Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data
Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data
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DOI：
10.1007/s10959-019-00958-1
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发表时间：
2019-10
影响因子：
0.8
通讯作者：
Tiefeng Jiang;Junshan Xie
中科院分区：
文献类型：
--
作者：
Tiefeng Jiang;Junshan Xie
Let, be a random sample of sizencoming from ap-dimensional population. For a fixed integer, consider a hypercubic random tensorofmth order and ranknwith $$\begin{aligned} \mathbf {{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big )_{1\le i_{1},\ldots , i_{m}\le p}. \end{aligned}$$Letbe the largest off-diagonal entry of. We derive the asymptotic distribution ofunder a suitable normalization for two cases. They are the ultra-high-dimension case withandand the high-dimension case withandwhere. The normalizing constant ofdepends onmand the limiting distribution ofis a Gumbel-type distribution involved with parameterm.
Let, be a random sample of sizencoming from ap-dimensional population. For a fixed integer, consider a hypercubic random tensorofmth order and ranknwith $$\begin{aligned} \mathbf {{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big )_{1\le i_{1},\ldots , i_{m}\le p}. \end{aligned}$$Letbe the largest off-diagonal entry of. We derive the asymptotic distribution ofunder a suitable normalization for two cases. They are the ultra-high-dimension case withandand the high-dimension case withandwhere. The normalizing constant ofdepends onmand the limiting distribution ofis a Gumbel-type distribution involved with parameterm.