论文标题
几乎肯定地限制了远程依赖性线性过程的重尾产品的定理
On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes
论文作者
论文摘要
Marcinkiewicz大的强大定律，$ {n^{ - \ frac1p}} \ sum_ {k = 1}^{n}（d_ {k} - d）\ rightarrow 0 \ $几乎肯定是$ p \ in（1,2）$开发了（1,1,2）$，是为$ d_k = _ prod_} $ d_k = \ prod_}其中$ x_k^{（r）} = \ sum_ {l = - \ \ iftty}^{\ infty} c_ {k-l}^{（r）}ξ_l^{（r）} $是带有系数$ \ \ \ \ \ {c_l^r）的两边线性过程\ Mathbb {Z}} $和I.I.D. Zero-Mean Innovations $ \ {ξ_l^{（r）} \} _ {l \ in \ Mathbb {z}} $。系数的衰减$ c_l^{（r）} $ as $ | l | \ to \ infty $，对于$ \ {x_k^{（r）} \} $，可以慢一点，而$ \ {d_k \ \} $可以具有沉重的尾巴。 $ \ {d_k \} $的远距离依赖性和沉重的尾巴是同时处理的，而去耦属性表明收敛速率取决于长期依赖和较重的尾巴的最差，但不是它们的组合。 Marcinkiewicz的强大规律也扩展到多元线性过程案例。
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.
