  In this article, we consider overlay space-air-ground integrated networks
(OSAGINs) where a low earth orbit (LEO) satellite communicates with ground
users (GUs) with the assistance of an energy-constrained coexisting air-to-air
(A2A) network. Particularly, a non-linear energy harvester with a hybrid SWIPT
utilizing both power-splitting and time-switching energy harvesting (EH)
techniques is employed at the aerial transmitter. Specifically, we take the
random locations of the satellite, ground and aerial receivers to investigate
the outage performance of both the satellite-to-ground and aerial networks
leveraging the stochastic tools. By taking into account the Shadowed-Rician
fading for satellite link, the Nakagami-\emph{m} for ground link, and the
Rician fading for aerial link, we derive analytical expressions for the outage
probability of these networks. For a comprehensive analysis of aerial network,
we consider both the perfect and imperfect successive interference cancellation
(SIC) scenarios. Through our analysis, we illustrate that, unlike linear EH,
the implementation of non-linear EH provides accurate figures for any target
rate, underscoring the significance of using non-linear EH models.
Additionally, the influence of key parameters is emphasized, providing
guidelines for the practical design of an energy-efficient as well as
spectrum-efficient future non-terrestrial networks. Monte Carlo simulations
validate the accuracy of our theoretical developments.
