Gaussian scaling for the critical spread-out contact process above the upper critical dimension

We consider the critical spread-out contact process in Zd with d=1, whose infection range is denoted by L=1.| The two-point function tt(x) is the probability that x in Zd is infected at time t by the infected individual located at the origin o in Zd at time 0. We prove Gaussian behaviour for the two-point function with L=L0 for some finite L0= L0(d) for d>4.| When d=4, we also perform a local mean-field limit to obtain Gaussian behaviour for ttT(x) with t>0 fixed and T tending to infinity when the infection range depends on T in such a way that LT=LTb for any b>(4-d)/2d. The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process.| We prove the existence of several critical exponents and show that they take on their respective mean-field values.| The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards those for the canonical measure of super-Brownian motion, which we defer to a sequel of this paper. The results in this paper also apply to oriented percolation, for which we reprove some of the results in [20] and extend the results to the local mean-field setting described above when d=4.