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arXiv 1810.04496math.PR
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关于平稳场的极大值

On maxima of stationary fields

Natalia Soja-Kukieła

备注：
Accepted for publication by the Applied Probability Trust in Journal of Applied Probability 56.4
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摘要：

设$\{X_{\mathbf{n}} : \mathbf{n}\in\mathbb{Z}^d\}$是一个弱相关的平稳场，对于有限的$A\subset\mathbb{Z}^d$有极大值$M_{A} := \sup\{X_{\mathbf{i}} : \mathbf{i}\in A\}$，对于$\mathbf{n}\in\mathbb{N}^d$有极大值$M_{\mathbf{n}} := \sup\{X_{\mathbf{i}} : \mathbf{1} \leq \mathbf{i} \leq \mathbf{n} \}$.在一般的设置，我们证明了$P(M_{(n,n,\ldots, n)} \leq v_n) = \exp(- n^d P(X_{\mathbf{0}} > v_n , M_{A_n} \leq v_n)) + o(1)$，对一些增加的序列集$A_n$的大小$ o(n^d)$。对于一类满足局部混合条件的域，包括依赖于$m$的域，用常数有限$A$代替$A_n$，定理成立。上述结果导出了随机场极值指数的新公式。

英文摘要：

Let $\{X_{\mathbf{n}} : \mathbf{n}\in\mathbb{Z}^d\}$ be a weakly dependent stationary field with maxima $M_{A} := \sup\{X_{\mathbf{i}} : \mathbf{i}\in A\}$ for finite $A\subset\mathbb{Z}^d$ and $M_{\mathbf{n}} := \sup\{X_{\mathbf{i}} : \mathbf{1} \leq \mathbf{i} \leq \mathbf{n} \}$ for $\mathbf{n}\in\mathbb{N}^d$. In a general setting we prove that $P(M_{(n,n,\ldots, n)} \leq v_n) = \exp(- n^d P(X_{\mathbf{0}} > v_n , M_{A_n} \leq v_n)) + o(1)$, for some increasing sequence of sets $A_n$ of size $ o(n^d)$. For a class of fields satisfying a local mixing condition, including $m$-dependent ones, the theorem holds with a constant finite $A$ replacing $A_n$. The above results lead to new formulas for the extremal index for random fields.

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。

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