Large deviations of empirical estimates in the stochastic programming problem for the homogeneous random field with a discrete parameter (Q2058688)

Let \(\xi=\xi(t_1,t_2)\) denote a homogeneous random random field in a certain metric space \(Y\), where \(t_1,t_2 \geq 0\) are nonnegative integer parameters. Consider then the mean value function \(F(x;t_1,t_2):= Ef(x,\xi(t_1,t_2))\), where \(f=f(x,y)\) denotes a continuous function on \(X \times Y\) with a compact domain \(X\) for the decision variable \(x\). The problem is to minimize F=\(F(x;0,0)\) on \(X\). The objective function \(F\) is then approximated by \(F_{T_1T_2}(x) := \frac{1}{T_1T_2} \sum\limits_{t_1=1}^{T_1} \sum\limits_{t_2=1}^{T_2}f(x,\xi(t_1,t_2))\). Let \(x(T_1,T_2)\) denote an optimal solution of the approximate problem. It is known that the optimal solution, the minimum value, resp., of the approximate problem converge with probability \(1\) to an optimal solution, the minimum value, resp., of the original problem if \(T_1,T_2 \rightarrow +\infty\). Under certain assumptions, estimations of the large deviation between the optimal solutions, optimal values, resp., of the original, the approximate problem, resp., are derived.