Continuous publication of statistics over user-generated streams can provide timely data monitoring and analysis for various applications. Nonetheless, such published statistics may reveal the details of individuals’ sensitive status or activities. To guarantee the privacy for event occurrences in data streams, based on the known privacy standard of $$\varepsilon $$ -differential privacy, w-event privacy has been proposed to hide multiple events occurring at continuous time instances. Nonetheless, the too strict requirement of w-event privacy makes it hard to achieve effective privacy protection with high data utility in many real-world scenarios. To this end, in this paper we propose a novel notion of average w-event privacy and the first Lyapunov optimization-based privacy-preserving scheme on infinite streams, aiming to obtain higher data utility while satisfying a relatively stable privacy guarantee for whole streams. In particular, we first formulate both our proposed privacy definition and the utility loss function of statistics publishing in a stream setting. We then design a Lyapunov optimization-based scheme with a detailed algorithm to maximize the publishing data utility under the requirement of our privacy notion. Finally, we conduct extensive experiments on both synthetic and real-world datasets to confirm the effectiveness of our scheme.