For a random field \(B(t_1, \ldots , t_d), t_i \in [0, T_i]\) with a reproducing kernel H and any function \(f\in H\) , we define the approximation error as \({\mathcal {E}}_{\bar{T}}(f, B) =\int \limits _0^{T_1}\ldots \int \limits _0^{T_d} (f(\bar{t}) - B(\bar{t}))^2 \, d\bar{t} + \lambda ^2 \Vert f\Vert _{H}^2.\) The first term defines the proximity of f to B and the second one defines the energy efficiency of f. The coefficient \(\lambda \) allows balancing between these two parts. The best approximation is \(f_{\text {opt}} = \underset{f\in H}{\arg \min }\ \,{\mathcal {E}}_{\bar{T}}(f, B).\) We prove the law of large numbers on the convergence of the optimal approximation error of a Brownian sheet in \(L^2\) and almost surely. Bibliography: 9 titles.
