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In this article, we consider the problem of periodic homogenization of aFeller process generated by a pseudo-differential operator, the so-calledL\'evy-type process. Under the assumptions that the generator has rapidlyperiodically oscillating coefficients, and that it admits "small jumps" only(that is, the jump kernel has finite second moment), we prove that theappropriately centered and scaled process converges weakly to a Brownian motionwith covariance matrix given in terms of the coefficients of the generator. Thepresented results generalize the classical and well-known results related toperiodic homogenization of a diffusion process.

Periodic Homogenization of a L\'evy-Type Process with Small Jumps