论文标题
具有固定数字的随机变量的表征
Characterization of random variables with stationary digits
论文作者
论文摘要
令$ q \ ge2 $为整数，$ \ {x_n \} _ {n \ geq 1} $带有状态空间$ \ {0，\ ldots，q-1 \} $的随机过程 $ \ sum_ {n = 1}^\ infty x_n q^{ - n} $。我们表明$ \ {x_n \} _ {n \ geq 1} $的平稳性等同于$ f $遵守的函数方程，并使用它来表征$ x $和 $ f $的结构就其Lebesgue分解而言。更确切地说，虽然$ f $的绝对连续组件只能是单位间隔上均匀的分布，但其离散组件只能是某些明确可计算的CDF的可数凸组合，以提供有限支持的概率分布。 我们还表明，$ \ mathrm {d} f $是Rajchman的措施，并且仅当$ f $是$ [0,1] $上的统一CDF。
Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$, and $F$ the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by $F$ and use this to characterize the characteristic function of $X$ and the structure of $F$ in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if $F $ is the uniform CDF on $[0,1]$.
