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Is studied asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in R d . The view of regular and singular parts of solution is found. Mathematics Subject Classification (2000): primary 60J25,secondary 35C20.

Obvious view of distribution function of Markovian random evolution is found in terms of Bessel functions of n + 1-th order. Mathematics Subject Classification (2000): 60K99.

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Let (ξ 1 , η 1 ), (ξ 2 , η 2 ), . . . be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J 1 -topology on the Skorokhod space of n −1/2 max 0≤k≤· (ξ 1 +. . .+ξ k +η k+1 ) was proved under the assumption that contributions of max 0≤k≤n (ξ 1 +. . .+ξ k ) and max 1≤k≤n η k to the limit are comparable and that n −1/2 (ξ 1 +. . .+ξ [n·] ) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ 1 + . . . + ξ [n·] , properly normalized without centering, is attracted to a centered stable Lévy process, a process with jumps. As a consequence, weak convergence normally holds in the M 1 -topology. We also provide sufficient conditions for the J 1 -convergence. For completeness, less interesting situations are discussed when one of the sequences max 0≤k≤n (ξ 1 + . . . + ξ k ) and max 1≤k≤n η k dominates the other.An application of our main results to divergent perpetuities with positive entries is given.

Regular and singular parts of asymptotic expansions of semi-Markov random evolutions are given. Regularity of boundary conditions is shown. An algorithm for calculation of initial conditions is proposed.

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