ON EXPECTED NUMBER OF LEVEL CROSSINGS OF A RANDOM HYPERBOLIC POLYNOMIAL

Let g(1)(w), g(2)(w), ..., g(n)(w) 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)(w) cosh x g(2)(w) cosh 2x + ... + g(n)(w) cosh nx crosses the line y = L, where L is a real number, is 1/pi log n + O(1) if L = o(root n) or L/root n = O(1), but decreases steadily as O(L) increases in magnitude and ultimately becomes negligible when n(-1) log L/root n -> infinity.