A Tetration Distribution
Abstract
We introduce a one-parameter family of continuous distributions on the unit interval whose cumulative distribution functions are given by the odd-height power towers x↑↑ n (n=1,3,5,…). The associated densities admit a simple recursive formula obtained by logarithmic differentiation. We prove that every such density is strictly positive on (0,1) and integrates to one, thereby establishing that the construction yields a genuine probability distribution. Numerical investigation reveals a characteristic peak--dip--rise shape: a sharp mode lying just below e−e for all examined heights, followed by a trough and a gradual recovery toward density value~1 at the right endpoint. We then identify the limiting distribution as the height tends to infinity through odd integers.
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BibTeX
@misc{thornton2026tetration,
title = {{A Tetration Distribution}},
author = {Micah Thornton},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03895},
eprint = {2611.03895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03895},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Tetration Distribution
AU - Micah Thornton
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03895
UR - https://arxiv.org/abs/2611.03895
AB - We introduce a one-parameter family of continuous distributions on the unit interval whose cumulative distribution functions are given by the odd-height power towers x↑↑ n (n=1,3,5,…). The associated densities admit a simple recursive formula obtained by logarithmic differentiation. We prove that every such density is strictly positive on (0,1) and integrates to one, thereby establishing that the construction yields a genuine probability distribution. Numerical investigation reveals a characteristic peak--dip--rise shape: a sharp mode lying just below e^−e for all examined heights, followed by a trough and a gradual recovery toward density value~1 at the right endpoint. We then identify the limiting distribution as the height tends to infinity through odd integers.
ER -