## On expected number of level crossings of a random hyperbolic polynomial.(English)Zbl 1337.60044

Summary: Let $$g_1(\omega),g_2(\omega),\dots, g_n(\omega)$$ be independent and normally distributed random variables with mean zero and variance one. We show that, for large values of $$n$$, the expected number of times the random hyperbolic polynomial $$y=g_1(\omega)\cosh x+ g_2(\omega)\cosh 2x+\cdots +g_n(\omega)\cosh nx$$ crosses the line $$y=L$$, where $$L$$ is a real number, is $$\frac{1}{\pi}\log n +O(1)$$ if $$L=o(\sqrt{n})$$ or $${L}/{\sqrt{n}} =O(1)$$, but decreases steadily as $$O(L)$$ increases in magnitude and ultimately becomes negligible when $$n^{-1}\log {L}/{\sqrt{n}}\to\infty$$.

### MSC:

 60F99 Limit theorems in probability theory 60H99 Stochastic analysis
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### References:

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