A functional law of the iterated logarithm for kernel-type estimators of the tail index

Under von Mises's condition, the kernel tail index estimators of Csörgő et al. (Ann. Statist. 13 (1985) 1050) have been generalized by Groeneboon et al. (Ann. Statist. 31 (2003) 1956) for all real tail indices. Weak consistency and asymptotic normality of their kernel estimators have been established. In this paper, we present a characterization of the almost sure behavior of these estimators and we show their strong consistency.