Previous research suggests that economy-wide poverty traps are rarely observed in the data. In this paper, we explore a related hypothesis: low-income countries rarely improve their position relative to the US. Using finite state Markov chains, we show that upwards mobility is indeed limited. Since capital-output ratios are similar across countries, and human capital is also converging, the persistence of low relative income seems to originate in the persistence of low relative TFP. We study the dynamics of relative TFP and how they interact with absolute levels of human capital, casting new light on the future of convergence.