  We establish a framework of distributed random inverse problems over network
graphs with online measurements, and propose a decentralized online learning
algorithm. This unifies the distributed parameter estimation in Hilbert spaces
and the least mean square problem in reproducing kernel Hilbert spaces
(RKHS-LMS). We transform the convergence of the algorithm into the asymptotic
stability of a class of inhomogeneous random difference equations in Hilbert
spaces with L2-bounded martingale difference terms and develop the L2
-asymptotic stability theory in Hilbert spaces. It is shown that if the network
graph is connected and the sequence of forward operators satisfies the
infinite-dimensional spatio-temporal persistence of excitation condition, then
the estimates of all nodes are mean square and almost surely strongly
consistent. Moreover, we propose a decentralized online learning algorithm in
RKHS based on non-stationary and non-independent online data streams, and prove
that the algorithm is mean square and almost surely strongly consistent if the
operators induced by the random input data satisfy the infinite-dimensional
spatio-temporal persistence of excitation condition.
