  We present a detailed study of the effects of L\'evy $\alpha$-stable disorder
distributions on the optical and localization properties of excitonic systems.
These distributions are a generalization of commonly studied Gaussian
randomness ($\alpha = 2$). However, the more general case ($\alpha < 2$)
includes also heavy-tailed behavior with a divergent second moment. These types
of distributions give rise to novel effects, such as exchange broadening of the
absorption spectra and anomalous localization of the excitonic states, which
has been found in various excitonic systems. We provide a thorough examination
of the localization behavior and the absorption spectra of excitons in chains
with L\'evy $\alpha$-stable diagonal disorder. Several regimes are considered
in detail: (i) weak disorder and small systems, where finite size effects
become significant; (ii) intermediate disorder and/or larger systems for which
exciton states around the exciton band edge spread over a number of monomers,
and (iii) strong disorder where the exciton wave functions resemble the
individual molecular states. We propose analytical approaches to describe the
disorder scaling of relevant quantities for these three regimes of
localization, which match excellently with numerical results. We also show the
appearance of short localization segments caused by outliers (monomers having
transition energies larger than the exciton band width). Such states give rise
to an additional structure in the absorption spectrum, density of states, and
in the nontrivial disorder scaling of the absorption band width.
