Long run convergence implies that the convergence hypothesis will be rejected if the income differential is not stationary. However, this definition is valid only if the catching-up process between the two countries is already over. If we take into account catching-up dynamics, then poorest countries should obtain a faster growth than developed countries. Thus, income gaps should integrate decreasing time trends. We formalise this hypothesis theoretically using a stochastic neoclassical growth model with heterogeneous technology. We then apply this model to the issue of per-capita GDP catching-up of eight MENA countries towards the level of income in Europe. We approximate the nonlinear deterministic trend by a linear function with breaks and apply panel unit root tests with breaks. The analysis reveals firstly that the periods of divergence outnumber the periods of convergence. Secondly, since the year 2000 all countries but Syria have been converging toward the European per-capita income level.