### 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 Ω((log 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√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} 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, ∞). Finally, we show that for the sequence k_{p} = 2_{p} the limit eigenvalue distribution is the famous Marchenko-Pastur distribution.

Original language | English |
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Pages (from-to) | 826-862 |

Number of pages | 37 |

Journal | Journal of Theoretical Probability |

Volume | 27 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2014 |

Externally published | Yes |

### Keywords

- Limit eigenvalue distribution
- Random matrices
- Vandermonde matrices