We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised $k$th order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised $k$th order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyze various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.
翻译:我们调查了一组随机序列的极端价值理论,该类随机序列是由总和自相近的高斯进程与趋势的全时超值来定义的。本项研究的动机是其在各个领域的潜在应用及其理论上的有趣性。 我们考虑趋势函数是否相同而获得的固定序列和非静止序列。 我们显示,一个适当正常的 $k$th 统计序列在分布到一个限制的随机变量, 这可能是一个负日志转换的 Erlang 分布随机变量、 一个普通随机变量或其中的混合, 根据模型参数所推导的三个条件。 值得注意的是, 这种现象类似于固定常态序列。 我们还显示, 正常的 $knkth 统计的不同时刻会与相应限制随机变量的时段汇合。 所获得的结果使我们能够分析这些随机序列的各种属性, 从而揭示了极值理论中随机序列的有趣特性 。