Asymptotic independence of regenerative processes with dependent cycles

We identify general conditions under which regenerative processes with dependent cycles and cycle lengths are asymptotically independent. The result is applied to various models. In particular, independent L\'evy processes with dependent secondary jumps at the origin (e.g., workloads of parallel M/G/1 queues with server vacations), the asymptotic performance of real-time status systems with multiple correlated sources measured by the stationary probability of an updated system and asymptotic results for clearing processes with dependent arrivals of inputs and clearings.