Ambient RF (Radio Frequency) energy harvesting technique has recently been proposed as a potential solution to provide proactive energy replenishment for wireless devices. This paper aims to analyze the performance of a battery-free wireless sensor powered by ambient RF energy harvesting using a stochastic-geometry approach. Specifically, we consider a random network model in which ambient RF sources are distributed as a Ginibre $\alpha$-determinantal point process which recovers the Poisson point process when alpha? approaches zero. We characterize the expected RF energy harvesting rate.We also perform a worst-case study which derives the upper bounds of both power outage and transmission outage probabilities. Numerical results show that our upper bounds are accurate and that better performance is achieved when the distribution of ambient sources exhibits stronger repulsion.