A functional form for the lower Lipschitz condition for the stable subordinator (Q1104630)

Let X be a completely asymmetric stable process with \(\alpha\in (0,1)\). The author characterized the set \(K_{\alpha}\) consisting of all limit points of the set \[ D_{\alpha}(h)=\{(2B(\alpha))^{-(1- \alpha)/\alpha} h^{-1/\alpha}(2\log (h^{-1}))^{(1- \alpha)/\alpha}(X(s+(\cdot)h)-X(s)): 0\leq s\leq 1-h\} \] (h\(>0)\) as \(h\downarrow 0\). This result is the functional version of \textit{J. Hawkes'} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 17, 23-32 (1971; Zbl 0193.450)] result.