Infinite random power towers

Abstract

We prove a probabilistic generalization of the classic result that infinite power towers, $c^{c^{⋰}}$, converge if and only if $c\in [e^{-e},e^{1∕e}]$. Given an i.i.d. sequence ${\{A_i\}}_{i\in \mathbb{N}}$, we find that convergence of the power tower $A_1^{A_2^{⋰}}$ is determined by the bounds of $A_1$’s support, $a=inf(\mathrm{supp}(A_1))$ and $b=sup(\mathrm{supp}(A_1))$. When $b\in [e^{-e},e^{1∕e}]$, $a<1<b$, or $a=0$, the power tower converges almost surely. When $b<e^{-e}$, 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 $b>e^{1∕e}$ 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 $A_1$ and the corresponding power tower $T=A_1^{A_2^{⋰}}$. For example, when $T\sim \mathrm{Unif}[0,1]$, then the corresponding distribution of $A_1$ is given by $UV$ where $U,V\sim \mathrm{Unif}[0,1]$ are independent. We generalize this example by showing that for $U\sim \mathrm{Unif}[\mathit{\alpha },\mathit{\beta }]$ and $r\in \mathbb{R}$, there exists an i.i.d. sequence ${\{A_i\}}_{i\in \mathbb{N}}$ such that $U^r\stackrel{d}{=}{A}_1^{A_2^{⋰}}$ if and only if $r\in [0,\frac{1}{1+log\mathit{\beta }}]$.