Infinite Random Power Towers

Abstract

We prove a probabilistic generalization of the classic result that infinite power towers, $c^{c^{\dots}}$, 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 $A_1^{A_2^{\dots}}$ 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 $T=A_1^{A_2^{\dots}}$. 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 $U\sim\mathrm{Unif}[\alpha,\beta]$ and $r\in\mathbb R$, there exists an i.i.d. sequence $\{A_i\}_{i\in\mathbb N}$ such that $U^r \stackrel{d}{=} A_1^{A_2^{\dots}}$ if and only if $r\in[0, \frac1{1+\log \beta}]$.}