与伯努利卷积相关的AR(1)过程的极大半稳定极端行为
Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions
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中文总结 AI 辅助
本文研究与伯努利卷积相关的AR(1)过程，给出其平稳分布的新结构表示，证明其平稳分布属于极大半稳定律的几何部分吸引域，并得到归一化最大值分布的收敛结果及与动力系统的关联。
中文摘要 AI 辅助
我们研究一类简单的自回归过程AR(1)，其平稳分布支撑在单位区间的一个子集上，且是伯努利卷积的仿射变换。本文给出该平稳分布的一种新结构表示：在原点附近，其可表示为幂函数与对数周期函数的乘积，结合特征泛函方程，该表示能帮助理解平稳分布在整个单位区间上的结构。这使得我们可以证明，这类AR(1)过程的平稳分布属于某极大半稳定律的几何部分吸引域。我们还证明，AR(1)过程归一化最大值的分布函数会依合并定理的精神一致收敛到该极大半稳定律的某一幂次，并指出其与确定性和随机动力系统的关联。
英文摘要
We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.
