Please use this identifier to cite or link to this item: https://hdl.handle.net/10356/96095
Title: The convergence of the empirical distribution of canonical correlation coefficients
Authors: Yang, Yanrong
Pan, Guangming
Issue Date: 2012
Source: Yang, Y. & Pan, G. (2012). The convergence of the empirical distribution of canonical correlation coefficients. Electronic Journal of Probability, 17(64), 1-13.
Series/Report no.: Electronic journal of probability
Abstract: Suppose that {Xij, j = 1,..., p1; k = 1,...,n} are independent and identically distributed (i.i.d) real random variables with EX11 = 0 and EX112 = 1, and that {Yjk, j = 1,..., p2; k = 1,..., n} are i.i.d real random variables with EY11 = 0 and EY112 = 1, and that {Xjk, j = 1,..., p1; k = 1,..., n} are independent of {Yjk, j = 1,..., p2; k = 1,..., n}. This paper investigates the canonical correlation coefficients r1 ≥ r2 ≥ ... ≥ rp1, whose squares λ1 = r12, λ2 = r22,..., λp1 = rp12 are the eigenvalues of the matrix Sxy = Ax-1AxyAy-1AxyT, where Ax = 1/n∑xkxkT, Ay = 1/n∑ykykT, Axy = 1/n∑xkykT, and xk = (X1k,..., Xp1k)T, yk = (Y1k,..., Yp2k)T, k = 1,..., n. When p1 → ∞, p2 → ∞ and n → ∞ with p1/n → c1, p2/n → c2, c1, c2 ∈ (0,1), it is proved that the empirical distribution of r1, r2,..., rp1 converges, with probability one, to a fixed distribution under the finite second moment condition.
URI: https://hdl.handle.net/10356/96095
http://hdl.handle.net/10220/10108
DOI: http://dx.doi.org/10.1214/EJP.v17-2239
Rights: © 2012 The Authors. This paper was published in Electronic Journal of Probability and is made available as an electronic reprint (preprint) with permission of The Authors. The paper can be found at the following official DOI: [http://dx.doi.org/10.1214/EJP.v17-2239].  One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.
Fulltext Permission: open
Fulltext Availability: With Fulltext
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