An Introduction to Probability Theory and Its Applications, by William Feller

By William Feller

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Major alterations during this version contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching tactics, Markov chains, and the De Moivre-Laplace theorem.

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Additional resources for An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition)

Sample text

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. Given a deterministic algebraic polynomial of degree n with real coefficients we associate the point a = (#o, #i, ·.

The following result is due to Arnold [1]. 3. Let F„(z, ω), z e D, be a random algebraic polynomial of degree n, and let ΐίί = [B : B C D, Nn(B, ώ) is a random variable}. Then (R(D) C 91. 26 2. Basic Definitions and Properties We will need the following lemma. 1. Let hn and h be holomorphic functions on D, with hn converging to h ^ 0 uniformly on every compact subset of D. If Nn and N are the number of zeros ofhn and Λ, then Nn converges vaguely^ to N. Proof. Let A be a continuity set of N. Since the zeros of h in A are interior points of A, whereas all other zeros of h are exterior points, it follows from Hurwitz's theorem* that Nn(A) = N(A) for all n ^ no.

3 is devoted to estimates to £>[Nn(B, ω)} when the coefficients of 49 50 4. 1) are real-valued random variables. 1) are complex-valued random variables. 2. ESTIMATES OF Nn(B, ω) A. Introduction Consider a random algebraic polynomial Fn(z, ω) of degree n. 2B we define the upper and lower bounds of Nn(B, ω), and present some estimates when the coefficients are normally distributed. These estimates are given in terms of probability measure, since Nn(B, ώ) is a random variable for every fixed B e (B(£>).