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...′W . If σmin(A) ≥ δ > 0, then L(ran(A), ran(B)) ≤ ‖R‖2/δ2 ≤ ‖E‖2/δ2. The last inequality holds as R = A′W = −E′W . Next, we present two probabilistic bounds on matrix norms. Proposition B.1 ([22] and =-=[8]-=-). Let Y be an n × p matrix with i.i.d. N(0, 1) entries. For tn = 6 √ log n/n and any fixed c > 0, there exist n0(c) > 0, such that for any n ≥ n0(c), P { ‖ 1 n Y ′Y − I‖ ≥ 2 √ p n + p n + ctn } ≤ 2n−...

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...n, let us mention two important recent results. In [37], Rudelson and Vershynin obtained strong tail estimates for the smallest singular value of random rectangular matrices of all possible sizes. In =-=[15]-=- Feldheim and Sodin considered the case when l is large, l = Θ(n) and proved universality for the distribution of the least singular value of random matrices with entries having sub-gaussian tails. (I...

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...s [36] etc. Also in similarity to the GUE case, it is widely conjectured that these statistics hold for a much larger class of random matrices. For some earlier results in this direction, we refer to =-=[43, 47, 3, 18]-=- and the references therein. The goal of this paper is to establish a Four Moment theorem for random covariance matrices, as an analogue of a recent result in [48]. This theorem asserts that all local...

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...he asymptotical region. ) (6) Novel results from RMT have very recently filled the gap in our knowledge on the limiting eigenvalue distributions in Wishart matrices. In particular, Feldheim and Sodin =-=[9]-=- found that the smallest eigenvalue also converges in distribution to the Tracy-Widom law as N → ∞, up to a proper rescaling factor. Thus, the random variable: Lmin = lmin − a µ converges to the Tracy...

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...ard moment method argument combined with refined combinatorics. Obtaining the bound (7.16) is fairly involved; it makes use of the Chebyshev polynomial representation first used by Feldheim and Sodin =-=[22,27]-=- in this context for a special distribution of hij , and extended in [5] to general symmetric entries. Proof of Theorem 7.3. We shall prove a lower bound on the smallest eigenvalue λ1 of H; the larges...

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... square regime m − n = O(1) (and under stronger moment assumptions), while Corollary 1.2 is valid for all dimensions m ≥ n. Another recent universality result was obtained by O. Feldheim and S. Sodin =-=[12]-=- for genuinely rectangular matrices, i.e. with aspect ratio m/n separated from 1 by a constant, and with subgaussian i.i.d. entries. In particular they proved the inequality (1.10) P ( smin(A) ≤ ( √ m...

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...ave the special form (2.1a) or (2.1b) is not necessary for our results to hold. We make it here because it greatly simplifies our proof. The reason for this is that, as observed by Feldheim and Sodin =-=[26, 42]-=-, the condition (2.2) allows one to obtain a simple algebraic expression for the nonbacktracking powers of H; see Lemma 5.2. In the forthcoming paper [16] we extend our results to random matrix ensemb...