An optimal transmission policy is considered for energy harvesting (EH) wireless point-to-point communications, wherein the source node is solar-powered and equipped with a finite-sized battery. The long-term outage probability is minimized by adapting the transmission power to the causal energy arrival information, battery energy amount and channel fading through a Markov decision process (MDP) framework. We reveal an interesting saturated structure of the expected outage probability for which it eventually converges to a battery empty probability in high signal-to-noise power ratio (SNR). This phenomenon that links outage probability with EH capability is derived based on a monotonic and bounded differential structure of the long-term reward and a threshold structure of the optimal policy. Furthermore, a saturation-free condition on the outage performance is presented as well. Simulations confirm the theoretical analysis and the superiority of the proposed policy.