In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \sim \text{DP}(\alpha \nu_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\alpha \mapsto \log \mathbb{E}_{X \sim \text{DP}(\alpha \nu_0)}[\exp( \mathbb{E}_X[\alpha f])]$, where $\mathbb{E}_X[f] = \int f dX$. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF $ \log\mathbb{E}_{X\sim \text{DP}(\alpha\nu_0)}{\exp(\mathbb{E}_{X}{[f]})} $ for any $\alpha > 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\alpha\mathrm{KL}(\nu_0\Vert \cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.

翻译：本文提出了一种利用超可加性来界定狄利克雷过程（DP）$X \sim \text{DP}(\alpha \nu_0)$ 累积生成函数（CGF）的新方法。具体而言，我们的核心技术贡献在于证明了 $\alpha \mapsto \log \mathbb{E}_{X \sim \text{DP}(\alpha \nu_0)}[\exp( \mathbb{E}_X[\alpha f])]$ 的超可加性，其中 $\mathbb{E}_X[f] = \int f dX$。该结果结合 Fekete 引理与 Varadhan 积分引理，将已知的渐近大偏差原理转化为对任意 $\alpha > 0$ 时 CGF $ \log\mathbb{E}_{X\sim \text{DP}(\alpha\nu_0)}{\exp(\mathbb{E}_{X}{[f]})} $ 的实用上界。该上界由缩放反向 Kullback-Leibler 散度 $\alpha\mathrm{KL}(\nu_0\Vert \cdot)$ 的凸共轭给出。这一新界为独立狄利克雷过程之和提供了特别有效的置信区域，使其在多个领域具有应用价值。