Topics in Random Polynomials presents a rigorous and comprehensive treatment of the mathematical behavior of different types of random polynomials. These polynomials-the subject of extensive recent research-have many applications in physics, economics, and statistics. The main results are presented in such a fashion that they can be understood and used by readers whose knowledge of probability incorporates little more than basic probability theory and stochastic processes.

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Random Polynomials focuses on a comprehensive treatment of random algebraic, orthogonal, and trigonometric polynomials. The publication first offers information on the basic definitions and properties of random algebraic polynomials and random matrices. Discussions focus on Newton's formula for random algebraic polynomials, random characteristic polynomials, measurability of the zeros of a random algebraic polynomial, and random power series and random algebraic polynomials. The text then elaborates on the number and expected number of real zeros of random algebraic polynomials; number and expected number of real zeros of other random polynomials; and variance of the number of real zeros of random algebraic polynomials. Topics include the expected number of real zeros of random orthogonal polynomials and the number and expected number of real zeros of trigonometric polynomials. The book takes a look at convergence and limit theorems for random polynomials and distribution of the zeros of random algebraic polynomials, including limit theorems for random algebraic polynomials and random companion matrices and distribution of the zeros of random algebraic polynomials. The publication is a dependable reference for probabilists, statisticians, physicists, engineers, and economists.

Examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients. This title presents a primer on modern techniques on the interface of probability and analysis.

In Part 2, we consider an n-step simple symmetric random walk {Sk} on Z2 with the final point Sn= (pn, q n), which is motivated by group theory. When n & rarr; infinity, we prove that with probability tending to 1 there exists a line l whose slope is qnpn such that S0, S 1 ..., Sn meet l once at a unique point. This answers an open conjecture from group theory, which is given by Sapir. In the last part, we consider the real random power series fU (z) = Sinfinityi=0 bizi with i.i.d. standard real normal coefficients {bn} and U = ( -l, 1). With a very simple proof, we obtain concise analytical expressions for n-point correlations between real zeros of fU (z) in the unit interval U = ( -1, 1). Consider the zero set of a Gaussian analytic function f(z) which is an at least 3-dimensional polynomial in C (its values form an at least 3 dimensional vector space as random variables). Virag conjectures that there are always two points z1 and z2 such that p(z1, z2)> p(z1)p(z2), where p(z) is the intensity of the zero process at z and p(z1, z2) is the joint intensity. In the first part, we prove that the above conjecture is true for f(z) = Snk=0 akbkzk where {an} are i.i.d. standard complex Gaussian coefficients and {bn} are non-random constants. We consider more general cases f(z) = A 0 + A1z + A 2z2 where (A0,A1,A2) are jointly Gaussian random variables, and prove that the above conjecture is also true. Furthermore, we consider f(z) = Snk=0 akzk. We get the rates of Convergence for hole probability (there is no zero of the polynomial in this disk) and full probability (all zeros of the polynomial are contained in this disk).