In the first part we consider partial sums where the positve integers are defined by the expansion of a real number which is chosen according to some probability measure . Under some regularity conditions imposed on the strictly monotone function we prove that the suitably normalized sum has an stable limit distribution ( ) and derive uniform bounds of the approximation error provided possesses a strictly positive, Lipschitz continuous Lebesgue density. Special emphasis is put on the case for which coincides with the power sum of the partial quotients and for (where ) of the continuous fraction expansion . In the second part we present large deviation relations (in the sense of H. Cramér) for the sequences and , where ( resp. ) is the denominator (resp. denumerator) of the th approximant of . The proofs of the results are based on a method developed in .
|||HEINRICH, L. (1996) Mixing properties and central limit theorem for a class of non-identical piecewise monotonic transformations. Math. Nachr. 182, 185 - 214.|