  A weighted Shiryaev-Roberts change detection procedure is shown to
approximately minimize the expected delay to detection as well as higher
moments of the detection delay among all change-point detection procedures with
the given low maximal local probability of a false alarm within a window of a
fixed length in pointwise and minimax settings for general non-i.i.d. data
models and for the composite post-change hypothesis when the post-change
parameter is unknown. We establish very general conditions for the models under
which the weighted Shiryaev-Roberts procedure is asymptotically optimal. These
conditions are formulated in terms of the rate of convergence in the strong law
of large numbers for the log-likelihood ratios between the "change" and
"no-change" hypotheses, and we also provide sufficient conditions for a large
class of ergodic Markov processes. Examples, where these conditions hold, are
given.
