ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES

Alexeev N, Götze F, Tikhomirov A (2010)
LITHUANIAN MATHEMATICAL JOURNAL 50(2): 121-132.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
Let x be a complex random variable such that Ex = 0, E|x|(2) = 1, and E|x|(4) < infinity. Let x(ij), i, j is an element of {1, 2,...}, be independent copies of x. Let X = ( N-(1/2) xij), 1 <= i, j <= N, be a random matrix. Writing X* for the adjoint matrix of X, consider the product (XX)-X-m*(m) with some m <= {1, 2,...}. The matrix (XX)-X-m*(m) is Hermitian positive semidefinite. Let lambda(1), lambda(2),...,lambda(N) be eigenvalues of (XX)-X-m*(m) (or squared singular values of the matrix X-m). In this paper, we find the asymptotic distribution function G((m))(x) = limN ->infinity EFN(m)(x) of the empirical distribution function F-N((m))(x) = N-1 Sigma(N)(k=1) I{lambda(k) <= x}, where I {A} stands for the indicator function of an event A. With m = 1, our result turns to a well- known result of Marchenko and Pastur [V. Marchenko and L. Pastur, The eigenvalue distribution in some ensembles of random matrices, Math. USSR Sb., 1:457- 483, 1967].
Erscheinungsjahr
Zeitschriftentitel
LITHUANIAN MATHEMATICAL JOURNAL
Band
50
Ausgabe
2
Seite(n)
121-132
ISSN
PUB-ID

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Alexeev N, Götze F, Tikhomirov A. ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES. LITHUANIAN MATHEMATICAL JOURNAL. 2010;50(2):121-132.
Alexeev, N., Götze, F., & Tikhomirov, A. (2010). ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES. LITHUANIAN MATHEMATICAL JOURNAL, 50(2), 121-132.
Alexeev, N., Götze, F., and Tikhomirov, A. (2010). ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES. LITHUANIAN MATHEMATICAL JOURNAL 50, 121-132.
Alexeev, N., Götze, F., & Tikhomirov, A., 2010. ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES. LITHUANIAN MATHEMATICAL JOURNAL, 50(2), p 121-132.
N. Alexeev, F. Götze, and A. Tikhomirov, “ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES”, LITHUANIAN MATHEMATICAL JOURNAL, vol. 50, 2010, pp. 121-132.
Alexeev, N., Götze, F., Tikhomirov, A.: ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES. LITHUANIAN MATHEMATICAL JOURNAL. 50, 121-132 (2010).
Alexeev, N., Götze, Friedrich, and Tikhomirov, Alexander. “ASYMPTOTIC DISTRIBUTION OF SINGULAR VALUES OF POWERS OF RANDOM MATRICES”. LITHUANIAN MATHEMATICAL JOURNAL 50.2 (2010): 121-132.