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 АГ^П^М- te[o,i] d, deN, 164 where (Ai) i6fj and (V'Ojgfj are all positive eigenvalues and eigenfunctions of the covariance operator of the process X\, (£fc) fc6£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] d) 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 d" 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 d - X^\\l td 2 E ||X d||^} and probabilistic approximation complexity п р/(е,6) := minjn G N : P (\\X d - Х л п)\\ 2 24 > e 2 E \\Х Л\\1 4)  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) i6fj is regularly and slowly decreasing to zero.