{"id":41194,"date":"2024-09-26T10:35:54","date_gmt":"2024-09-26T10:35:54","guid":{"rendered":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%b4%d8%b1%d8%b7-%d9%84%d8%a7%d8%b2%d9%85-%d9%88-%da%a9%d8%a7%d9%81%db%8c-%d8%a8%d8%b1%d8%a7%db%8c-clt\/"},"modified":"2024-09-26T10:35:56","modified_gmt":"2024-09-26T10:35:56","slug":"%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%b4%d8%b1%d8%b7-%d9%84%d8%a7%d8%b2%d9%85-%d9%88-%da%a9%d8%a7%d9%81%db%8c-%d8%a8%d8%b1%d8%a7%db%8c-clt","status":"publish","type":"product","link":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-%d8%b4%d8%b1%d8%b7-%d9%84%d8%a7%d8%b2%d9%85-%d9%88-%da%a9%d8%a7%d9%81%db%8c-%d8%a8%d8%b1%d8%a7%db%8c-clt\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0634\u0631\u0637 \u0644\u0627\u0632\u0645 \u0648 \u06a9\u0627\u0641\u06cc \u0628\u0631\u0627\u06cc CLT \u0622\u0645\u0627\u0631 \u0637\u06cc\u0641\u06cc \u062e\u0637\u06cc \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647"},"content":{"rendered":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nNecessary and sufficient condition for CLT of linear spectral statistics of sample correlation matrices<\/td>\n<\/tr>\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0634\u0631\u0637 \u0644\u0627\u0632\u0645 \u0648 \u06a9\u0627\u0641\u06cc \u0628\u0631\u0627\u06cc CLT \u0622\u0645\u0627\u0631 \u0637\u06cc\u0641\u06cc \u062e\u0637\u06cc \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nYanpeng Li, Guangming Pan, Jiahui Xie, Wang Zhou<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nPDF<\/td>\n<\/tr>\n
\u0632\u0628\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u062a\u062d\u0648\u06cc\u0644\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u062a\u0631\u062c\u0645\u0647 \u0634\u062f\u0647 <\/td>\n\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\n<\/tr>\n
\u0646\u062d\u0648\u0647 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 <\/td>\n\u062f\u0648 \u062a\u0627 \u0633\u0647 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634 (\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u062f\u0627\u0646\u0644\u0648\u062f\u06cc)<\/td>\n<\/tr>\n
\u062a\u0639\u062f\u0627\u062f \u0635\u0641\u062d\u0627\u062a<\/td>\n112<\/td>\n<\/tr>\n
\u0644\u06cc\u0646\u06a9 \u062f\u0627\u0646\u0644\u0648\u062f \u0631\u0627\u06cc\u06af\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc<\/td>\n\u062f\u0627\u0646\u0644\u0648\u062f \u0645\u0642\u0627\u0644\u0647<\/a><\/td>\n<\/tr>\n
\u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0645\u0648\u0636\u0648\u0639\u0627\u062a <\/td>\nProbability,Statistics Theory,\u0627\u062d\u062a\u0645\u0627\u0644 , \u062a\u0626\u0648\u0631\u06cc \u0622\u0645\u0627\u0631 ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 19 September, 2024; originally announced September 2024. , Comments: 112 pages , MSC Class: 60B20; 60F05; 62E20; 62H20; 15B52<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u0627\u0631\u0633\u0627\u0644 \u0634\u062f\u0647 \u062f\u0631 19 \u0633\u067e\u062a\u0627\u0645\u0628\u0631 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0633\u067e\u062a\u0627\u0645\u0628\u0631 2024 \u0627\u0639\u0644\u0627\u0645 \u0634\u062f \u060c \u0646\u0638\u0631\u0627\u062a: 112 \u0635\u0641\u062d\u0647 \u060c \u06a9\u0644\u0627\u0633 MSC: 60B20 \u061b60F05 \u061b62e20 \u061b62H20 \u061b15b52<\/td>\n<\/tr>\n
\u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0628\u06cc\u0634\u062a\u0631 \u0627\u0632 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u062f\u0631 \u067e\u0627\u06cc\u06af\u0627\u0647 \u0647\u0627\u06cc \u0639\u0644\u0645\u06cc <\/td>\n\n INSPIRE HEP<\/a>
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\n NASA ADS<\/a>
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\n Google Scholar<\/a>
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\n Semantic Scholar<\/a>
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\n arXiv<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n

