Central limit theorems for additive functionals of long-range zero-range processes

In this paper, we extend the central limit theorem of the additive functional of the nearest-neighbor zero-range process given in \cite{Quastel2002} to the long-range case. Our main results show that in several cases the limit processes are driven by fractional Brownian motions with Hurst parameters in $(1\mathrm{/}2,3\mathrm{/}4]$. A local central limit theorem of the long-range random walk and a relaxation to equilibrium theorem of the long-range zero-range process play the key roles in the proofs of our main results.