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This article deals with the limit distribution for a stochastic differentialequation driven by a non-symmetric cylindrical $\alpha$-stable process. Undersuitable conditions, it is proved that the solution of this equation convergesweakly to that of a stochastic differential equation driven by a Brownianmotion in the Skorohod space as $\alpha\rightarrow2$. Also, the rate of weakconvergence, which depends on $ 2-\alpha$, for the solution towards thesolution of the limit equation is obtained. For illustration, the results areapplied to a simple one-dimensional stochastic differential equation, whichimplies the rate of weak convergence is optimal.

On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric $\alpha$-stable L\'{e}vy processes