Before revision:  We provide necessary and sufficient conditions on the assignment on portfolios to guarantee that, for all such evolutions in a  sufficiently volatile market, the actively traded portfolio outperforms the buy-and-hold portfolio in the long run.
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.