Weak randomness and Kamae's theorem on normal numbers

A function from sequences to their subsequences is called selection function. A selection function is called admissible (with respect to normal numbers) if for all normal numbers, their subsequences obtained by the selection function are normal numbers. In Kamae (1973) selection functions that are not depend on sequences (depend only on coordinates) are studied, and their necessary and sufficient condition for admissibility is given. In this paper we introduce a notion of weak randomness and study an algorithmic analogy to the Kamae's theorem.