Strong limit theorems for large and small increments of \(\ell^ p\)- valued Gaussian processes
Let \(\{Y(t), t \geq 0\}=\{X_ k(t), t \geq 0\}^ \infty_{k=1}\) be a sequence of independent Gaussian processes with \(\mathbb{E} X_ k(t)=0\) and stationary increments \(\sigma^ 2_ k (h)=\mathbb{E}(X_ k (t+h)-X_ k(t))^ 2\). The authors are interested in the limit behaviour of the increments of the process \(Y(t)\). In fact they consider \[ I(T)=\sup_{0 \leq t \leq T} \sup_{0 \leq s \leq a_ T} \| Y(t+s)-Y(t) \|_{\ell^ p}, \] where \(a_ T\) is a positive continuous function. The main results claim that with a suitable normalizing factor \(\gamma_ T\) the almost sure limit of the process \(\gamma_ TI(T)\) is 1 provided that some regularity conditions hold true.
scientific article; zbMATH DE number 528628