  We consider a long-range version of self-avoiding walk in dimension $d >
2(\alpha \wedge 2)$, where $d$ denotes dimension and $\alpha$ the power-law
decay exponent of the coupling function. Under appropriate scaling we prove
convergence to Brownian motion for $\alpha \ge 2$, and to $\alpha$-stable
L\'evy motion for $\alpha < 2$. This complements results by Slade (1988), who
proves convergence to Brownian motion for nearest-neighbor self-avoiding walk
in high dimension.
