Universal portfolios with short sales and margin

In the horizon-free universal portfolio and the fixed horizon minimax regret portfolio, the only allowed portfolios are those that allocate a nonnegative proportion of wealth to each of the assets. It was shown that a Dirichlet(1/2)-weighting of constant rebalanced portfolios was asymptotically minimax regret in growth rate and therefore universal. We extend these results to bounded stocks and portfolios allowing margin and short sales.