root. 

Yap and Speed (2005) systematically reproduced the results in Yang and Roberts (1995) and Huelsenbeck et al. (2002) and, with some small discrepancies with the earlier studies, found again that a non-stationary model was able to make statisti- cally significant inferences about the location of the root, but that a non-reversible model did little better than a reversible model. This also was empirical research that left unanswered questions about whether any of the models were theoretically able to identify the location of the root. 

The contribution of the present work is to constructively prove that there is a non-stationary substitution process that identifies the location of the root that can be statistically consistently estimated from data. Indeed, the model is shown to be consistently estimable for two taxa. This is not possible for general non-stationary processes (Chang 1996; Bonhomme et al. 2014), so we make the additional assump- tion that the process is strand-symmetric ; that the process of evolution is identical on the sense and antisense strands of DNA. The conditions of the proof ensure that the process is non-stationary, and we show that a non-reversible, stationary model is not identifiable so cannot be consistently estimated. This observation sheds some light on the success of non-stationary processes and the failure of non-reversible, stationary processes at detecting the root in the literature. 

Much has been written about the biological mechanisms that result in nucleotide substitution processes being strand asymmetric and there is now substantial empiri- cal evidence to support strand asymmetry’s existence in nature. Touchon and Rocha (2008) provide a good review of the subject. Strand asymmetry seems to be a lo- calised phenomenon, existing on the scale of genes rather than genomes, and ap- pears to be common in prokaryote and organelle genomes but not in eukaryotes. Nucleotide compositional asymmetry is the most common measure used for statisti- cal inference. Under very loose assumptions (Lobry 1995; Lobry and Lobry 1999) a strand-symmetric process should result in the proportions of As and Ts being equal 

