  We study the statistical properties of Ising spin chains with finite
(although arbitrary large) range of interaction between the elements. We
examine mesoscopic subsystems (fragments of an Ising chain) with the lengths
comparable with the interaction range. The equivalence of the Ising chains and
the multi-step Markov sequences is used for calculating different non-additive
statistical quantities of a chain and its fragments. In particular, we study
the variance of fluctuating magnetization of fragments, magnetization of the
chain in the external magnetic field, etc. Asymptotical expressions for the
non-additive energy and entropy of the mesoscopic fragments are derived in the
limiting cases of weak and strong interactions.
