The Rauzy induction is a dynamical system acting on the space of interval exchange transformations which is introduced by Rauzy in  and used by Veech to give an affirmative answer to the Keane Conjecture in  and . The results in  enable us to construct induced transformations and jump transformations to the various sets. In this article the dynamical systems obtained by composing some of those transformations are called renormalized Rauzy inductions. Note that the two fold iteration of continued fraction transformation can be regarded as a classical example of renormalized Rauzy inductions via appropriate conjugacy. Our present goal is to establish the same kinds of central limit theorems as obtained in  for a class of renormalized Rauzy inductions.