  The paper deals with homogenization problem for a non-local linear operator
with a kernel of convolution type in a medium with a periodic structure. We
consider the natural diffusive scaling of this operator and study the limit
behaviour of the rescaled operators as the scaling parameter tends to 0. More
precisely we show that in the topology of resolvent convergence the family of
rescaled operators converges to a second order elliptic operator with constant
coefficients. We also prove the convergence of the corresponding semigroups
both in $L^2$ space and the space of continuous functions, and show that for
the related family of Markov processes the invariance principle holds.
