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We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{\\lfloor N t \\rfloor})_{t \\in \\mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t \\in \\mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N \\to \\infty$. 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