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Building on the work of Frankel (1979) and Froot & Rogoff (1995), this article proposes a more general model to determine long-term national inflation. It combines the assumptions underlying "weak" relative purchasing power parity (PPP) with those underlying the uncovered interest rate parity (UIRP). Using this model, we endeavour to determine, first, whether monetary convergence - within the meaning of the Maastricht Treaty - linking these variables has occurred and, secondly, whether achievement of such convergence bears out the PPP and UIRP assumptions. Since Maastricht-based convergence and PPP both appear to be long-term economic relations, we are using the fractional cointegration to test these relations. Fractional cointegration allows us to determine, first, whether convergence towards a long-term equilibrium has occurred and, secondly, at which speed the economic system returns to this equilibrium after an exogenous shock. After defining fractional cointegration and giving an economic interpretation of this method, we apply it to three pairs of "in" countries. We show that the Netherlands and France have achieved monetary convergence with Germany within the meaning of the Maastricht Treaty. We conclude that the weak form of PPP and UIRP is working between the Netherlands and Germany. The speed of Maastricht-based convergence with Germany is fast for the Netherlands and slower for France. Italy does not appear to have converged with Germany.