Open Access
2024 Infinite random power towers
Mark Dalthorp
Author Affiliations +
Electron. J. Probab. 29: 1-27 (2024). DOI: 10.1214/24-EJP1074


We prove a probabilistic generalization of the classic result that infinite power towers, cc, converge if and only if c[ee,e1e]. Given an i.i.d. sequence {Ai}iN, we find that convergence of the power tower A1A2 is determined by the bounds of A1’s support, a=inf(supp(A1)) and b=sup(supp(A1)). When b[ee,e1e], a<1<b, or a=0, the power tower converges almost surely. When b<ee, we define a special function B such that almost sure convergence is equivalent to a<B(b). Only in the case when a=1 and b>e1e are the values of a and b insufficient to determine convergence. We show a rather complicated necessary and sufficient condition for convergence when a=1 and b is finite.

We also briefly discuss the relationship between the distribution of A1 and the corresponding power tower T=A1A2. For example, when TUnif[0,1], then the corresponding distribution of A1 is given by UV where U,VUnif[0,1] are independent. We generalize this example by showing that for UUnif[α,β] and rR, there exists an i.i.d. sequence {Ai}iN such that Ur=dA1A2 if and only if r[0,11+logβ].


I would like to thank Laurent Saloff-Coste, Persi Diaconis, Dan Dalthorp, and Lisa Madsen for their support and helpful comments. I would especially like to thank Dan Dalthorp for his work on the figures. I would also like to thank the anonymous reviewer for many helpful suggestions, especially on how to improve the introduction.


Download Citation

Mark Dalthorp. "Infinite random power towers." Electron. J. Probab. 29 1 - 27, 2024.


Received: 25 October 2022; Accepted: 3 January 2024; Published: 2024
First available in Project Euclid: 6 February 2024

Digital Object Identifier: 10.1214/24-EJP1074

Primary: 60J05

Keywords: Markov process , power towers , Random dynamical system , tetration

Vol.29 • 2024
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