  Iterative distributed optimization algorithms involve multiple agents that
communicate with each other, over time, in order to minimize/maximize a global
objective. In the presence of unreliable communication networks, the
Age-of-Information (AoI), which measures the freshness of data received, may be
large and hence hinder algorithmic convergence. In this paper, we study the
convergence of general distributed gradient-based optimization algorithms in
the presence of communication that neither happens periodically nor at
stochastically independent points in time. We show that convergence is
guaranteed provided the random variables associated with the AoI processes are
stochastically dominated by a random variable with finite first moment. This
improves on previous requirements of boundedness of more than the first moment.
We then introduce stochastically strongly connected (SSC) networks, a new
stochastic form of strong connectedness for time-varying networks. We show: If
for any $p \ge0$ the processes that describe the success of communication
between agents in a SSC network are $\alpha$-mixing with $n^{p-1}\alpha(n)$
summable, then the associated AoI processes are stochastically dominated by a
random variable with finite $p$-th moment. In combination with our first
contribution, this implies that distributed stochastic gradient descend
converges in the presence of AoI, if $\alpha(n)$ is summable.
