论文标题
具有固定数字的随机变量的单数分布函数
Singular distribution functions for random variables with stationary digits
论文作者
论文摘要
令$ f $为基本的累积分布函数（CDF），$ q $扩展$ \ sum_ {n = 1}^\ infty x_n q^{ - n} $，其中$ q \ ge2 $是整数，$ \ \ \ {x_n \ \ \ \ \ \ geq 1} $ \ {0，\ ldots，q-1 \} $。在上一篇论文中，我们表征了$ f $的绝对连续和离散组件。在本文中，我们研究了模型的特殊案例，包括固定的马尔可夫链和任何固定续订点流程，在那里我们建立了纯类型的定律：$ f $是$ [0,1] $的统一或单数CDF。此外，我们研究了此类模型的混合物。在大多数情况下，给出了$ f $的表达式和图。
Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and $\{X_n\}_{n\geq 1}$ is a stationary stochastic process with state space $\{0,\ldots,q-1\}$. In a previous paper we characterized the absolutely continuous and the discrete components of $F$. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: $F$ is then either a uniform or a singular CDF on $[0,1]$. Moreover, we study mixtures of such models. In most cases expressions and plots of $F$ are given.
