On Distribution of Zeros of Random Polynomials in Complex Plane
Let \(G_{n}(z) = \xi _{0} + \xi _{1}z + \cdots + \xi _{n}{z}^{n}\) be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of G n (z) are uniformly distributed in [0, 2π] asymptotically as \(n\,\rightarro
- PDF / 231,192 Bytes
- 21 Pages / 439.36 x 666.15 pts Page_size
- 5 Downloads / 221 Views
Abstract Let Gn .z/ D 0 C 1 z C C n zn be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of Gn .z/ are uniformly distributed in Œ0; 2 asymptotically as n ! 1. We also prove that the condition E ln.1 C j0 j/ < 1 is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference. Keywords Roots of random polynomial • Roots concentration • Random analytic function
Mathematics Subject Classification (2010): 60-XX, 30C15
1 Inroduction: Problem and Results Let fk g1 kD0 be a sequence of independent identically distributed real- or complexvalued random variables. It is always supposed that P .0 D 0/ < 1. Consider the sequence of random polynomials Gn .z/ D 0 C 1 z C C n1 zn1 C n zn : By z1n ; : : : ; znn denote the zeros of Gn . It is not hard to show (see [1]) that there exists an indexing of the zeros such that for each k D 1; : : : ; n the k-th zero zk n is a one-valued random variable. For any measurable subset A of complex plain
I. Ibragimov () D. Zaporozhets St. Petersburg Department of Steklov Mathematical Institute RAS 27 Fontanka, St. Petersburg 191023, Russia e-mail: [email protected]; [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 18, © Springer-Verlag Berlin Heidelberg 2013
303
304
I. Ibragimov and D. Zaporozhets
C put Nn .A/ D #fzk n W zk n 2 Ag. Then Nn .A/=n is a probability measure on the plane (the empirical distribution of the zeros of Gn ). For any a; b such that 0 6 a < b 6 1 put Rn .a; b/ D Nn .fz W a 6 jzj 6 bg/ and for any ˛; ˇ such that 0 6 ˛ < ˇ 6 2 put Sn .˛; ˇ/ D Nn .fz W ˛ 6 arg z 6 ˇg/. Thus Rn =n and Sn =n define the empiric distributions of jzk n j and arg zk n . In this paper we study the limit distributions of Nn ; Rn ; Sn as n ! 1. The question of the distribution of the complex roots of Gn have been originated by Hammersley in [1]. The asymptotic study of Rn ; Sn has been initiated by Shparo and Shur in [16]. To describe their results let us introduce the function 31C"
2
7 6 f .t/ D 4logC logC : : : logC t 5 „ ƒ‚ … mC1
m Y i D1
logC logC : : : logC t ; „ ƒ‚ … i
where logC s D max.1; log s/. We assume that " > 0; m 2 ZC and f .t/ D .logC t/1C" for m D 0. Shparo and Shur have proved in [16] that if E f .j0 j/ < 1 for some " > 0; m 2 ZC , then for any ı 2 .0; 1/ and ˛; ˇ such that 0 6 ˛ < ˇ 6 2 1 P Rn .1 ı; 1 C ı/ ! 1; n
n ! 1;
1 P ˇ˛ Sn .˛; ˇ/ ! ; n ! 1: n 2 The first relation means that under quite weak constraints imposed on the coefficients of a random polynomial, almost all its roots “concentrate uniformly” near the unit circumference with high probability; the second relation means that the arguments of the roots are asymptotically uniformly distributed. Later Shepp and Vanderbei [15] and Ibragimov and Zeitouni [5] under additional conditions imposed on the coefficients of Gn got more precise asymptotic formulas for
Data Loading...
