## 1 Introduction

[2] The tail of the frequency-magnitude distribution (FMD) of earthquakes is the most desirable part to know, because it predicts the frequency of the largest events. It is broadly accepted that the Gutenberg-Richter scaling relation cannot be extrapolated to arbitrarily large magnitudes, because this results in a conflict with basic physical principles, e.g., conservation of energy. A simple truncated version, the so-called doubly truncated Gutenberg-Richter (GR) law defined by probability density function (PDF)

with magnitudes *m*∈*m*_{0};*M*and the Richter *b*-value has been criticized, because the sharp cutoff at the maximum magnitude *M*is considered to be unphysical. As a compromise, the tapered Pareto distribution (*Kagan and Jackson* [2000]), also called “modified Gutenberg-Richter” (MGR) distribution became popular in the recent past. The PDF, which is usually given in terms of seismic moment , is

and can be transformed by , with measured in nanometers, into a PDF for earthquake magnitudes *m*:

the tapering is described by the function

[3] In equations (2) and (3), *β* is the shape parameter and the sharp cutoff at an ultimate maximum magnitude *M* in the truncated distribution (equation (1)) is replaced by smooth tapering characterized by the corner magnitude *m*_{c}. Although probabilities for large earthquakes are reduced significantly, this model still allows for the occurrence of infinitely large events.

[4] In their recent analysis, *Bell et al.* [2013] claim that in the periods between the largest global earthquakes, “the preferred model gradually converges to the tapered GR relation” (equations (2) and (3)) and that “the form of the convergence cannot be explained by random sampling of an unbounded GR distribution.” In the present analysis, I demonstrate that such a conclusion cannot be justified solely from an analysis of the global catalog. Performing an objective comparison of various state-of-the-art models for the FMD, the limits of catalog-based studies on the tail of the FMD are identified. This will be carried out in the following steps: in section 2, I show that the time dependence of the estimated FMDs is governed by the strongly biased estimation of the tail parameter *m*_{c} or *M*, which, in turn, results in a misleading comparison of models with different degrees of undersampling, and eventually to the conclusion of a converging FMD. Taking this into account, I calculate Bayesian posterior probabilities to compare various state-of-the-art models for the FMD of the global CMT catalog (section 3). For this analysis, I use the same catalog as *Bell et al.* [2013], namely the global CMT catalog [*Ekström et al.*, 2012] with magnitudes *m*≥5.75 from 1977 until mid-2012. Finally, conclusions with respect to potential future studies are drawn.