True Random Number Generators (TRNGs) based on ring oscillators require rigorous statistical validation to ensure cryptographic quality. While the Mauduit-Sárközy $k$-th order correlation measure $C_k$ provides theoretical bounds on pseudorandomness, and Maurer's Universal Statistical Test offers empirical entropy assessment, no prior work has correlated these metrics. This paper presents the first joint validation framework linking Maurer's Z-score to off-peak 2nd-order correlation $C_2$. We also derive the mathematical relationship between the previous two measures and high-order Markov chain transition probabilities in counter-based TRNGs over oscillator sampling architectures. Our results are validated computationally using OpenTRNG implementations, and demonstrate that practical implementations achieve Schmidt's improved bound. The initial results suggest a strong positive correlation between Maurer Z-score and $C_2$. Therefore, the results suggest a unified metric for TRNG quality-assessment can be achieve as a combination of these metrics, simplifying the study of new designs.

翻译：基于环形振荡器的真随机数生成器需要严格的统计验证以确保其密码学质量。虽然Mauduit-Sárközy $k$阶相关性度量$C_k$为伪随机性提供了理论界，且Maurer通用统计检验提供了经验熵评估，但先前工作尚未将这些指标关联起来。本文提出了首个将Maurer Z分数与非峰值二阶相关性$C_2$相联系的联合验证框架。我们还推导了上述两个度量与基于计数器的真随机数生成器在振荡器采样架构下的高阶马尔可夫链转移概率之间的数学关系。我们的结果通过OpenTRNG实现进行了计算验证，并证明实际实现达到了Schmidt改进界。初步结果表明Maurer Z分数与$C_2$之间存在强正相关性。因此，这些结果表明可将这些度量组合形成统一的真随机数生成器质量评估指标，从而简化新设计的研究。