Reproduction in these stars takes two forms sexual reproduction and asexual reproduction by splitting or fission.
There are a couple of things to realize about them that will influence how you interpret their reproduction. Small animals are generally pretty cryptic and hard to see (if they aren't something bigger will eat them). So even if your stars are reproducing it may take quite a while before there are enough of them around to notice. Secondly if they are sexually reproducing, it will depend on whether or not they have a planktonic larval stage. If they do, you may never see successful reproduction. If they don't (many brittles brood their young) you will probably see reproduction, but only after a year or so.
There are numerous species of these small brittle stars that arrive in aquarists tanks, and most of them seem to do pretty well. I generally assume that they are reproducing sexually; asexual reproduction to produce a big population takes a lot of time. In my tanks it seemed to take about a year after the introduction of some before they were abundant. Now, I could generally find a few here and there before that, but it took quite a while until they were common.
Once they are doing well, they should be around for a while. Like all echinoderms they have no old age or senescence and are, in effect, immortal. So unless they starve, get eaten, or get diseased they have the potential of lasting a very long time.
Cheers, Ron