Portfolio choice based on the empirical distribution

Let X ∈ R denote a random stock market return vector, where Xj is the value of a one unit investment in stock j at the end of the trading day. We require that Xj ≥ 0 for j = 1, 2, . . . , m, that is, an investor cannot lose more than the invested capital. Let b, bj ≥ 0, ∑m j=1 bj = 1, denote a portfolio, that is, an allocation of investor’s capital across the investment alternatives. Let B denote the set of such portfolios. Thus bj is the proportion of current capital invested in stock j. The resulting wealth is S = ∑m j=1 bjXj = bX. This is the wealth resulting from a unit investment allocated to the m stocks according to portfolio b. If the current capital is reallocated according to portfolio bi at time i in repeated investments against stock vectors X1, X2, . . . then the wealth Sn at time n is given by