In this study, convergence between Turkey and the EU has been examined by using time series econometrics in the context of per capita incomes. Firstly stochastic convergence has been summarized. Then, it has been investigated whether Turkey is catching up the EU or not. Both average incomes of the EU13 and individual income of each country in the EU13 have been considered in order to analysis Turkeys convergence process to the EU for the period 1950-2010. According to results, there have no strong convergence tendency in the income differences of Turkey EU13. Turkey is catching up only a country while falling behind of EU13 average and other twelve countries. These results indicate that Turkey must have better economic performance in the process of economic integration in order to catch up living standards of the EU.