Local maxima of a random algebraic polynomial

Abstract

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(log n) on the intervals (−∞, −1) and (1, ∞).2000 Mathematics Subject Classification. Primary 60H99, 26C99. Introduction.A significant amount has been written concerning the mean number of crossings of a fixed level, by both stationary and nonstationary normal processes. Farahmand [5] has given a formula for the expected number of maxima, below a level u, for a normal process under certain assumptions. The initial interest in the stationary case can be traced back to Rice [13], and was subsequently investigated by Ito [7] and Ylvisarer [15]. These works concentrated on level crossings and have been reviewed in the comprehensive book by Cramér and Leadbetter [2]. In this book, Cramér and Leadbetter have also given a formula for the expected number of local maxima. In addition, their method enabled them to find the distribution function for the height of a local maximum. Leadbetter [9], in his treatment of the nonstationary case, gives a result for the mean number of level crossings by a normal process. In his work Leadbetter assumes that the random process has continuous sample functions, with probability one.In what follows we consider ξ(t) to be a real valued normal process. We assume ξ(t), and its first and second derivatives ξ (t) and ξ (t), posses, with probability one, continuous one dimensional distributions, such that the mean number of crossings of any level by ξ(t), and the zero level by ξ (t) are finite. In addition, we assume that the mean of ξ(t), ξ (t), and ξ (t) are all zero. ξ(t) has a local maximum at t = t i , if ξ (t) has a down crossing of the level zero at t i . The local maxima which are of interest here, are those that occur when ξ(t) is also below the level u. The total number of down crossings of the level zero by ξ (t) in (α, β) is defined as M(α, β), and these occur at the points α < t 1 < t 2 < ··· < t M(α,β) < β. We define M u (α, β) as the number of zero down crossings by ξ (t), where 0 ≤ i ≤ M(α, β) and ξ(t i ) ≤ u. The corollary presented below is a corollary to Theorem 1.1 in [5]. The corollary is proved in Section 2.

Algebraic polynomials with non-identical random coefficients

“…For those distributions that do not belong to the above domain, Logan and Shepp [9] and [10] show there is a slight increase to the coefficient (2/π) obtained for EN n (−∞, ∞). Wilkins [14] obtained an interesting result by showing that the error term for the asymptotic formula is small and is in fact independent of n. Other interesting results in this direction are due to Sambandham [12] and [13] and Farahmand and Hannigan [5] and [6]. These results and other related topics are discussed in some detail in two books, one by Bharucha-Reid and Sambandham [1], the other by Farahmand [3].…”