\u0686\u06a9\u06cc\u062f\u0647<\/h2>\n

In this paper, we establish the central limit theorem (CLT) for the linear spectral statistics (LSS) of sample correlation matrix $R$, constructed from a $p\\times n$ data matrix $X$ with independent and identically distributed (i.i.d.) entries having mean zero, variance one, and infinite fourth moments in the high-dimensional regime $n\/p\\rightarrow \u03c6\\in \\mathbb{R}_+\\backslash \\{1\\}$. We derive a necessary and sufficient condition for the CLT. More precisely, under the assumption that the identical distribution $\u03be$ of the entries in $X$ satisfies $\\mathbb{P}(|\u03be|>x)\\sim l(x)x^{-\u03b1}$ when $x\\rightarrow \\infty$ for $\u03b1\\in (2,4]$, where $l(x)$ is a slowly varying function, we conclude that: (i). When $\u03b1\\in(3,4]$, the universal asymptotic normality for the LSS of sample correlation matrix holds, with the same asymptotic mean and variance as in the finite fourth moment scenario; (ii) We identify a necessary and sufficient condition $\\lim_{x\\rightarrow\\infty}x^3\\mathbb{P}(|\u03be|>x)=0$ for the universal CLT; (iii) We establish a local law for $\u03b1\\in (2, 4]$. Overall, our proof strategy follows the routine of the matrix resampling, intermediate local law, Green function comparison, and characteristic function estimation. In various parts of the proof, we are required to come up with new approaches and ideas to solve the challenges posed by the special structure of sample correlation matrix. Our results also demonstrate that the symmetry condition is unnecessary for the CLT of LSS for sample correlation matrix, but the tail index $\u03b1$ plays a crucial role in determining the asymptotic behaviors of LSS for $\u03b1\\in (2, 3)$.<\/p>\n

\u0686\u06a9\u06cc\u062f\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc (\u062a\u0631\u062c\u0645\u0647 \u0645\u0627\u0634\u06cc\u0646\u06cc)<\/h2>\n

