In this paper, we consider an overlay cognitive radio network, in which a secondary transmitter (ST) is willing to relay the information of a primary transmitter toward a primary receiver. In return, ST can access the licensed band to send its own information superimposed with the primary signal to a secondary receiver. The power-limited ST uses a power splitting protocol to harvest energy from its received signal to increase its transmit power. We analyze the performance of the primary and the secondary systems under independent Nakagami-m fading by deriving their corresponding outage probabilities in integral-based expressions. In addition, by considering the high signal-to-noise ratio, we obtain very tight closed-form approximations of the outage probabilities. Thereafter, by further analyzing the approximations, we reveal novel insights on the diversity orders and coding gains of the two systems. Our analytical results are validated through extensive Monte-Carlo simulations.