101 research outputs found

    How many entries of a typical orthogonal matrix can be approximated by independent normals?

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    We solve an open problem of Diaconis that asks what are the largest orders of pnp_n and qnq_n such that Zn,Z_n, the pn×qnp_n\times q_n upper left block of a random matrix Γn\boldsymbol{\Gamma}_n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of ZnZ_n and that of pnqnp_nq_n independent standard normals goes to zero provided pn=o(n)p_n=o(\sqrt{n}) and qn=o(n)q_n=o(\sqrt{n}). We also show that the above variation distance does not go to zero if pn=[xn]p_n=[x\sqrt{n} ] and qn=[yn]q_n=[y\sqrt{n} ] for any positive numbers xx and yy. This says that the largest orders of pnp_n and qnq_n are o(n1/2)o(n^{1/2}) in the sense of the above approximation. Second, suppose Γn=(γij)n×n\boldsymbol{\Gamma}_n=(\gamma_{ij})_{n\times n} is generated by performing the Gram--Schmidt algorithm on the columns of Yn=(yij)n×n\bold{Y}_n=(y_{ij})_{n\times n}, where {yij;1i,jn}\{y_{ij};1\leq i,j\leq n\} are i.i.d. standard normals. We show that ϵn(m):=max1in,1jmnγijyij\epsilon_n(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\cdot\gamma_{ij}-y_{ij}| goes to zero in probability as long as m=mn=o(n/logn)m=m_n=o(n/\log n). We also prove that ϵn(mn)2α\epsilon_n(m_n)\to 2\sqrt{\alpha} in probability when mn=[nα/logn]m_n=[n\alpha/\log n] for any α>0.\alpha>0. This says that mn=o(n/logn)m_n=o(n/\log n) is the largest order such that the entries of the first mnm_n columns of Γn\boldsymbol{\Gamma}_n can be approximated simultaneously by independent standard normals.Comment: Published at http://dx.doi.org/10.1214/009117906000000205 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    The asymptotic distributions of the largest entries of sample correlation matrices

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    Let X_n=(x_{ij}) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let R_n=(\rho_{ij}) be the p\times p sample correlation matrix of X_n; that is, the entry \rho_{ij} is the usual Pearson's correlation coefficient between the ith column of X_n and jth column of X_n. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H_0: the p variates of the population are uncorrelated. A test statistic is chosen as L_n=max_{i\ne j}|\rho_{ij}|. The asymptotic distribution of L_n is derived by using the Chen-Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived

    Approximation of Rectangular Beta-Laguerre Ensembles and Large Deviations

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    We investigate the random eigenvalues coming from the beta-Laguerre ensemble with parameter p, which is a generalization of the real, complex and quaternion Wishart matrices of parameter (n,p). In the case that the sample size n is much smaller than the dimension of the population distribution p, a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when n is much smaller than p, we show that the largest and smallest eigenvalues of the complex Wishart matrix are asymptotically independent; we obtain the limiting distribution of the condition numbers as a sum of two i.i.d. random variables with a Tracy-Widom distribution, which is much different from the exact square case that n=p by Edelman (1988); we propose a test procedure for a spherical hypothesis test. By the same approximation tool, we obtain the asymptotic distribution of the smallest eigenvalue of the beta-Laguerre ensemble. In the second part of the paper, under the assumption that n is much smaller than p in a certain scale, we prove the large deviation principles for three basic statistics: the largest eigenvalue, the smallest eigenvalue and the empirical distribution of eigenvalues, where the last large deviation is derived by using a non-standard method

    Random restricted partitions

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    We study two types of probability measures on the set of integer partitions of nn with at most mm parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as nn tends to infinity while mm is fixed or tends to infinity. In particular, if mm goes to infinity not fast enough, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We derive the asymptotic distributions of the parts jointly and that of the largest part by taking limit of nn and mm in the same manner as that in the first probability measure.Comment: 32 page

    Moments of traces of circular beta-ensembles

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    Let θ1,,θn\theta_1,\ldots,\theta_n be random variables from Dyson's circular β\beta-ensemble with probability density function Const1j<kneiθjeiθkβ\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_j}-e^{i\theta _k}|^{\beta}. For each n2n\geq2 and β>0\beta>0, we obtain some inequalities on E[pμ(Zn)pν(Zn)ˉ]\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}], where Zn=(eiθ1,,eiθn)Z_n=(e^{i\theta_1},\ldots,e^{i\theta_n}) and pμp_{\mu} is the power-sum symmetric function for partition μ\mu. When β=2\beta=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: limnE[pμ(Zn)pν(Zn)ˉ]=δμν(2β)l(μ)zμ \lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}]= \delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu} for any β>0\beta>0 and partitions μ,ν\mu,\nu; limmE[pm(Zn)2]=n\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n for any β>0\beta>0 and n2n\geq2, where l(μ)l(\mu) is the length of μ\mu and zμz_{\mu} is explicit on μ\mu. These results apply to the three important ensembles: COE (β=1\beta=1), CUE (β=2\beta=2) and CSE (β=4\beta=4). We further examine the nonasymptotic behavior of E[pm(Zn)2]\mathbb{E}[|p_m(Z_n)|^2] for β=1,4\beta=1,4. The central limit theorems of j=1ng(eiθj)\sum_{j=1}^ng(e^{i\theta_j}) are obtained when (i) g(z)g(z) is a polynomial and β>0\beta>0 is arbitrary, or (ii) g(z)g(z) has a Fourier expansion and β=1,4\beta=1,4. The main tool is the Jack function.Comment: Published at http://dx.doi.org/10.1214/14-AOP960 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
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