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- Title
ON THE MOMENTS OF ROOTS OF LAGUERRE-POLYNOMIALS AND THE MARCHENKO-PASTUR LAW.
- Authors
Kornyik, Miklós; Michaletzky, György
- Abstract
In this paper we compute the leading terms in the sum of the kth power of the roots of Lp(a) the Laguerre-polynomial of degree p with parameter a. 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 (a + p) of the sum of the kth power of the roots of Lp(a) coincides with the kth moment of the Marchenko-Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a XXT type random covariance matrix, where X is a p × n random matrix with iid elements, is ℓp(a-p), i.e. the monic version of the pth Laguerre polynomial with parameter n -- p.
- Subjects
LAGUERRE polynomials; ROOTS of equations; MOMENTS method (Statistics); MARCHENKO equation; DISTRIBUTION (Probability theory)
- Publication
Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, 2017, Vol 46, p137
- ISSN
0138-9491
- Publication type
Article