arXiv 2007.060832020-07math.PRmath.STstat.TH

关于长程相依线性过程重尾乘积的几乎必然极限定理

On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes

Michael A. Kouritzin, Sounak Paul

备注：: 38 pages
发表信息：: Stoch. Process. Their Appl. 152 (2022) 208-232
DOI：: 10.1016/j.spa.2022.06.021

摘要：

本文对乘积$d_k=\prod_{r=1}^s x_k^{(r)}$建立了Marcinkiewicz强大数定律${n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ $几乎必然与$p\in(1,2)$，其中$x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}ξ_l^{(r)}$是系数为$\{c_l^{(r)}\}_{l\in \mathbb{Z}}$和i.i.d.的双边线性过程.零均值新息$\{ξ_l^{(r)}\}_{l\in \mathbb{Z}}$。系数$c_l^{(r)}$作为$|l|\to\infty$的衰减可以足够慢以使$\{x_k^{(r)}\}$具有长记忆，而$\{d_k\}$可以具有重尾。$\{d_k\}$的长程相关性和重尾同时处理，解耦特性表明收敛速度取决于最坏的长程相关性和重尾，而不是它们的组合。Marcinkiewicz强大数定律也推广到多元线性过程的情况。

英文摘要：

Marcinkiewicz strong law of large numbers, ${n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\ $ almost surely with $p\in(1,2)$, are developed for products $d_k=\prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}ξ_l^{(r)}$ are two-sided linear processes with coefficients $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d. zero-mean innovations $\{ξ_l^{(r)}\}_{l\in \mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough for $\{x_k^{(r)}\}$ to have long memory while $\{d_k\}$ can have heavy tails. The long-range dependence and heavy tails for $\{d_k\}$ are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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