We consider finite blocklength lossy compression of information sources whose components are independent but non-identically distributed. Crucially, Gaussian sources with memory and quadratic distortion can be cast in this form. We show that under the operational constraint of exceeding distortion $d$ with probability at most $ε$, the minimum achievable rate at blocklength $n$ satisfies $R(n, d, ε)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(ε)+O \left(\frac{\log n}{n}\right)$, where $Q^{-1}(\cdot)$ is the inverse $Q$-function, while $\mathbb{R}_n(d)$ and $\mathbb{V}_n(d)$ are fundamental characteristics of the source computed using its $n$-letter joint distribution and the distortion measure, called the $n$th-order informational rate-distortion function and the source dispersion, respectively. Our result generalizes the existing dispersion result for abstract sources with i.i.d. components. It also sharpens and extends the only known dispersion result for a source with memory, namely, the scalar Gauss-Markov source. The key novel technical tool in our analysis is the point-mass product proxy measure, which enables the construction of typical sets. This proxy generalizes the empirical distribution beyond the i.i.d. setting by preserving additivity across coordinates and facilitating a typicality analysis for sums of independent, non-identical terms.

翻译：本文研究信息源在有限块长下的有损压缩问题，该信息源各分量独立但非同分布。关键在于，具有记忆的高斯信源在二次失真度量下可转化为该形式。我们证明，在失真超过$d$的概率不超过$ε$的操作约束下，块长$n$时可达到的最小速率满足$R(n, d, ε)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(ε)+O \left(\frac{\log n}{n}\right)$，其中$Q^{-1}(\cdot)$为逆$Q$函数，而$\mathbb{R}_n(d)$与$\mathbb{V}_n(d)$是通过信源的$n$维联合分布与失真度量计算的基本特征量，分别称为$n$阶信息率失真函数与信源弥散度。我们的结果推广了现有关于分量独立同分布抽象信源的弥散性结论，同时改进并扩展了目前唯一已知的具有记忆信源（即标量高斯-马尔可夫信源）的弥散性结果。分析中的关键新技术工具是点质量乘积代理测度，该工具使得典型集的构造成为可能。此代理方法通过保持坐标间的可加性，将经验分布推广至独立同分布设定之外，并为独立非同分布项的和提供了典型性分析框架。