converge to a single fixed point, but only at a polynomial rate**]{}, in both the RRN and FRN cases (\[thm:p\_q\_linear\] and \[thm:full\_res\_l\_g\_recurr\]); we can even explicitly compute the fixed point and the rate of convergence: For FRN, there is a [**unique stable fixed point**]{} ${\mathbf{e}}^* < 1$ determined by the equation $${\mathbf{e}}^* = \f 1 {\sigma_v^2 + \sigma_a^2}[\sigma_v^2 \f 2 \pi \arcsin\lp {\mathbf{e}}^* \rp + \sigma_a^2],$$ and $|{\mathbf{e}}^* - \p {\mathbf{e}}l|$ decreases like $l^{-\delta^*}$, where $$\delta^* := 1 - \f 2 \pi \f 1 {\sqrt{1 - ({\mathbf{e}}^*)^2}} \f{\sigma_v^2 }{\sigma_v^2 + \sigma_a^2}.$$ Since ${\mathbf{e}}^* < 1$, ${\mathbf{s}}= (1 - {\mathbf{e}}) {\mathbf{p}}= \Theta({\mathbf{p}}) = \Theta(l).$ The case of RRN can be viewed as a special case of the above, setting $\sigma_v^2 = 1$ and $\sigma_a^2 = 0$, which yields ${\mathbf{e}}^* = 0$ and $\delta^* = 1 - \f 2 \pi$. We observe that both ${\mathbf{e}}^*$ and $\delta^*$ only depend on the ratio $\rho := \sigma_a/\sigma_v$, so in \[fig:edelta\_plot\] we graph these two quantities as a function of $\rho$. ${\mathbf{e}}^*$ and $\delta^*$ both increase with $\rho$ and asymptotically approach 1 and $\nicefrac 1 2$ respectively from below. When $\rho = \sigma_a = 0$, ${\mathbf{e}}^* = 0$ and $\delta^* = 1 - \f 2 