The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable

Ira W. Herbst (), Jesper Møller () and Anne Marie Svane ()
Additional contact information
Ira W. Herbst: University of Virginia
Jesper Møller: Aalborg University
Anne Marie Svane: Aalborg University

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 1, 1-13

Abstract: Abstract Let $$X=\sum _{k=1}^\infty X_k \beta ^{-k}$$ X = ∑ k = 1 ∞ X k β - k be the (greedy) base- $$\beta $$ β expansion of a continuous random variable X on the unit interval where $$\beta $$ β is the positive solution to $$\beta ^n = 1 + \beta + \cdots + \beta ^{n-1}$$ β n = 1 + β + ⋯ + β n - 1 for an integer $$n\geqslant 2$$ n ⩾ 2 (i.e., $$\beta $$ β is a generalization of the golden mean corresponding to $$n=2$$ n = 2 ). We study the asymptotic distribution and convergence rate of the scaled remainder $$\sum _{k=1}^\infty X_{m+k} \beta ^{-k}$$ ∑ k = 1 ∞ X m + k β - k when m tends to infinity.

Keywords: Asymptotic distribution; $$\beta $$ β -expansions; Invariant distribution; Pseudo golden mean; Scaled remainder; 37A50; 62E17; 60F25 (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s11009-025-10137-x Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:metcap:v:27:y:2025:i:1:d:10.1007_s11009-025-10137-x

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/11009

DOI: 10.1007/s11009-025-10137-x

Access Statistics for this article

Methodology and Computing in Applied Probability is currently edited by Joseph Glaz

More articles in Methodology and Computing in Applied Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().