Advanced Search

We consider the random recursion $X{ₙ}^x=MₙX_{n-1}^x+Qₙ+Nₙ\left(X_{n-1}^x\right)$, 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 $Nₙ\left(X_{n-1}^x\right)$ is smaller at infinity, than $MₙX_{n-1}^x$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S{ₙ}^x={\sum }_{k=0}^n{X}_k^x$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.