arXiv subjects
Publications and source records attributed to György Michaletzky.
Given a distribution in the unite square and having iid sample from it the first question what a statistician might do to test the hypothesis that the sample is iid. For this purpose an extension of the Plancherel measure is introduced. Recent literature on asymptotic behavior of Plancherel measure is discussed with extension to the new set up. Models for random permutations are described and the power of different tests is compared.
In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, where $c(z)$ is the generating function of the Catalan numbers $C_k$. In the second proof we compute the leading and the second leading term of the $k^{th}$ moments (as a polynomial in $n$) of $H_n$ and show that the first one coincides with $C_{k/2}$, the $(k/2)^{\rm th}$ Catalan number, where $k$ is even and the second one is given by $-(2^{2k-1}-\binom{2k-1}{k})$. We also mention the known result that the expectation of the characteristic polynomial ($p_n$) of a Wigner random matrix is exactly the Hermite polynomial ($H_n$), i.e. $Ep_n(x)=H_n(x)$, which suggest the presence of a deep connection between the Hermite polynomials and Wigner matrices.
In this paper we compute the leading terms in the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$, the Laguerre-polynomial of degree $p$ with parameter $α$. The connection between the Laguerre-polynomials and the Marchenko-Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko-Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that the leading term in $p$ and $(α+p)$ of the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$ coincides with the $k^{th}$ moment of the Marchenko-Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a $XX^T$ type random covariance matrix, where $X$ is a $p\times n$ random matrix with iid elements, is $\ell^{(n-p)}_p$, i.e. the monic version of the $p^{th}$ Laguerre polynomial with parameter $n-p$.