## 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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