The Bayesian `star paradox' persists for long finite sequences

The `star paradox' in phylogenetics is the tendency for a particular resolved tree to be sometimes strongly supported even when the data is generated by an unresolved (`star') tree. There have been contrary claims as to whether this phenomenon persists when very long sequences are considered. This note settles one aspect of this debate by proving mathematically that there is always a chance that a resolved tree could be strongly supported, even as the length of the sequences becomes very large.