  We present a new proof of $|x|^{-(d-2)}$ decay of critical two-point
functions for spread-out statistical mechanical models on $\mathbb{Z}^d$ above
the upper critical dimension, based on the lace expansion and assuming
appropriate diagrammatic estimates. Applications include spread-out models of
the Ising model and self-avoiding walk in dimensions $d>4$, and spread-out
percolation for $d>6$. The proof is based on an extension of the new Gaussian
deconvolution theorem we obtained in a recent paper. It provides a technically
simpler and conceptually more transparent approach than the method of Hara, van
der Hofstad and Slade (2003).
