Abstract
Let {X k,i; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p n ; n ≥ 1} be a sequence of positive integers such that n/p n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry \({L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}\) of the sample correlation matrix \({{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}\) where \({\hat{\rho}^{(n)}_{i,j}}\) denotes the Pearson correlation coefficient between (X 1,i , ..., X n,i)′ and (X 1,j ,...,X n,j)′. Write \({F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}\) , \({W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}\) , and \({W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}\) . Under the assumption that \({\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}\) for some δ > 0, we show that the following six statements are equivalent:
where \({\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}\) , and a n = 4 log p n − log log p n . The equivalences between (i), (ii), (iii), and (v) assume that only \({\mathbb{E}X_{1,1}^{2} < \infty}\) . Weak laws of large numbers for W n and L n , n ≥ 1, are also established and these are of the form \({W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)\) and \({n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)\), respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L n obtained by Jiang. Some open problems are also posed.
References
Araujo A., Giné E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York (1980)
Cuzick J., Giné E., Zinn J.: Laws of large numbers for quadratic forms, maxima of products and truncated sums of i.i.d. random variables. Ann. Probab. 23, 292–333 (1995)
Giné E., Götze F., Mason D.M.: When is the Student t-statistic asymptotically standard normal?. Ann. Probab. 25, 1514–1531 (1997)
Jiang T.: The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14, 865–880 (2004)
Latała R., Zinn J.: Necessary and sufficient conditions for the strong law of large numbers for U-statistics. Ann. Probab. 28, 1908–1924 (2000)
Ledoux M., Talagrand M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991)
Li D., Rosalsky A.: Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16, 423–447 (2006)
Liu W.-D., Lin Z., Shao Q.-M.: The asymptotic distribution and Berry–Esseen bound of a new test for independence in high dimension with an application to stochastic optimization. Ann. Appl. Probab. 18, 2337–2366 (2008)
Nagaev S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)
Petrov V.V.: Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford (1995)
Zhou W.: Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Am. Math. Soc. 359, 5345–5363 (2007)
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Li, D., Liu, WD. & Rosalsky, A. Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Relat. Fields 148, 5–35 (2010). https://doi.org/10.1007/s00440-009-0220-z
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DOI: https://doi.org/10.1007/s00440-009-0220-z
Keywords
- Asymptotic distribution
- Largest entries of sample correlation matrices
- Weak law of large numbers
- Weak law of the logarithm
Mathematics Subject Classification (2000)
- Primary: 60F05
- 60F10
- Secondary: 62H99