Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{ k+1} = Y^N_{ k} + a_N(Y^N_k) X_{ k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. 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 $N \to \infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.
Categories: math.PR