Gaussian chaos and functional laws of the iterated logarithm for Itô- Wiener integrals
The authors study random samples \(X_ 1, X_ 2,\dots\) of centered \(B\)- valued Gaussian chaos \(X= \sum_{i,j\geq 1} a_{ij} (g_ i g_ j- \delta_{ij})\), where \(a_{ij}\) are nonrandom elements of a separable Banach space \(B\) and \(g_ i\), \(i= 1,2,\dots\), is an i.i.d. sequence of stationary Gaussian random variables. It turns out that the random set \(E_ n= \{X_ 1,\dots, X_ n\}/2 \log n\) converges a.s. to a non- random compact set \(\Sigma\subset B\). The authors provide rates of convergence in a Hausdorff metric. Applications are given to multiple Wiener-Itô integrals, where \(B= C[0,T]\).
scientific article; zbMATH DE number 566211