Traditional empirical strategies for studying convergence---more generally, the dynamics and determinants of economic growth---can be misleading if important, underlying permanent or growth components are stochastically time-varying. This paper documents the degree to which this instability characterizes the data, and then offers an alternative empirical framework. This alternative---directly modelling the dynamics of the evolving cross-section distributions---applied to cross-country income data yields some interesting insights: economies across the world seem to be converging to a distribution where many remain wealthy, and many remain poor. Those economies able to make the transition from low to high income levels are primarily small and sparsely-populated; middle-income ones, by contrast, are a vanishing class.