\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u060c \u0645\u0627 \u0642\u0636\u06cc\u0647 \u0645\u062d\u062f\u0648\u062f\u06cc\u062a \u0645\u0631\u06a9\u0632\u06cc (CLT) \u0631\u0627 \u0628\u0631\u0627\u06cc \u0622\u0645\u0627\u0631 \u0637\u06cc\u0641\u06cc \u062e\u0637\u06cc (LSS) \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647 $ r $ \u060c \u0633\u0627\u062e\u062a\u0647 \u0634\u062f\u0647 \u0627\u0632 \u06cc\u06a9 \u0645\u0627\u062a\u0631\u06cc\u0633 \u062f\u0627\u062f\u0647 $ p \\ times n $ $ x $ \u0628\u0627 \u0645\u0633\u062a\u0642\u0644 \u0648 \u06cc\u06a9\u0633\u0627\u0646 \u062a\u0648\u0632\u06cc\u0639 \u0645\u06cc \u06a9\u0646\u06cc\u0645 (i.i.d.) \u0648\u0631\u0648\u062f\u06cc \u0647\u0627\u06cc\u06cc \u06a9\u0647 \u062f\u0627\u0631\u0627\u06cc \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0635\u0641\u0631 \u060c \u0648\u0627\u0631\u06cc\u0627\u0646\u0633 \u06cc\u06a9 \u0648 \u0644\u062d\u0638\u0647 \u0647\u0627\u06cc \u0686\u0647\u0627\u0631\u0645 \u0628\u06cc \u0646\u0647\u0627\u06cc\u062a \u062f\u0631 \u0631\u0698\u06cc\u0645 \u0628\u0627\u0644\u0627 $ n\/p \\ \u0631\u0627\u0633\u062a \u03c6 \\ in \\ mathbb {r} _+\\ backslash \\ {1 \\} $ \u0647\u0633\u062a\u0646\u062f.\u0645\u0627 \u06cc\u06a9 \u0634\u0631\u0637 \u0644\u0627\u0632\u0645 \u0648 \u06a9\u0627\u0641\u06cc \u0628\u0631\u0627\u06cc CLT \u0628\u0647 \u062f\u0633\u062a \u0645\u06cc \u0622\u0648\u0631\u06cc\u0645.\u0628\u0647 \u0637\u0648\u0631 \u062f\u0642\u06cc\u0642 \u062a\u0631 \u060c \u0628\u0627 \u0627\u06cc\u0646 \u0641\u0631\u0636 \u06a9\u0647 \u062a\u0648\u0632\u06cc\u0639 \u06cc\u06a9\u0633\u0627\u0646 $ \u03be $ \u0627\u0632 \u0648\u0631\u0648\u062f\u06cc \u0647\u0627\u06cc $ x $ $ \\ Mathbb {p} (| \u03be |> x) \\ sim l (x) x^{-\u03b1} $ $ x \u0631\u0627 \u0628\u0631\u0622\u0648\u0631\u062f\u0647 \u0645\u06cc \u06a9\u0646\u062f.\\ Rightarrow \\ infty $ \u0628\u0631\u0627\u06cc $ \u03b1 \\ in (2\u060c4] $ \u060c \u062c\u0627\u06cc\u06cc \u06a9\u0647 $ l (x) $ \u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0628\u0647 \u0622\u0631\u0627\u0645\u06cc \u0645\u062a\u0641\u0627\u0648\u062a \u0627\u0633\u062a \u060c \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u0645 \u06a9\u0647: (i) \u0648\u0642\u062a\u06cc $ \u03b1 \\ in (3\u060c4] $ \u060c\u0646\u0631\u0645\u0627\u0644 \u0628\u0648\u062f\u0646 \u0628\u062f\u0648\u0646 \u0639\u0644\u0627\u0645\u062a \u062c\u0647\u0627\u0646\u06cc \u0628\u0631\u0627\u06cc LSS \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647 \u060c \u0628\u0627 \u0647\u0645\u0627\u0646 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0648 \u0648\u0627\u0631\u06cc\u0627\u0646\u0633 \u06cc\u06a9\u0633\u0627\u0646 \u0645\u0627\u0646\u0646\u062f \u0633\u0646\u0627\u0631\u06cc\u0648\u06cc \u0644\u062d\u0638\u0647 \u0686\u0647\u0627\u0631\u0645 \u061b3 \\ Mathbb {P} (| \u03be |> x) = 0 $ \u0628\u0631\u0627\u06cc CLT \u062c\u0647\u0627\u0646\u06cc \u061b\u0628\u0627\u0632\u06af\u0631\u062f\u0627\u0646\u062f\u0646 \u0645\u0627\u062a\u0631\u06cc\u0633 \u060c \u0642\u0627\u0646\u0648\u0646 \u0645\u062d\u0644\u06cc \u0645\u062a\u0648\u0633\u0637 \u200b\u200b\u060c \u0645\u0642\u0627\u06cc\u0633\u0647 \u0639\u0645\u0644\u06a9\u0631\u062f \u0633\u0628\u0632 \u0648 \u0628\u0631\u0622\u0648\u0631\u062f \u0639\u0645\u0644\u06a9\u0631\u062f \u0645\u0634\u062e\u0635\u0647 \u060c \u0645\u0627 \u0628\u0627\u06cc\u062f \u0631\u0648\u06cc\u06a9\u0631\u062f\u0647\u0627\u06cc \u0648 \u0627\u06cc\u062f\u0647 \u0647\u0627\u06cc \u062c\u062f\u06cc\u062f\u06cc \u0631\u0627 \u0628\u0631\u0627\u06cc \u062d\u0644 \u0686\u0627\u0644\u0634 \u0647\u0627\u06cc \u0627\u06cc\u062c\u0627\u062f \u0634\u062f\u0647 \u062a\u0648\u0633\u0637 \u0633\u0627\u062e\u062a\u0627\u0631 \u062e\u0627\u0635 \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647 \u0627\u0631\u0627\u0626\u0647 \u062f\u0647\u06cc\u0645.\u0646\u062a\u0627\u06cc\u062c \u0647\u0645\u0686\u0646\u06cc\u0646 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0634\u0631\u0627\u06cc\u0637 \u062a\u0642\u0627\u0631\u0646 \u0628\u0631\u0627\u06cc CLT LSS \u0628\u0631\u0627\u06cc \u0645\u0627\u062a\u0631\u06cc\u0633 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0646\u0645\u0648\u0646\u0647 \u0636\u0631\u0648\u0631\u06cc \u0646\u06cc\u0633\u062a \u060c \u0627\u0645\u0627 \u0634\u0627\u062e\u0635 \u062f\u0645 $ \u03b1 $ \u0646\u0642\u0634 \u0645\u0647\u0645\u06cc \u062f\u0631 \u062a\u0639\u06cc\u06cc\u0646 \u0631\u0641\u062a\u0627\u0631\u0647\u0627\u06cc \u0628\u062f\u0648\u0646 \u0639\u0644\u0627\u0645\u062a LSS \u0628\u0631\u0627\u06cc \u03b1 \u03b1 \\ \u062f\u0631 (2 \u060c 3) $ \u062f\u0627\u0631\u062f.<\/p>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n","protected":false},"excerpt":{"rendered":"

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