  This paper investigates the performance of multiple reconfigurable
intelligent surfaces (multi-RIS) communication systems where the RIS link with
the highest signal-to-noise-ratio (SNR) is selected at the destination. In
practice, all the RISs will not have the same number of reflecting elements.
Hence, selecting the RIS link with the highest SNR will involve characterizing
the distribution of the maximum of independent, non-identically distributed
(i.n.i.d.) SNR random variables (RVs). Using extreme value theory (EVT), we
derive the asymptotic distribution of the normalized maximum of i.n.i.d.
non-central chi-square (NCCS) distributed SNR RVs with one degree of freedom
(d.o.f) and then extend the results for k-th order statistics. Using these
asymptotic results, the outage capacity and average throughput expressions are
derived for the multi-RIS system. The results for independent and identically
distributed (i.i.d.) SNR RVs are then derived as a special case of i.n.i.d.
RVs. All the derivations are validated through extensive Monte Carlo
simulations, and their utility is discussed.
