Norm minimizing estimation in the set-indexed stochastic processes

In this article, we present the norm minimizing estimation of a set indexed stochastic process Y={Y_A: A∈A} (by a linear and a nonlinear function of another set indexed stochastic process X={X_A: A∈A})of the future value Y_T in terms of limits of its past X_An and Y_An. In addition, we present the orthogonality principle. We prove with some assumptions that aset indexed norm minimizing estimation of Y is X if and only if Y−αX, X are orthogonal, when α = .