Age of Information Process under Strongly Mixing Communication -- Moment Bound, Mixing Rate and Strong Law

The decentralized nature of multi-agent systems requires continuous data exchange to achieve global objectives. In such scenarios, Age of Information (AoI) has become an important metric of the freshness of exchanged data due to the error-proneness and delays of communication systems. Communication systems usually possess dependencies: the process describing the success or failure of communication is highly correlated when these attempts are ``close'' in some domain (e.g. in time, frequency, space or code as in wireless communication) and is, in general, non-stationary. To study AoI in such scenarios, we consider an abstract event-based AoI process $\Delta(n)$, expressing time since the last update: If, at time $n$, a monitoring node receives a status update from a source node (event $A(n-1)$ occurs), then $\Delta(n)$ is reset to one; otherwise, $\Delta(n)$ grows linearly in time. This AoI process can thus be viewed as a special random walk with resets. The event process $A(n)$ may be nonstationary and we merely assume that its temporal dependencies decay sufficiently, described by $\alpha$-mixing. We calculate moment bounds for the resulting AoI process as a function of the mixing rate of $A(n)$. Furthermore, we prove that the AoI process $\Delta(n)$ is itself $\alpha$-mixing from which we conclude a strong law of large numbers for $\Delta(n)$. These results are new, since AoI processes have not been studied so far in this general strongly mixing setting. This opens up future work on renewal processes with non-independent interarrival times.