Evolution of shares in a proof-of-stake cryptocurrency

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, 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