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Level crossings of a random polynomial with hyperbolic elements. (English) Zbl 0802.60049

This paper provides an asymptotic estimate for the expected number of \(K\)-level crossings of a random hyperbolic polynomial \(g_ 1 \cosh x+ g_ 2 \cosh 2x+ \dots + g_ n \cosh nx\), where \(g_ j\) \((j=1,2,\dots,n)\) are independent normally distributed random variables with mean zero, variance one and \(K\) is any constant independent of \(x\). It is shown that the result for \(K=0\) remains valid as long as \(K\equiv K_ n= O(\sqrt{n})\). The results for the cases of \(\mu\neq 0\) and \(K=0\) are also discussed and compared with the random algebraic and trigonometric polynomials.

MSC:

60G99 Stochastic processes
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