自回归随机场的聚集与各向异性长程相关
Aggregation of autoregressive random fields and anisotropic long-range dependence
摘要：
我们引入了定常随机场$Y$在$\mathbb {Z}^2$上的标度跃迁和分布长程依赖的概念，对于任意的$γ>0$，其在边以速率$O(n)$和$O(n^γ)$增长的矩形上的归一化部分和趋向于一个标度$\mathbb {R}^2$上的随机场$V_γ$的算子.缩放转换的特征在于存在唯一的$γ_0>0$，使得缩放极限$V_γ$不同，并且对于$γ>γ_0$和$γ<γ_0$不依赖于$γ$。证明了在$\mathbb{Z}^2$上一类$α$-稳定的$(1<α\le2)$聚集最近邻自回归随机场的标度转换以及各向异性和各向同性分布长程相关性质的存在性，其中标量随机系数$A$在$A=1$的"单位根"附近具有规则变化的概率密度.
英文摘要：
We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on $\mathbb {Z}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^γ)$ tend to an operator scaling random field $V_γ$ on $\mathbb {R}^2$, for any $γ>0$. The scaling transition is characterized by the fact that there exists a unique $γ_0>0$ such that the scaling limits $V_γ$ are different and do not depend on $γ$ for $γ>γ_0$ and $γ<γ_0$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $α$-stable $(1<α\le2)$ aggregated nearest-neighbor autoregressive random fields on $\mathbb{Z}^2$ with a scalar random coefficient $A$ having a regularly varying probability density near the "unit root" $A=1$.
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