3 research outputs found
Spectral measure of large random Hankel, Markov and Toeplitz matrices
We study the limiting spectral measure of large symmetric random matrices of
linear algebraic structure. For Hankel and Toeplitz matrices generated by
i.i.d. random variables of unit variance, and for symmetric Markov
matrices generated by i.i.d. random variables of zero mean
and unit variance, scaling the eigenvalues by we prove the almost
sure, weak convergence of the spectral measures to universal, nonrandom,
symmetric distributions , and of unbounded
support. The moments of and are the sum of volumes of
solids related to Eulerian numbers, whereas has a bounded smooth
density given by the free convolution of the semicircle and normal densities.
For symmetric Markov matrices generated by i.i.d. random variables
of mean and finite variance, scaling the eigenvalues by
we prove the almost sure, weak convergence of the spectral measures to
the atomic measure at . If , and the fourth moment is finite, we prove
that the spectral norm of scaled by converges
almost surely to 1.Comment: Published at http://dx.doi.org/10.1214/009117905000000495 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org