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The random split tree introduced by Devroye (1999) is considered. We derive asecond order expansion for the mean of its internal path length and furthermoreobtain a limit law by the contraction method. As an assumption we need thesplitter having a Lebesgue density and mass in every neighborhood of 1. We useproperly stopped homogeneous Markov chains, for which limit results in totalvariation distance as well as renewal theory are used. Furthermore, we extendthis method to obtain the corresponding results for the Wiener index.

On the asymptotic internal path length and the asymptotic Wiener index of random split trees