Implicit renewal theory in the arithmetic case

We extend Goldie’s implicit renewal theorem to the arithmetic case, which allows us to determine the tail behavior of the solution of various random fixed point equations. It turns out that the arithmetic and nonarithmetic cases are very different. Under appropriate conditions we obtain that the tail of the solution $X$ of the fixed point equations $XAX+B$, $XAX\vee B$ is $\mathrm{\ell }(x)q(x)x^{-\kappa }$, where $q$ is a logarithmically periodic function $q(x{e}^h)=q(x)$, $x>0$, with $h$ being the span of the arithmetic distribution of $\log A$, and $\mathrm{\ell }$ is a slowly varying function. In particular, the tail is not necessarily regularly varying. We use the renewal theoretic approach developed by Grincevi\v{c}ius and Goldie.