In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}\sum_{i=1}^N c_i(g(\sigma_i)-\mathbb{E}_N[g(\sigma_i)|\sigma_j,j\neq i])$$ where $(\sigma_1,\ldots ,\sigma_N)$ are sampled from a dependent random field, and $g$ is some bounded function. Our first main result shows that under weak smoothness assumptions on the conditional means (which cover both sparse and dense interactions), the above statistic converges to a Gaussian \emph{scale mixture} with a random scale determined by a \emph{quadratic variance} and an \emph{interaction component}. We also show that under appropriate studentization, the limit becomes a pivotal Gaussian. We leverage this theory to develop a general asymptotic framework for maximum pseudolikelihood (MPLE) inference in dependent random fields. We apply our results to Ising models with pairwise as well as higher-order interactions and exponential random graph models (ERGMs). In particular, we obtain a joint central limit theorem for the inverse temperature and magnetization parameters via the joint MPLE (to our knowledge, the first such result in dense, irregular regimes), and we derive conditionally centered edge CLTs and marginal MPLE CLTs for ERGMs without restricting to the ``sub-critical" region. Our proof is based on a method of moments approach via combinatorial decision-tree pruning, which may be of independent interest.

翻译：本文研究条件中心化统计量的波动性，其形式为 $$N^{-1/2}\sum_{i=1}^N c_i(g(\sigma_i)-\mathbb{E}_N[g(\sigma_i)|\sigma_j,j\neq i])$$，其中 $(\sigma_1,\ldots ,\sigma_N)$ 采样自依赖随机场，$g$ 为有界函数。我们的第一个主要结果表明，在条件均值的弱光滑性假设下（该假设同时涵盖稀疏与稠密相互作用），上述统计量收敛于一个高斯尺度混合分布，其随机尺度由二次方差与相互作用分量共同决定。我们还证明，在适当的标准化处理后，极限分布将转化为一个关键高斯分布。我们利用该理论为依赖随机场中的最大伪似然估计推断建立了一个通用的渐近分析框架。我们将所得结果应用于具有成对及高阶相互作用的伊辛模型以及指数随机图模型。特别地，我们通过联合最大伪似然估计获得了逆温度参数与磁化参数的联合中心极限定理（据我们所知，这是稠密非规则区域内的首个此类结果），并且在不局限于“亚临界”区域的条件下，推导了指数随机图模型的条件中心化边中心极限定理与边缘最大伪似然估计中心极限定理。我们的证明基于通过组合决策树剪枝实现的矩方法，该方法可能具有独立的学术价值。