The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable
Let X = ∑ k = 1 ∞ X k β - k be the (greedy) base- β expansion of a continuous random variable X on the unit interval where β is the positive solution to β n = 1 + β + ⋯ + β n - 1 for an integer n ⩾ 2 (i.e., β is a generalization of the golden mean corresponding to n = 2 ). We study the asymptotic di...
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| Veröffentlicht in: | Methodology and computing in applied probability Jg. 27; H. 1; S. 10 |
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.03.2025
Springer Nature B.V |
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| ISSN: | 1387-5841, 1573-7713 |
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| Abstract | Let
X
=
∑
k
=
1
∞
X
k
β
-
k
be the (greedy) base-
β
expansion of a continuous random variable
X
on the unit interval where
β
is the positive solution to
β
n
=
1
+
β
+
⋯
+
β
n
-
1
for an integer
n
⩾
2
(i.e.,
β
is a generalization of the golden mean corresponding to
n
=
2
). We study the asymptotic distribution and convergence rate of the scaled remainder
∑
k
=
1
∞
X
m
+
k
β
-
k
when
m
tends to infinity. |
|---|---|
| AbstractList | Let X=∑k=1∞Xkβ-k be the (greedy) base-β expansion of a continuous random variable X on the unit interval where β is the positive solution to βn=1+β+⋯+βn-1 for an integer n⩾2 (i.e., β is a generalization of the golden mean corresponding to n=2). We study the asymptotic distribution and convergence rate of the scaled remainder ∑k=1∞Xm+kβ-k when m tends to infinity. Let X = ∑ k = 1 ∞ X k β - k be the (greedy) base- β expansion of a continuous random variable X on the unit interval where β is the positive solution to β n = 1 + β + ⋯ + β n - 1 for an integer n ⩾ 2 (i.e., β is a generalization of the golden mean corresponding to n = 2 ). We study the asymptotic distribution and convergence rate of the scaled remainder ∑ k = 1 ∞ X m + k β - k when m tends to infinity. 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. |
| ArticleNumber | 10 |
| Author | Møller, Jesper Svane, Anne Marie Herbst, Ira W. |
| Author_xml | – sequence: 1 givenname: Ira W. surname: Herbst fullname: Herbst, Ira W. organization: Department of Mathematics, University of Virginia – sequence: 2 givenname: Jesper surname: Møller fullname: Møller, Jesper email: jm@math.aau.dk organization: Department of Mathematical Sciences, Aalborg University – sequence: 3 givenname: Anne Marie surname: Svane fullname: Svane, Anne Marie organization: Department of Mathematical Sciences, Aalborg University |
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| Keywords | 37A50 60F25 Scaled remainder Asymptotic distribution Pseudo golden mean expansions Invariant distribution 62E17 |
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| References | K Dajani (10137_CR1) 2007; 7 W Parry (10137_CR5) 1960; 11 10137_CR3 10137_CR2 A Tysbakov (10137_CR8) 2009 EP Miles (10137_CR4) 1960; 67 F Schweiger (10137_CR7) 1995 A Rényi (10137_CR6) 1957; 8 |
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| Snippet | Let
X
=
∑
k
=
1
∞
X
k
β
-
k
be the (greedy) base-
β
expansion of a continuous random variable
X
on the unit interval where
β
is the positive solution to
β
n
=... 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... Let X=∑k=1∞Xkβ-k be the (greedy) base-β expansion of a continuous random variable X on the unit interval where β is the positive solution to βn=1+β+⋯+βn-1 for... |
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| SubjectTerms | Asymptotic properties Asymptotic series Business and Management Continuity (mathematics) Economics Electrical Engineering Life Sciences Mathematics and Statistics Random variables Statistics |
| Title | The Asymptotic Distribution of the Scaled Remainder for Pseudo Golden Ratio Expansions of a Continuous Random Variable |
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