Large deviations of condition numbers of random matrices

Abstract: Random matrix theory has found many applications in various fields such as physics, statistics, number theory and so on. One important approach of studying random matrices is based on their spectral properties. In this thesis, we investigate the limiting behaviors of condition numbers of suitable random matrices in terms of large deviations.The thesis is divided into two parts. Part I, provides to the readers an short introduction on the theory of large deviations, some spectral properties of random matrices, and a summary of the results we derived, and in Part II, two papers are appended. In the first paper, we study the limiting behaviors of the 2-norm condition number of p x n random matrix in terms of large deviations for large n and p being fixed or p = p(n) → ∞ with p(n) = o(n). The entries of the random matrix are assumed to be i.i.d. whose distribution is quite general (namely sub- Gaussian distribution). When the entries are i.i.d. normal random variables, we even obtain an application in statistical inference. The second paper deals with the β-Laguerre (or Wishart) ensembles with a general parameter β > 0. There are three special cases β = 1, β = 2 and β = 4 which are called, separately, as real, complex and quaternion Wishart matrices. In the paper, large deviations of the condition number are achieved as n → ∞, while p is either fixed or p = p(n) → ∞ with p(n) = o(n/ln(n)).

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