For a wide class of Feller processes the limit behaviour as \(t\to 0\) and \(t\to\infty\) is investigated. For that certain indices are introduced, in case of Lévy processes coinciding with the Blumenthal-Getoor indices [see R. M. Blumenthal and R. K. Getoor, J. Math. Mech. 10, 493-516 (1961; Zbl 0097.33703)]. The results extend the Lévy processes case considered by W. Pruitt [Ann. Probab. 9, 948-956 (1981; Zbl 0477.60033)].