重尾长程相依数据外积的Marcinkiewicz Law of Large Numbers
Marcinkiewicz Law of Large Numbers for Outer-products of Heavy-tailed, Long-range Dependent Data
摘要：
Marcinkiewicz Strong Law，$\displaystyle\lim_{n\to\infty}\frac{1}{n^{\frac1p}}\sum_{k=1}^n (D_{k}- D)=0$ a.s.其中$p\in(1,2)$均为双边（多元）线性过程（系数矩阵为$(C_l), (\overline{C}_l)$和i.i.d.）零均值新息$\{Ξ\}$、$\{\overlineΞ\}$）。矩阵序列$C_l$和$\overline{C}_l$可以足够慢地衰减（如$|l|\to\infty$），使得$\{X_k,\overline{X}_k\}$具有长程相关性，而$\{D_k\}$可以具有重尾。特别是，重尾和长程依赖现象$\{D_k\}$同时处理和一个新的解耦性能证明，表明收敛速度是由最坏的重尾或长程依赖，但不是组合。主要结果被应用于随机逼近、非线性函数形式和自协方差的Marcinkiewicz强大数定律。
英文摘要：
The Marcinkiewicz Strong Law, $\displaystyle\lim_{n\to\infty}\frac{1}{n^{\frac1p}}\sum_{k=1}^n (D_{k}- D)=0$ a.s. with $p\in(1,2)$, is studied for outer products $D_k=X_k\overline{X}_k^T$, where $\{X_k\},\{\overline{X}_k\}$ are both two-sided (multivariate) linear processes ( with coefficient matrices $(C_l), (\overline{C}_l)$ and i.i.d.\ zero-mean innovations $\{Ξ\}$, $\{\overlineΞ\}$). Matrix sequences $C_l$ and $\overline{C}_l$ can decay slowly enough (as $|l|\to\infty$) that $\{X_k,\overline{X}_k\}$ have long-range dependence while $\{D_k\}$ can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for $\{D_k\}$ are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy-tails or the long-range dependence, but not the combination. The main result is applied to obtain Marcinkiewicz Strong Law of Large Numbers for stochastic approximation, non-linear functions forms and autocovariances.
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