Convergence Results forr-Iterated Means of the Denominators of the Lüroth Series

Abstract

In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.

Convergence for weighted sums of Luroth type random variables

“…The theorem that follows is a convergence result for the sequence of random variables Y n defined above. Its proof is motivated by the proof of Theorem 3 in [10] which is a convergence result for the Luroth random variables.…”

“…In the case of the Lüroth sequence {D n } n≥1 , Theorems 1.1 and 1.2 have been extended in [10] by considering sums of the type…”

“…where {a k,n } n≥1 k≤n is an array of positive numbers satisfying a suitable set of assumptions. The particular class of {a k,n } n≥1 k≤n allows to apply the results of [10] to the r-iterated α-weighted means of {D n } n≥1 , i.e. to sequences built as follows: for α < 1, the r-iterated α-weighted means of D n are defined inductively by…”