Before revision:  We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a diverse and sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run . This class of `relative arbitrage' portfolios were discovered by Fernholz and are called functionally generated portfolios.
After revision:  We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a diverse and sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made.