The shared resource access wireless communication system is regarded as a paradigm that allows the mobile network operators (MNOs) to have extended coverage and allows them to satisfy their subscribers' high capacity demands while keeping the capital and operational expenditure in check. On the other hand, dual connectivity (DC), a small cell enhancement feature, allows the subscribers to have two simultaneous connections increasing throughput and enhancing mobility robustness. For the DC mechanism, the user needs to be under the coverage of multiple BSs concurrently. Because of random nature of the wireless links and distribution of BSs in a network, the practicability/operability of DC for a typical user under such random operating conditions (both topological and wireless link randomness) needs to be properly evaluated. In this paper, we perform stochastic analysis of DC in shared resource access high capacity wireless networks. We consider a fading-averaged signal propagation loss process, where the selection of the serving BS is perturbed shadowing, not by fading. The DC coverage probability is analyzed in terms of tractable, integral expressions. We have observed that the DC coverage probability is greatly affected by the density of the small cell BSs and the variance of the log-normal shadowing.