On minimal singular values of random matrices with correlated entries

Götze F, Naumov A, Tikhomirov A (2015)
RANDOM MATRICES-THEORY AND APPLICATIONS 4(2): 30.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Let X be a random matrix whose pairs of entries X-jk and X-kj are correlated and vectors (X-jk, X-kj), for 1 <= j < k <= n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that EXjk = 0, EX (2)(jk) = 1, for any j, k = 1,..., n and EXjkXkj = rho for 1 <= j < k = n. Let M-n be a non-random nxn matrix with parallel to M-n parallel to <= K-n(Q), for some positive constants K > 0 and Q >= 0. Let s(n) (X + M-n) denote the least singular value of the matrix X + M-n. It is shown that there exist positive constants A and B depending on K, Q, rho only such that P(sn(X + M-n) <= n (A)) = n (B). As an application of this result we prove the elliptic law for this class of matrices with non-identically distributed correlated entries.
Erscheinungsjahr
Zeitschriftentitel
RANDOM MATRICES-THEORY AND APPLICATIONS
Band
4
Ausgabe
2
Art.-Nr.
30
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Götze F, Naumov A, Tikhomirov A. On minimal singular values of random matrices with correlated entries. RANDOM MATRICES-THEORY AND APPLICATIONS. 2015;4(2): 30.
Götze, F., Naumov, A., & Tikhomirov, A. (2015). On minimal singular values of random matrices with correlated entries. RANDOM MATRICES-THEORY AND APPLICATIONS, 4(2), 30. doi:10.1142/S2010326315500069
Götze, F., Naumov, A., and Tikhomirov, A. (2015). On minimal singular values of random matrices with correlated entries. RANDOM MATRICES-THEORY AND APPLICATIONS 4:30.
Götze, F., Naumov, A., & Tikhomirov, A., 2015. On minimal singular values of random matrices with correlated entries. RANDOM MATRICES-THEORY AND APPLICATIONS, 4(2): 30.
F. Götze, A. Naumov, and A. Tikhomirov, “On minimal singular values of random matrices with correlated entries”, RANDOM MATRICES-THEORY AND APPLICATIONS, vol. 4, 2015, : 30.
Götze, F., Naumov, A., Tikhomirov, A.: On minimal singular values of random matrices with correlated entries. RANDOM MATRICES-THEORY AND APPLICATIONS. 4, : 30 (2015).
Götze, Friedrich, Naumov, A., and Tikhomirov, A. “On minimal singular values of random matrices with correlated entries”. RANDOM MATRICES-THEORY AND APPLICATIONS 4.2 (2015): 30.