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Non-uniform coverage is a central challenge in UAV-assisted wireless networks, yet most existing methods either assume uniform demand or depend on precise user locations. This letter proposes a distribution-centric trajectory design that uses only statistical demand maps to generate ergodic UAV paths, guaranteeing that time-averaged presence converges to the prescribed spatial distribution. The scheme modulates UAV speed via a time-warping function for smooth transitions between high- and low-density regions, uses continuous angular adjustments, and provides rigorous convergence and complexity analyses. Simulations demonstrate that our method improves coverage fairness and distribution matching by 4.7× and 5.8×, respectively, relative to a deep-reinforcement-learning baseline, while also achieving 1.1× and 1.5× gains over an optimaltransport approach. It reduces outage probability by up to 65 % compared with uniform coverage. Experiments with the Telecom Italia Milano dataset show that the resulting coverage closely matches real urban demand patterns under wind and sensor disturbances.