Abstract
We consider the random recursion $X_{n}^{x}=M_nX_{n-1}^{x}+Q_n+N_n(X_{n-1}^{x})$, where $x\in\mathbb R$ and $(M_n, Q_n, N_n)$ are i.i.d., $Q_n$ has a heavy tail with exponent $\alpha>0$, the tail of $M_n$ is lighter and $N_n(X_{n-1}^{x})$ is smaller at in
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