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On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart Matrix, in preparation
, 2010
"... The ratio of the largest eigenvalue divided by the trace of a p × p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate expli ..."
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The ratio of the largest eigenvalue divided by the trace of a p × p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p, n → ∞ with p/n → c. Our analysis reveals that even though asymptotically in this limit the ratio follows a TracyWidom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is nonnegligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p, n. 1
DOI 10.1007/s1095501101555 Asymptotics for the Covariance of the Airy2 Process
"... this paper we compute some of the higher order terms in the asymptotic behavior of the two point function P(A2(0) ≤ s1, A2(t) ≤ s2), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005)and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We pr ..."
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this paper we compute some of the higher order terms in the asymptotic behavior of the two point function P(A2(0) ≤ s1, A2(t) ≤ s2), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005)and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Painlevé II function q and its derivative q ′. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the TracyWidom GUE density function f2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the TracyWidom GUE distribution.