论文标题
无限随机电塔
Infinite Random Power Towers
论文作者
论文摘要
我们证明了经典结果的概率概括，即Infinite Power Towers $ C^{C^{c^{\ dots}} $，当且仅当$ c \ in [e^{ - e}，e^{1/e}] $时收敛。给定I.I.D.序列$ \ {a_i \} _ {i \ in \ mathbb n} $，我们发现动力塔的收敛$ a_1^{a_2^{\ dots}} $由$ a_1 $ support的界限确定$ b = \ sup（\ mathrm {spep}（a_1））$。当$ b \ in [e^{ - e}，e^{1/e}] $，$ a <1 <b $或$ a = 0 $时，电源塔几乎肯定会收敛。当$ b <e^{ - e} $时，我们定义一个特殊的函数$ b $，以便几乎可以肯定的收敛等于$ a <b（b）$。仅在$ a = 1 $和$ b> e^{1/e} $的情况下，$ a $和$ b $的值不足以确定收敛。当$ a = 1 $和$ b $有限时，我们显示出相当复杂的收敛条件。 我们还简要讨论了$ a_1 $的分布与相应的电源塔$ t = a_1^{a_2^{\ dots}} $之间的关系。例如，当$ t \ sim \ mathrm {unif} [0,1] $时，$ a_1 $的相应分布由$ uv $给出，其中$ u，v \ sim \ sim \ mathrm {unif} [0,1] $是独立的。我们通过证明$ \ sim \ mathrm {unif} [α，β] $和$ r \ in \ mathbb r $中的$ r \ r $来概括这个示例。序列$ \ {a_i \} _ {i \ in \ mathbb n} $，使得$ u^r \ stackrel {d} {=} a_1^{a_2^{a_2^{\ dots}}} $ IF
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 $U\sim\mathrm{Unif}[α,β]$ 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 $r\in[0, \frac1{1+\log β}]$.}
