Universality results for largest eigenvalues of some sample covariance matrix ensembles
By S. P'ech'e. 2007
116 citations recorded by the source corpus.
For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/cea683b9a1caad6eca1ea69cdfa71e263f8d4372
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Discrete Mathematics and Combinatorics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Limits of spiked random matrices I (2010, methodology): “Remark 1.6 The remarks following the previous theorem apply also to this theorem; the universality issue is discussed in Féral and Péché [25].”
- On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, background): “the proof of the circular law in a general form [36] as well as significant weakening of the assumptions leading to the Tracy-Widom distribution for the operator norm [33, 30] or the sine law for local eigenvalues statistics (see e.”
- Random matrix theory in statistics: A review (2014, background): “Péché (2009) extended the results of Soshnikov (2002a) by allowing the dimension to sample size ratio to approach any nonnegative value.”
- Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, background): “It was established in the null case in [36, 33, 21] for symmetric distributions and subsequently in [41] for distributions with vanishing third moment.”
- Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “…including likelihood ratio test in Bai et al. (2009) and Jiang et al. (2012), largest eigenvalue test in Johnstone (2001), Soshnikov (2002) and Peche (2009), Frobenius distance test in Ledoit and Wolf (2002), Srivastava (2005), Birke and Dette (2005), and Chen et al. (2010), and maximum…”
- The Surprising Mathematics of Longest Increasing Subsequences (2015, background): “This was proved in various levels of generality in [97], [99], [122], [144].”
- Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs (2017, methodology): “The moment method was further developed in [30, 50] and [54].”
- Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “See Johnstone (2001); Soshnikov (2002); Peche (2009) for further details.”
- Universality of covariance matrices (2011, methodology): “Using similar assumptions as in [34] and [31], Feldheim and Sodin [19] showed that the smallest eigenvalues (appropriately rescaled) converge to the Tracy-Widom distribution for limN→∞ dN 6= 1.”
- TESTING HYPOTHESES ABOUT THE NUMBER OF FACTORS IN LARGE FACTOR MODELS (2009, background): “important universality results were recently obtained by Soshnikov (2002) and Péché (2009). Further development of such results remains a challenge for mathematicians.”
- Factor modeling for high-dimensional time series: inference for the number of factors (2012, unknown): “See also conditions (i′)–(iv′) of [17].”
- The local relaxation flow approach to universality of the local statistics for random matrices (2009, unknown): “Since the works of Sinai and Soshnikov [35, 37], the edge universality is commonly approached via the fairly robust moment method [33, 22, 36, 38, 34]; very recently an alternative approach was given in [40].”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “The readers can refer to [43, 46, 49, 56] for some representative developments on this topic.”