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Mean number of real zeros of a random trigonometric polynomial. IV. (English) Zbl 0880.60057

Summary: [For parts I and II (by the author) see Proc. Am. Math. Soc. 111, No. 3, 851-863 (1991; Zbl 0722.60047) and in: Topics in polynomials of one and several variables and their applications, 581-594 (1993; Zbl 0857.60047), respectively, and for part III (by the author and S. A. Souter) see J. Appl. Math. Stochastic Anal. 8, No. 3, 299-317 (1995; Zbl 0828.60035).]
If \(a_j\) \((j=1,2,\dots,n)\) are independent, normally distributed random variables with mean 0 and variance 1, if \(p\) is one half of any odd positive integer except one, and if \(\nu_{np}\) is the mean number of zeros on \((0,2\pi)\) of the trigonometric polynomial \(a_1\cos x+ 2^pa_2\cos 2x+\cdots+ n^pa_n\cos nx\), then \[ \nu_{np}= \mu_p\{(2n+1) D_{1p}+ (2n+1)^{-1}D_{2p}+(2n+ 1)^{-2}D_{3p}\}+ O\{(2n+1)^{-3}\}, \] in which \(\mu_p= \{(2p+1)/(2p+ 3)\}^{1/2}\), and \(D_{1p}\), \(D_{2p}\) and \(D_{3p}\) are explicitly stated constants.

MSC:

60G99 Stochastic processes
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