Metrical properties of some random variables connected with the continued fraction expansion

AbstractFor certain random variables X1,X2,… which can be expressed by means of the natural extension of the Gauss transformation of a continued fraction, we prove that for all z ϵ R and for all probability measures on [0, 1] absolutely continuous with respect to the Lebesgue measure, the probabilities P(Xn ≤ z) converge when n → ∞. The limit can be determined explicitly