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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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