  The present paper investigates the effects of tempering the power law kernel
of moving average representation of a fractional Brownian motion (fBm) on some
local and global properties of this Gaussian stochastic process. Tempered
fractional Brownian motion (TFBM) and tempered fractional Brownian motion of
the second kind (TFBMII) are the processes that are considered in order to
investigate the role of tempering. Tempering does not change the local
properties of fBm including the sample paths and p-variation, but it has a
strong impact on the Breuer-Major theorem, asymptotic behavior of the 3rd and
4th cumulants of fBm and the optimal fourth moment theorem.
