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We prove that the Fourier transform of the properly-scaled normalizedtwo-point function for sufficiently spread-out long-range oriented percolationwith index \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for someC\in(0,\infty) above the upper-critical dimension 2(\alpha\wedge2). Thisanswers the open question remained in the previous paper [arXiv:math/0703455].Moreover, we show that the constant C exhibits crossover at \alpha=2, which isa result of interactions among occupied paths. The proof is based on a newmethod of estimating fractional moments for the spatial variable of thelace-expansion coefficients.

Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation