Abstract

We consider the difference approximation of the one-dimensional Liouville equation for modeling the evolution of the sample distribution density (of the nonstationary time series) estimated by a histogram. It is shown that the change in the sample density of the distribution over a certain period of time can be numerically described as a solution of the Liouville equation if the initial density distribution is strictly positive in the internal class intervals. The algorithm for determining the corresponding rate is constructed and its mechanical and statistical meaning is shown as a semigroup equivalent in the Chernoff sense to the average semigroup, which generates the evolution of the distribution function.

中文翻译：

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使用Liouville方程对随机变量样本分布的演化进行建模

摘要

我们考虑一维Liouville方程的差分近似，以对直方图估计的样本分布密度（非平稳时间序列）的演化进行建模。结果表明，如果内部类区间中的初始密度分布严格为正，则可以将一定时间内分布的样本密度的变化描述为Liouville方程的解。构造了用于确定相应速率的算法，其机械和统计意义在切尔诺夫意义上表示为与平均半群等效的半群等效项，从而产生了分布函数的演化。