Convergence Rates for the Optimal Values of Allocation Processes

The optimal allocation process is of the following type. A sequence of functions (production functions) f₁, f₂, …, is given, each mapping R₊ⁿ into [0, ∞). Along with it a vector q > 0 in R₊ⁿ (vector of resources) is provided. For each integer k we consider the problem

Here R₊ⁿ is the nonnegative orthant of the n-dimensional Euclidean space. Inequalities between vectors are taken coordinatewise.

In this paper we consider the case where the functions f_i are subject to randomness. Specifically, they are generated as independent random drawings from a common distribution. In particular f_i = f_i(x, ω) with ω being in a probability space, and the solutions x_i might depend on u, as well as on k and q. The detailed technical conditions are displayed in §2. We are interested in limit properties, as k → ∞, of the optimal values of the allocation process.