= a > 0. Then, as

Proof Assertion (i) is the Erdos-Darling theorem, for the proof see, e.g., Shorack and Wellner (1986). The result in Eq. (9.3) is proved in Chen (1988).

Pn)(>l of Theorem 2.1. This is the case with the fixed change. For simplicity assume that 1J = 1/2. The estimator JJILS(l/2) can be equivalently defined as