Abstract We consider asymptotic properties of the empirical process of tempered moving average with memory parameter d ∈ [ 0 , 1 ∕ 2 ) and tempering parameter λ N → 0 , centered by the marginal distribution function of the corresponding untempered stationary process. We prove that under long memory ( 0 d 1 ∕ 2 ) and strong tempering ( N λ N → ∞ ) the above empirical process has a faster rate of convergence than under weak or moderate tempering. Moreover, for 0 d 1 ∕ 2 and λ N = o ( N − 1 ∕ ( 1 − 2 d ) ) a uniform reduction principle holds and weak convergence to a degenerate Gaussian process is obtained.

中文翻译：

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缓和移动平均线的经验过程

摘要 我们考虑具有记忆参数 d ∈ [ 0 , 1 ∕ 2 ) 和调节参数 λ N → 0 的缓和移动平均经验过程的渐近特性，以相应的非缓和平稳过程的边际分布函数为中心。我们证明，在长记忆（0 d 1 ∕ 2 ）和强回火（N λ N → ∞ ）下，上述经验过程比弱回火或中等回火下具有更快的收敛速度。此外，对于 0 d 1 ∕ 2 和 λ N = o ( N − 1 ∕ ( 1 − 2 d ) ) 一致归约原理成立，并获得了对退化高斯过程的弱收敛。