  In this paper, we define a compound generalized fractional counting process
(CGFCP) which is a generalization of the compound versions of several
well-known fractional counting processes. We obtain its mean, variance, and the
fractional differential equation governing the probability law. Motivated by
Kumar et al. (2020), we introduce a fractional risk process by considering
CGFCP as the surplus process and call it generalized fractional risk process
(GFRP). We study the martingale property of the GFRP and show that GFRP and the
associated increment process exhibit the long-range dependence (LRD) and the
short-range dependence (SRD) property, respectively. We also define an
alternative to GFRP, namely AGFRP which is premium wise different from the
GFRP. Finally, the asymptotic structure of the ruin probability for the AGFRP
is established in case of light-tailed and heavy-tailed claim sizes.
