Algorithmic analogies to kamae-Weiss theorem on normal numbers

In this paper we study subsequences of random numbers. In Kamae (1973), selection functions that depend only on coordinates are studied, and their necessary and sufficient condition for the selected sequences to be normal numbers is given. In van Lambalgen (1987), an algorithmic analogy to the theorem is conjectured in terms of algorithmic randomness and Kolmogorov complexity. In this paper, we show different algorithmic analogies to the theorem.