CONVERGENCE RATES OF SUMS OF α -MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES
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α 混合三角阵列和的收敛速率及连续时间漂移函数估计应用
- 研究问题
- 当时间依赖较强或高阶矩不存在，以致中心极限定理可能失效时，异质 α 混合三角阵列之和仍能以怎样的速率收敛？
- 方法与思路
- 作者分别考虑固定观测间隔与间隔趋零的渐近设定，允许混合系数任意缓慢衰减，推导收敛速度与依赖程度之间的权衡。
- 材料陈述的贡献
- 摘要称，文章补充此类阵列和的收敛速率理论，并利用间隔趋零结果分析连续时间过程估计的离散化偏差及相关动态例子。
- 阅读关注点
- 应核对矩条件、混合衰减与速率之间的具体关系，区分固定间隔和加密采样，并检查漂移估计应用所需的额外过程条件。
- 阅读边界
- 仅据所给 Crossref 版本摘要，不能写出具体速率或判定最优性；允许慢混合不意味着所有依赖强度下均有相同收敛结论。
原文摘要
The convergence rates of the sums of α -mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α -mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n ; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.