Kesten–McKay Law for Random Subensembles of Paley Equiangular Tight Frames

Moments of Subsets of General Equiangular Tight Frames

“…Note that for γ = 0.5, s = 0 and we get that for odd cycle, i.e., odd l i , A li = 0, and for even l i , A li = (−1) li/2−1 • C li/2−1 . Thus if G π contains any odd cycle, then V n (π) → 0 otherwise V n (π) converges to a product of Catalan numbers, in agreement with Lemma 16 in [1].…”

“…Note that x/(n − 1) is the Welch bound, i.e., the squared absolute cross correlation between the frame vectors. For γ = 0.5, we have x = 1, s = 0, so S = √ n − 1 • (F ′ F − I n ), and S ′ S = (n − 1)I, in agreement with the properties of the conference matrix in [1]. § Now at Amazon.…”

“…We extend the proof in [1] to derive the expression for moments of the random submatrix X S = P SP defined as…”

“…The main difference from the KM case will be the computation of V n (π). Lemma 16 in [1] will be replaced by Lemma 1. For every non-crossing partition π ∈ (k, t), the following holds: if the edges of G π partition into m simple cycles of sizes l 1 , .…”

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This note outlines the steps for proving that the moments of a randomly-selected subset of a general ETF (complex, with aspect ratio 0 < γ < 1) converge to the corresponding MANOVA moments. We bring here an extension for the proof of the 'Kesten-Mckay' moments (real ETF, γ = 1/2) [1]. In particular, we establish a recursive computation of the rth moment, for r = 1, 2, .

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