Mathematics – Probability
Scientific paper
2012-02-15
Mathematics
Probability
arXiv admin note: text overlap with arXiv:math/0307330 by other authors
Scientific paper
This work 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 $O(\log N^d)$ and $O(\log N^{d} /\log \log N^d)$ respectively where $N$ is the dimension of the matrix, generalizing the results in \cite{TW}. We further study the behavior of the minimum singular value of a random Vandermonde matrix. In particular, we prove that the minimum singular value is at most $N^2\exp(-C\sqrt{N}))$ where $N$ is the dimension of the matrix and $C$ is a constant. 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 with 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. We believe that this has independent mathematical interest. Lastly, for each sequence of positive integers $\{k_p\}_{p=1}^{\infty}$ we present a generalized version of the previously discussed random Vandermonde matrices. The classical random Vandermonde matrix corresponds to the sequence $k_{p}=p-1$. We find a combinatorial formula for their moments and we 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.
Tucci Gabriel H.
Whiting Philip A.
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