Yes, pistils turning brown is a normal process. At the base of each pair of pistils and located inside the female calyx, is an ovule, that when fertilized, will grow into a seed. The white pistils indicate that the ovule at their base is still viable and fresh enough for successful pollination and fertilization. As the female flower ages, the ovules, which have also aged, have become less viable for successful fertilization, as indicated by the brown colour of the pistils. This is an adaptive response so that any pollen is more likely to land and be "trapped" on/by a plump and "juicy" white pistil, where fertilization will favour a fresh and viable ovule, rather than landing on/being trapped by a dried up brown pistil with an aged and less viable ovule at their base. This ensure pollen is not "wasted" by landing on pistils that have less viable ovules at their base and the pollen is most likely to be "trapped" / "caught" by the plump "juicier" white pistils, that have fresh and viable ovules at their base.
Fertilization, by the way, takes about 20-30 minutes from when the pollen grain(s) land on the pistil and the pollen tube or "sperm" grows down the pistil to reach the female ovule at the base of the pistils and for fertilization to occur and start the process of a seed developing. Fascinating stuff!
Hope this helps,.......
Organoman.