Balanced random and Toeplitz matrices

Aniran Basak (Stanford University)
Arup Bose (Indian Statistical Institute)

Abstract


Except for the Toeplitz and Hankel matrices, the common patterned matrices for which the limiting spectral distribution (LSD) are known to exist share a common property–the number of times each random variable appears in the matrix is (more or less) the same across the variables. Thus it seems natural to ask what happens to the spectrum of the Toeplitz and Hankel matrices when each entry is scaled by the square root of the number of times that entry appears in the matrix instead of the uniform scaling by $n^{−1/2}$. We show that the LSD of these balanced matrices exist and derive integral formulae for the moments of the limit distribution. Curiously, it is not clear if these moments define a unique distribution

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 134-148

Publication Date: April 27, 2010

DOI: 10.1214/ECP.v15-1537

References

  1. Bai, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices, a review. Statistica Sinica 9, 611--677 (with discussions).
  2. Bai, Z. D. and Silverstein, J. (2006). Spectral Analysis of Large Dimensional Random Matrices. Science Press, Beijing.
  3. Basak, Anirban (2009). Large dimensional random matrices. M.Stat. Project Report, May 2009. Indian Statitstical Institute, Kolkata.
  4. Bhamidi, Shankar; Evans, Steven N. and Sen, Arnab (2009). Spectra of large random Trees. Available at http://front.math.ucdavis.edu/0903.3589. %
  5. Bhattacharya, R. N.; Ranga Rao, R. Normal approximation and asymptotic expansions.Wiley Series in Probability and Mathematical Statistics.John Wiley \& Sons, New York-London-Sydney, 1976. xiv+274 pp. MR0436272 (55 #9219)
  6. Bose, Arup; Gangopadhyay, Sreela and Sen, Arnab (2009). Limiting spectral distribution of $XX^{\prime}$ matrices. Annales de l'Institut Henri PoincarÈ. To appear. Currently available at http://imstat.org/aihp/accepted.html.
  7. Bose, Arup; Sen, Arnab. Another look at the moment method for large dimensional random matrices. Electron. J. Probab. 13 (2008), no. 21, 588--628. MR2399292 (2009d:60049)
  8. Bryc, Wƚodzimierz; Dembo, Amir; Jiang, Tiefeng. Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 (2006), no. 1, 1--38. MR2206341 (2007c:60039)
  9. Hammond, Christopher; Miller, Steven J. Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theoret. Probab. 18 (2005), no. 3, 537--566. MR2167641 (2006h:15023)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.