ASYMPTOTIC EXPANSIONS IN LIMIT-THEOREMS FOR STOCHASTIC-PROCESSES .1.

For some families of locally infinitely divisible Markov processes eta (epsilon)(t),0 less than or equal to t less than or equal to T, with f requent small jumps, limit theorems for expectations of functionals F( eta(epsilon)[0, T]) are proved of the form \E(epsilon)F(eta(epsilon)[0 , T]) - E(0)F(eta(0)[0, T])\ less than or equal to const . k(epsilon), E(epsilon)F(eta(epsilon)[0, T]) = E(0)[F(eta(0)[0, T]) + k(epsilon) . A(1)F(eta(0)[0, T])] + o(k(epsilon)) (epsilon down arrow 0), where A( 1) is a linear differential operator acting on functionals, and the co nstant is expressed in terms of the local characteristics of the proce sses eta(epsilon)(t) and the norms of the derivatives of the functiona l F.