EN
We consider the random recursion $Xₙ^{x} = MₙX_{n-1}^{x} + Qₙ + Nₙ(X_{n-1}^{x})$, where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $Nₙ(X_{n-1}^{x})$ is smaller at infinity, than $MₙX_{n-1}^{x}$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $Sₙ^{x} = ∑_{k=0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.