The persistency of excitation property is crucial for the stability analysis of parameter identification algorithms and adaptive control loops. In this paper, we propose a relaxed definition of persistency of excitation and we use it to establish uniform global asymptotic stability (UGAS) for a large class of nonlinear, time-varying systems. Our relaxed definition is conceptually equivalent to the definition introduced in [25], but is easier to verify. It is useful for stability analysis of nonlinear systems that arise, for example, in nonlinear adaptive control, stabilization of nonholonomic systems, and tracking control. Our proof of UGAS relies on some integral characterizations of UGAS recently established in [45]. These characterizations also streamline the proof of UGAS in the presence of (uniform) classical persistency of excitation.