Approximation complexity of tensor product-type random fields with heavy spectrum
Consider a sequence of Gaussian tensor product-type random fields, given by d d ^Й=ЕП <sup>А</sup>Г^П^М- te[o,i] <sup>d</sup>, deN, 164 where (Ai) <sub>i6</sub>fj and (V'Ojgfj are all positive eigenvalues and eigenfunctions of the covariance operator of the process X\, (£fc) <sub>fc6</sub>£j are standard Gaussian random variables, and N is a subset of natural numbers. For any d G IN, sample paths of Xd almost surely belong to L2QO, l] <sup>d</sup>) supplied with the norm || • ||2,rf. The d tuples {\к,"Фк) '·= (Пг=1 -^п Пг=1 ^Ph), k € Ы, are eigenpairs of the covariance operator of Xd. We approximate Xd, d G IN, with finite sum X <sub>d</sub>" corresponding to n maximal eigenvalues Xk, к G N. We investigate the logarithmic asymptotics of the average approximation complexity пГМ := min{n G N : E \\X <sub>d</sub> - X^\\l <sub>td</sub> < e <sup>2</sup> E ||X <sub>d</sub>||^} and probabilistic approximation complexity п <sup>р</sup>/(е,6) := minjn G N : P (\\X <sub>d</sub> - Х <sub>л</sub> <sup>п)</sup>\\ <sup>2</sup> <sub>24</sub> > e <sup>2</sup> E \\Х <sub>Л</sub>\\1 <sub>4</sub>) < <j}, when the parametric dimension d —> 00, the error threshold e G (0,1) is fixed, and the confidence level S = 5d,£ may go to zero. Complementing recent results of M. A. Lifshits and E. V. Tulyakova we consider the unexplored case, when the sequence (Ai) <sub>i6</sub>fj is regularly and slowly decreasing to zero.