We continue the line of research of random walks with barrier initiated by Iksanov and Möhle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent −α, α ∈ (0, 1), we prove that the number V n of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for V n proved in Iksanov and Negadailov (2008).

Let (X k , ξ k ) k∈N be a sequence of independent copies of a pair (X, ξ) where X is a random process with paths in the Skorokhod space D[0, ∞) and ξ is a positive random variable. The random process with immigration (Y (u)) u∈R is defined as the a.s. finite sum Y (u) = k≥0 X k+1 (u−ξ1 −· · ·−ξ k ) 1 {ξ 1 +···+ξ k ≤u} . We obtain a functional limit theorem for the process (Y (ut)) u≥0 , as t → ∞, when the law of ξ belongs to the domain of attraction of an α-stable law with α ∈ (0, 1), and the process X oscillates moderately around its mean E[X(t)]. In this situation the process (Y (ut)) u≥0 , when scaled appropriately, converges weakly in the Skorokhod space D(0, ∞) to a fractionally integrated inverse stable subordinator.Keywords Fractionally integrated inverse stable subordinators, random process with immigration, shot noise process 2010 MSC Primary 60F05, Secondary 60K05

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