Satellite communication (SatCom) is an essential component of next-generation wireless communication to achieve a goal of ubiquitous connectivity on globe. The outage event of SatCom link connecting to a network is more critical in an infrastructure-deficient remote area. In this paper, we analyze outage probability (OP) and symbol error rate (SER) over SatCom downlink channels when the users are randomly located in single beam and multibeam area. The downlink beams will suffer from propagation loss and the shadowed-Rician fading depending on the user location which is assumed to follow a Poisson point process. For mathematically tractable, informative, and insightful interpretation, we obtain the asymptotic OP and SER expressions of user link under several channel conditions in the high power regime. Finally, numerical results are presented to verify the analysis and show the accuracy of the asymptotic results.