Semi-general asymptotic formula for fixed-gain amplify-and-forward multihop outage probability

Consider an $N$-hop fixed-gain amplify-and-forward link, where the PDF of the channel power for each hop decays exponentially and its Mellin transform is well-behaved as $\left\vert \operatorname{Im}(s)\right\vert\to\infty$. Let $m$ be the minimum shape parameter among the $N$ channel-power distributions, let $\mu$ be the multiplicity of $m$, and let $\boldsymbol{\ell}=(\ell_1,\ldots,\ell_{N-1})$. The constants $A$ and $B_{\boldsymbol{\ell}}$ depend on the distributional parameters, with $B_{\boldsymbol{\ell}}$ also depending on the indices in $\boldsymbol{\ell}$. Define

$$C_r(s)=(s-m)^r I(s)\left(\frac{\bar{\gamma}}{A}\right)^s,$$

where $I(s)$ is the integrand with all terms independent of $s$ removed.

Semi-general asymptotic formula. The outage probability satisfies

$$P_o\sim A^m\sum_{r=1}^{\mu}\sum_{\{\ell_1,\ldots,\ell_{N-1}\}\in\mathcal{L}_r}(-1)^{\lambda_{N-1}}B_{\boldsymbol{\ell}}\sum_{l=0}^{r-1}\frac{C_r^{(r-1-l)}(m)}{(r-1-l)!l!}\sum_{p=0}^{l}\binom{l}{p}(-1)^p(\log A)^{l-p}\frac{(\log\bar{\gamma})^p}{\bar{\gamma}^m}+o(\bar{\gamma}^{-m})$$

as $\bar{\gamma}\to\infty$.

This formula is intended as a semi-general asymptotic expression for fixed-gain amplify-and-forward multihop links with exponentially decaying channel-power PDFs. It includes the case $m=1$ and $\mu=N$ when all hops are Rayleigh, Rician, or Hoyt, but the source does not provide evidence that the stated result has been proved or resolved beyond this formulation.