关于随机过程的信息维
On the Information Dimension of Stochastic Processes
摘要：
1959年，Rényi提出了信息维和$d$维熵来度量一般随机变量的信息含量。本文将信息维推广到随机过程，定义信息维率为均匀量化随机过程的熵率除以量化器步长$1/m$的对数之差，其极限为$m\to\infty$。证明了信息维数率与率失真维数率一致，率失真维数定义为随机过程的率失真函数$R(D)$的两倍除以$-\log(D)$，极限为$D\downarrow 0$ .进一步证明了在具有给定（矩阵值）谱分布函数（SDF）的所有多元平稳过程中，高斯过程具有最大的信息维率，多元平稳高斯过程的信息维率由SDF的导数的平均秩给出.结果表明，通过可压缩信号追踪和几乎无损模拟压缩的几乎零失真恢复的基本限制是不同的。
英文摘要：
In 1959, Rényi proposed the information dimension and the $d$-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly-quantized stochastic process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $m\to\infty$. It is demonstrated that the information dimension rate coincides with the rate-distortion dimension, defined as twice the rate-distortion function $R(D)$ of the stochastic process divided by $-\log(D)$ in the limit as $D\downarrow 0$. It is further shown that, among all multivariate stationary processes with a given (matrix-valued) spectral distribution function (SDF), the Gaussian process has the largest information dimension rate, and that the information dimension rate of multivariate stationary Gaussian processes is given by the average rank of the derivative of the SDF. The presented results reveal that the fundamental limits of almost zero-distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
