On large increments of \(l^ p\)-valued Gaussian processes (Q1362877)

Let \((Y(t), t\geq 0)= (X_k(t), t\geq 0)^\infty_{k=1}\) be a sequence of independent Gaussian processes with \(EX_k(t)= 0\) and stationary increments \(\sigma^2_k(h)= E(X_k(t+h)- X_k(t))^2\), where \(\sigma_k(h)\) is assumed to be a nondecreasing continuous function for each \(k\geq 1\). Put \(\sigma(p,h)= \left(\sum^\infty_{k=1}\sigma^p_k(h)\right)^{1/p}\), \(p\geq 1\). If \(\sigma(p,h)<\infty\), then \(Y(t+h)- Y(t)\in\ell^p\), \(1\leq p<\infty\), almost surely for fixed \(t\) and \(h\). The author establishes the large increment results for \((Y(t), t\geq 0)\) with bounded \(\sigma(p,h)\). For example, \(\sigma(p,h)\) is bounded for the Ornstein-Uhlenbeck processes.