Stochastic recursions: between Kesten's and Grincevi\v{c}ius-Grey's assumptions

We study the stochastic recursion $X_n=\Psi_n(X_{n-1})$, where $(\Psi_n)_{n\geq 1}$ is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation $x\mapsto Ax+B$. We describe the tail behaviour of the stationary solution $X$ under the assumption that there exists $\alpha>0$ such that $\mathbb{E} |A|^{\alpha}=1$ and the tail of $B$ is regularly varying with index $-\alpha<0$. We also find the second order asymptotics of the tail of $X$ when $\Psi(x)=Ax+B$.