We develop the limit theory of the quantilogram and cross-quantilogram under long memory. We establish the sub-root-n central limit theorems for quantilograms that depend on nuisance parameters. We propose a moving block bootstrap (MBB) procedure for inference and establish its consistency, thereby enabling a consistent confidence interval construction for the quantilograms. The newly developed reduction principles for the quantilograms serve as the main technical devices used to derive the asymptotics and establish the validity of MBB. We report some simulation evidence that our methods work satisfactorily. We apply our method to quantile predictive relations between financial returns and long-memory predictors.

中文翻译：

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强依赖下的量子图

我们发展了长记忆下的分位数图和交叉分位数图的极限理论。我们为依赖于讨厌参数的分位数建立了次根-n 中心极限定理。我们提出了一种用于推理的移动块引导 (MBB) 程序并建立其一致性，从而为分位数图实现一致的置信区间构造。新开发的分位数图归约原理是用于推导渐近线和建立 MBB 有效性的主要技术手段。我们报告了一些模拟证据，证明我们的方法可以令人满意地工作。我们将我们的方法应用于财务回报和长记忆预测变量之间的分位数预测关系。