Do the rich always get richer by investing in a cryptocurrency for which new coins are issued according to a proof-of-stake (PoS) protocol? We answer this question in the negative: Without trading, the investor shares in the cryptocurrency are martingales that converge to a well-defined limiting distribution and, hence, are stable in the long run. This result is robust to allowing trading when investors are risk neutral. Then, investors have no incentive to accumulate coins and gamble on the PoS protocol but weakly prefer not to trade. This paper was accepted by Kay Giesecke, finance.