Self-intersections and local nondeterminism of Gaussian processes
Let X(t), \(t\geq 0\), be a vector Gaussian process in \({\mathbb{R}}^ m\) whose components are i.i.d. copies of a real Gaussian process \(\xi\) (t) with stationary increments. For r closed disjoint intervals \(I_ 1,...,I_ r\) define the process \((X(t_ 2)-X(t_ 1),...,X(t_ r)-X(t_{r-1})),\) \(t_ i\in I_ i\), \(i=1,...,r\), from \(I_ 1\times...\times I_ r\) into \(R^{m(r-1)}\). Under conditions on the spectral distribution of \(\xi\) (t) it is shown that the local time of this process is jointly continuous. This is used as a key step to obtain estimations from above and below of the Hausdorff dimension of the r-multiple self-intersection set \(\{(t_ 1,...,t_ r):\;t_ 1<t_ 2<...<t_ r,\quad X(t_ 1)=...=X(t_ r)\}.\) The main tool in all these results is the concept of local nondeterminism introduced and developed by the author [Stochastic Processes Appl. 27, No.1, 73-84 (1987; Zbl 0633.60055)]. An extension of the results to random Fourier series with coefficients which are symmetric stable random variables is also given.
scientific article; zbMATH DE number 4203369