A necessary and sufficient condition for convergence of the zeros of random polynomials

Zeros of Random Polynomial Mappings in Several Complex Variables

“…Suppose d = 1. The condition that {q j } j be asymptotically Chebyshev for K is the same as the notion in [9] of asymptotically minimal for the case p = ∞. Moreover, it is straightforward to see that {q j } j asymptotically Chebyshev for K is equivalent to…”

“…4. A strong converse to Theorem 4.1 in C was given by Dauvergne in [9]: for univariate random polynomials H n (z) = n j=0 a j p j (z) where {p j } are asymptotically minimal for K ⊂ C (see Remark 2.3) and {a j } are nondegenerate i.i.d. random variables, if E log(1 + |a j | = ∞, then the sequence µ Hn has no almost sure limit (in the space of probability measures on C).…”

“…First, we allow {p j } to be an asymptotically Chebyshev family which need not come from orthonormal polynomials; and second, we strive for the weakest conditions on the i.i.d. complex random variables a j in order to have the appropriate convergence as, e.g., in [9] and [6].…”

“…Pritsker and Ramachandran [14] gave some almost sure convergence results for special polynomial bases on certain compacta in C. Bloom and Dauvergne [6] showed that for a nonpolar compact set K ⊂ C with equilibrium measure µ K , given a Bernstein-Markov measure µ on K and an orthonormal basis of polynomials {p j } for L 2 (µ), if P(|a j | > e |z| ) = o(1/|z|), then µ Hn → µ K weakly in probability. Dauvergne generalized these univariate results in [9]: for his class of asymptotically minimal polynomials {p j } for K, which includes orthonormal polynomials for a Bernstein-Markov measure µ on K and essentially all "classical" sets of polynomials (Chebyshev, Fekete, Fejer, etc. ), the condition E(log(1 + |a j )) < ∞ is a necessary and sufficient condition for the sequence µ Hn → µ K weakly almost surely while the condition P(|a j | > e |z| ) = o(1/|z|) is a necessary and sufficient condition for the sequence µ Hn → µ K weakly in probability.…”

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

“…Many subsequent results on random polynomials are concerned with relaxing the conditions on random coefficients, see, for example, [13,18,10], or the behavior of the counting measures of zeros of random polynomials as in [21,6,14,5,19,2,20,17,4,9]. Our primary interest lies in studying the expected number of real zeros when the basis is a family of orthogonal polynomials in the spirit of [7,8,26,16].…”