Brigitte Chauvin ; Quansheng Liu ; Nicolas Pouyanne

Support and density of the limit $m$ary search trees distribution
dmtcs:2994 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2012,
DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)

https://doi.org/10.46298/dmtcs.2994
Support and density of the limit $m$ary search trees distribution
Authors: Brigitte Chauvin ; Quansheng Liu ; Nicolas Pouyanne
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Brigitte Chauvin;Quansheng Liu;Nicolas Pouyanne
The space requirements of an $m$ary search tree satisfies a wellknown 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 complexvalued 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 \overset{\mathcal{L}}{=} \sum_{k=1}^mV_k^{\lambda}W_k$, where $V_1, ..., V_m$ are the spacings of $(m1)$ 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.
Volume: DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)