support of W is the entire complex plane.

W has finite exponential moments.

Abstract

The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number.