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Additional resources for A Bayesian forecasting model: predicting U.S. male mortality (2006)(en)(21s)

Sample text

One of the first things we learn in linear algebra is that the set of eigenvalues of M is the same as the solution set of F„(A) = 0. The same result holds almost surely for the random eigenvalues of random matrices and the solution sets of the associated random characteristic equations Fn(A, ω) = 0. We are of the opinion that the best way to determine the eigenvalues of a random matrix is to use simulation to generate random matrices whose elements have a given distribution, and then use a polynomial solver to obtain the zeros of the associated random characteristic poly­ nomials.

Hence d(œ) ^ n almost surely ; and since μ([ω : αη(ω) ^ 0)) > 0 we have d* = n. We will use the notation Fn(z, ώ) for a random algebraic polynomial of degree n. 22 2. 3. 2 random algebraic polynomials were defined via random power series. We now give some other definitions of random algebraic polynomials, several of which are based on notions from probabilistic functional analysis. , αη(ω)): Ω -► (R„+i, (B(R„+i)). To be more precise, Kac [10] has considered the following "model" for a random algebraic polynomial.

Addison-Wesley, Reading, Massachusetts, 1972. 19. 20. 21. 22. 23. 24. 1. 1) be a random algebraic polynomial of degree n, as defined in Chapter 1. As before, let Nn(B, ω) denote the number of zeros of Fn(z, ω) (equivalently, the number of solutions of the random algebraic equation Fn(z, ω) = 0) in a Borei set B C D where D C Z is in the domain of F„(z, ω). During the past 30-40 years, most of the studies on random algebraic polynomials have been concerned with the estimation of Nn(R, ω), the number of real zeros, v„(R), the expected number of real zeros, and Vn[R], the variance of the number of real zeros, and the distribution of the number of real zeros.

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