Local Maxima of a Random Algebraic Polynomial
نویسندگان
چکیده
We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a levelu. In the derivation we assume chiefly that ξ(t), ξ(t), and ξ(t) have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to above is then used to find the expected number of maxima below the level u for the random algebraic polynomial. This result highlights the very pronounced difference in the behaviour of the random algebraic polynomial on the interval (−1,1) compared with the intervals (−∞,−1) and (1,∞). It is also shown that the number of maxima below the zero level is no longer O(logn) on the intervals (−∞,−1) and (1,∞). 2000 Mathematics Subject Classification. Primary 60H99, 26C99.
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