Infinite Random Power Towers

TL;DR

This paper extends the classic convergence criteria of infinite power towers to probabilistic settings, analyzing how the support bounds of i.i.d. sequences influence almost sure convergence and exploring distributional relationships.

Contribution

It provides a probabilistic generalization of power tower convergence conditions, introduces a new function for bounds, and characterizes distributional equivalences in this context.

Findings

Convergence depends on support bounds of the initial random variable.

A new function B determines convergence when the upper support bound is below e^{-e}.

Distributional relationships between initial variables and power towers are characterized.

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 $r\in[0, \frac1{1+\log \beta}]$.}