The Annals of Statistics

The middle-scale asymptotics of Wishart matrices

Didier Chételat and Martin T. Wells

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We study the behavior of a real $p$-dimensional Wishart random matrix with $n$ degrees of freedom when $n,p\rightarrow\infty$ but $p/n\rightarrow0$. We establish the existence of phase transitions when $p$ grows at the order $n^{(K+1)/(K+3)}$ for every $K\in\mathbb{N}$, and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the $\mathcal{F}$-conjugate of an absolutely continuous distribution, which is obtained from the Fourier transform of the square root of its density. In the case of the normalized Wishart distribution, this represents an extension of the $t$-distribution to the space of real symmetric matrices.

Article information

Ann. Statist., Volume 47, Number 5 (2019), 2639-2670.

Received: July 2017
Revised: August 2018
First available in Project Euclid: 3 August 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B10: Convergence of probability measures
Secondary: 60E10: Characteristic functions; other transforms

Covariance estimation Gaussian orthogonal ensemble high-dimensional asymptotics phase transitions random matric theory


Chételat, Didier; Wells, Martin T. The middle-scale asymptotics of Wishart matrices. Ann. Statist. 47 (2019), no. 5, 2639--2670. doi:10.1214/18-AOS1760.

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  • Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • Bartlett, M. S. (1933). On the theory of statistical regression. Proc. Roy. Soc. Edinburgh Sect. A 53 260–283.
  • Bubeck, S. and Ganguly, S. (2018). Entropic CLT and phase transition in high-dimensional Wishart matrices. Int. Math. Res. Not. IMRN 2 588–606.
  • Bubeck, S., Ding, J., Eldan, R. and Rácz, M. Z. (2016). Testing for high-dimensional geometry in random graphs. Random Structures Algorithms 49 503–532.
  • Hardy, G. H., Littlewood, J. E. and Pólya, G. (1967). Inequalities, 2nd ed. Cambridge University Press.
  • Hua, L. K. (1963). Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Amer. Math. Soc., Providence, RI.
  • Jiang, T. and Li, D. (2015). Approximation of rectangular beta-Laguerre ensembles and large deviations. J. Theoret. Probab. 28 804–847.
  • Knuth, D. (1976). Big omicron and big omega and big theta. SIGACT News 18–24.
  • Ledoux, M. (2009). A recursion formula for the moments of the Gaussian orthogonal ensemble. Ann. Inst. Henri Poincaré Probab. Stat. 45 754–769.
  • Letac, G. and Massam, H. (2004). All invariant moments of the Wishart distribution. Scand. J. Stat. 31 295–318.
  • Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72 507–536.
  • Matsumoto, S. (2012). General moments of the inverse real Wishart distribution and orthogonal Weingarten functions. J. Theoret. Probab. 25 798–822.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • Rácz, M. Z. and Richey, J. (2018). A smooth transition from Wishart to GOE. J. Theoret. Probab. 1–9.
  • Stein, E. M. and Weiss, G. (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series 32. Princeton Univ. Press, Princeton, NJ.
  • Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A 32–52.