Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

Gabriel H. Tucci () and Philip A. Whiting ()
Additional contact information
Gabriel H. Tucci: Alcatel–Lucent
Philip A. Whiting: Alcatel–Lucent

Journal of Theoretical Probability, 2014, vol. 27, issue 3, 826-862

Abstract: Abstract This study examines various statistical distributions in connection with random Vandermonde matrices and their extension to $$d$$ -dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be $$O(\log ^{1/2}{N^{d}})$$ and $$\Omega ((\log N^{d} /(\log \log N^d))^{1/2})$$ , respectively, where $$N$$ is the dimension of the matrix, generalizing the results in Tucci and Whiting (IEEE Trans Inf Theory 57(6):3938–3954, 2011). We further study the behavior of the minimum singular value of these random matrices. In particular, we prove that the minimum singular value is at most $$N\exp (-C\sqrt{N}))$$ with high probability where $$C$$ is a constant independent of $$N$$ . Furthermore, the value of the constant $$C$$ is determined explicitly. The main result is obtained in two different ways. One approach uses techniques from stochastic processes and in particular a construction related to the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a lower bound for the maximum absolute value of a random complex polynomial on the unit circle, which may be of independent mathematical interest. Lastly, for each sequence of positive integers $$\{k_p\}_{p=1}^{\infty }$$ we present a generalized version of the previously discussed matrices. The classical random Vandermonde matrix corresponds to the sequence $$k_{p}=p-1$$ . We find a combinatorial formula for their moments and show that the limit eigenvalue distribution converges to a probability measure supported on $$[0,\infty )$$ . Finally, we show that for the sequence $$k_p=2^{p}$$ the limit eigenvalue distribution is the famous Marchenko–Pastur distribution.

Keywords: Random matrices; Limit eigenvalue distribution; Vandermonde matrices; 15B52; 15B51; 60B20 (search for similar items in EconPapers)
Date: 2014
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-012-0466-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:27:y:2014:i:3:d:10.1007_s10959-012-0466-8

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-012-0466-8

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().