International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 5, Pages 331-343
doi:10.1155/S016117120100391X

Local maxima of a random algebraic polynomial

Abstract

We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. 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,∞).