  We introduce an observation-matrix-based framework for fully asynchronous
online Federated Learning (FL) with adversaries. In this work, we demonstrate
its effectiveness in estimating the mean of a random vector. Our main result is
that the proposed algorithm almost surely converges to the desired mean $\mu.$
This makes ours the first asynchronous FL method to have an a.s. convergence
guarantee in the presence of adversaries. We derive this convergence using a
novel differential-inclusion-based two-timescale analysis. Two other highlights
of our proof include (a) the use of a novel Lyapunov function to show that
$\mu$ is the unique global attractor for our algorithm's limiting dynamics, and
(b) the use of martingale and stopping-time theory to show that our algorithm's
iterates are almost surely bounded.
