Claims about distributions of time series are often unproven assertions instead of substantiated conclusions for lack of hypotheses testing tools. In this work, Kolmogorov–Smirnov type simultaneous confidence bands (SCBs) are constructed based on simple random samples (SRSs) drawn from realizations of time series, together with smooth SCBs using kernel distribution estimator (KDE) instead of empirical cumulative distribution function of the SRS. All SCBs are shown to enjoy the same limiting distribution as the standard Kolmogorov–Smirnov for i.i.d. sample, which is validated in simulation experiments on various time series. Computing these SCBs for the standardized S&P 500 daily returns data leads to some rather unexpected findings, i.e., student’s t-distributions with degrees of freedom no less than 3 and the normal distribution are all acceptable versions of the standardized daily returns series’ distribution, with proper rescaling. These findings present challenges to the long held belief that daily financial returns distribution is fat-tailed and leptokurtic.

由于缺乏假设测试工具，关于时间序列分布的声明通常是未经证实的断言，而不是经过证实的结论。在这项工作中，Kolmogorov-Smirnov 型同时置信带 (SCB) 是基于从时间序列的实现中抽取的简单随机样本 (SRS) 以及使用核分布估计器 (KDE) 而不是经验累积分布函数的平滑 SCB 构建的。 SRS。所有 SCB 都显示出与 iid 样本的标准 Kolmogorov-Smirnov 相同的极限分布，这在各种时间序列的模拟实验中得到了验证。为标准化的标准普尔 500 指数每日回报数据计算这些 SCB 会导致一些相当意外的发现，即学生的t- 自由度不小于 3 的分布和正态分布都是标准化每日收益序列分布的可接受版本，并进行了适当的重新缩放。这些发现对长期以来认为每日财务回报分布是肥尾和瘦高的观点提出了挑战。