Sample function behavior of increasing processes of class \(L\) (Q1912570)

A stochastic process \(\{X(t), t \geq 0\}\) is said to be a process of class \(L\) with exponent \(H\) if (i) \(X(t)\) is self-similar, that is, for every \(c > 0\), \(\{X (ct)\}\) and \(\{c^H X(t)\}\) have the same finite-dimensional distributions, (ii) \(\{X(t)\}\) has independent increments and (iii) \(X(t)\) is right continuous for \(t \geq 0\) with left limits for \(t > 0\) almost surely. Suppose the process \(\{X(t)\}\) is increasing and of class \(L\) and let \(h(t)\) be an increasing positive function on \((0, \infty)\) with \(h(0+) = 0\) and \(h (\infty) = \infty\). The author gives necessary and sufficient conditions for the existence of a function \(h(t)\) such that \(\lim_{t \to \infty} \sup {X(t) \over h(t)} = C\) or \(\lim_{t \to \infty} \inf {X(t) \over h(t)} = c\) where \(C = c = 1\). He explicitly constructs such \(h(t)\) in case \(h(t)\) exists with \(c = 1\) or \(C = 1\). A criterion to classify functions \(h(t)\) with \(c = 0\) or \(C = 0\) or with \(c = \infty\) or \(C = \infty\) is also given in case \(h(t)\) does not exist with \(c = 1\) or \(C = 1\). Results extend earlier work of \textit{K.-i. Sato} [ibid. 89, No. 3, 285-300 (1991; Zbl 0725.60034)].