Energy efficient approximations of Brownian Sheet

N. A. Karagodin

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 141-155

Résumé

For a random field $B(t_1, \ldots, t_d), t_i \in [0, T_i]$ with a reproducing kernel $H$ and any function $f\in H$ define approximation error as $$ \mathcal{E}_{\bar T}(f, B) =\int\limits_0^{T_1}\ldots \int\limits_0^{T_d} (f(\bar t) - B(\bar t))^2 d\bar t + \lambda^2 \|f\|_{H}^2. $$ The first term defines proximity of $f$ to $B$ and the second one defines energy efficiency of $f$. Coefficient $\lambda$ allows to balance between these two parts. The best approximation is $$ f_{\mathrm{opt}} = \underset{f\in H}{\arg\min}\, \mathcal{E}_{\bar T}(f, B). $$ We prove the law of large numbers on convergence of optimal approximation error of Brownian Sheet in $L^2$ and almost surely.