Approximation in Probability of Tensor Product-Type Random Fields of Increasing Parametric Dimension

We consider the sequence of Gaussian tensor product-type random fields X_d given by a multiparametric Karhunen-Loéve expansion

$$ {X}_d(t)={\displaystyle \sum_{k\in {\mathrm{\mathbb{N}}}^d}{\displaystyle \prod_{l=1}^d{\uplambda}_{k_l}^{1/2}{\xi}_k}{\displaystyle \prod_{l=1}^d{\psi}_{k_l}\left({t}_l\right),\kern0.36em t\in {\left[0,1\right]}^d},} $$

where (ξk)_k∈ℕ ^dare standard Gaussian random variables and (λ_i)_i∈ℕ and (ψ_i)_i∈ℕ are the eigenvalues and eigenfunctions of the covariance operator of the process X₁. We approximate X_d by finite sums X ⁽ⁿ⁾ _d of the series using L₂([0, 1]^d)-norm ‖ ⋅ ‖ _2,d and study the exact asymptotics of the probabilistic approximation complexity

$$ {n}_d^{\mathrm{pr}}\left(\varepsilon, \delta \right):= \min \left\{n\in \mathbb{N}:\mathrm{\mathbb{P}}\left({\left\Vert {X}_d-{X}_d^{(n)}\right\Vert}_{2,d}^2>{\varepsilon}^2\mathbb{E}{\left\Vert {X}_d\right\Vert}_{2,d}^2\right)\le \delta \right\} $$

in the case where the error threshold ε ∈ (0, 1) is fixed, the parametric dimension d→∞, and the confidence level δ = δ_d,ε may depend on ε and d. We show that under some conditions on (λ_i)_i∈ℕ, the probabilistic complexity is asymptotically equivalent to the average approximation complexity

$$ {n}_d^{\mathrm{avg}}\left(\varepsilon \right):= \min \left\{n\in \mathbb{N}:\mathbb{E}{\left\Vert {X}_d-{X}_d^{(n)}\right\Vert}_{2,d}^2\le {\varepsilon}^2\mathbb{E}{\left\Vert {X}_d\right\Vert}_{2,d}^2\right\} $$

The result depends on the existence of a lattice structure of the sequence (λ_i)_i∈ℕ. Bibliography: 10 titles.