  We study the convergence of recursive regularized learning algorithms in the
reproducing kernel Hilbert space (RKHS) with dependent and non-stationary
online data streams. Firstly, we study the mean square asymptotic stability of
a class of random difference equations in RKHS, whose non-homogeneous terms are
martingale difference sequences dependent on the homogeneous ones. Secondly, we
introduce the concept of random Tikhonov regularization path, and show that if
the regularization path is slowly time-varying in some sense, then the output
of the algorithm is consistent with the regularization path in mean square.
Furthermore, if the data streams also satisfy the RKHS persistence of
excitation condition, i.e. there exists a fixed length of time period, such
that the conditional expectation of the operators induced by the input data
accumulated over every time period has a uniformly strictly positive compact
lower bound in the sense of the operator order with respect to time, then the
output of the algorithm is consistent with the unknown function in mean square.
Finally, for the case with independent and non-identically distributed data
streams, the algorithm achieves the mean square consistency provided the
marginal probability measures induced by the input data are slowly time-varying
and the average measure over each fixed-length time period has a uniformly
strictly positive lower bound.
