Random entire functions from random polynomials with real zeros

We point out a simple criterion for convergence of polynomials to concrete entire function in the Laguerre-Pólya ( LP ) class (of all functions arising as uniform limits with only real roots). then use this show that any random can be obtained limit rescaled characteristic principal submatrices an infinite unitarily invariant Hermitian matrix. Conversely, matrix converge uniformly function. This result also has natural extension β -ensembles. Distinguished cases include associated -Sine, and more generally -Hua-Pickrell, -Bessel -Airy processes studied literature.