LOCAL MAXIMA OF A RANDOM ALGEBRAIC POLYNOMIAL

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.


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.
Corollary 1.1.For any random process ξ(t) satisfying these conditions we denote the variances of ξ(t), ξ (t), and ξ (t) by A 2 ≡ A 2 (t), B 2 ≡ B 2 (t), and C 2 ≡ C 2 (t), respectively, and let D ≡ D(t), E ≡ E(t), and F ≡ F(t) represent cov{ξ(t), ξ (t)}, cov{ξ(t), ξ (t)}, and cov{ξ (t), ξ (t)}, respectively.Then the expected number of maxima below the level u, where E{ξ(t)} = E{ξ (t)} = E{ξ (t)} = 0, is given by The above corollary is useful for determining the total expected number of maxima of a normal process and also for finding the expected number of maxima of a random polynomial, in both cases these maxima occur below a level u.The result derived by Farahmand [5] for the random trigonometric polynomial is particularly suited to this treatment.
In this paper, we concentrate on the random algebraic polynomial We assume a 0 (ω), a 1 (ω),...,a n−1 (ω) is a sequence of independent, normally distributed random variables defined on a probability space (Ω,A,Pr), each having mean zero and variance one.Let M u (α, β) be the number of maxima of the polynomial T (x) in the interval (α, β) below the level u, and let EM u (α, β) be its mathematical expectation.We list a small number of relevant results for random polynomials below; a more complete reference can be found in the book by Bharucha-Reid and Sambandham [1].Littlewood and Offord [10,11] initiated the study of random algebraic polynomials and Offord [12] continues this work in related fields.Kac [8] found the leading behaviour and an error term for the expected number of zero crossings of the random polynomial (1.3).Farahmand [4] found the expected number of K level crossings in the interval (−1, 1) is asymptotic to (1/π ) log(n/K 2 ), and in the intervals (−∞, −1) and (1, ∞) is asymptotic to (1/2π)log n, provided K 2 /n → 0 as n → ∞.When K 2 /n tends to a nonzero constant Farahmand found the expected number of crossings to be asymptotic to (1/π ) log n for the interval (−∞, ∞).Wilkins [14] in recent times using a delicate method reduced the error term in Kac's result significantly.Das [3] considered the polynomial (1.3) with the same assumptions on the random coefficients.In this work, Das showed that the expected number of local maxima in the whole real line is asymptotic to ( √ 3 + 1) log n/2π .Farahmand and Hannigan [6] showed that if the random coefficients have nonzero mean, µ, then the expected number of maxima is asymptotically half that of the Das result, this result remains valid when µ → 0 and µ → ∞.The expected number of maxima below a level u, for a random trigonometric polynomial with the same random coefficients as (1.3), was considered previously by Farahmand [5].He showed that only about an eighth of the maxima appear below the level zero, or any level u such that u 2 /n → 0 as n → ∞.
In Theorem 1.2, we show that asymptotically all the maxima on the interval (−1, 1) occur below the level u, where u ≥ √ n.Also, we see that asymptotically around a fifth of the maxima occur below the level zero, or any level u such that u/ log n → 0 as n → ∞.On the intervals (−∞, −1) and (1, ∞) the result is very different.In this case, asymptotically all the maxima on the intervals occur below the level u ≥ exp(n/ log n).
A seemingly strange result at first is that the number of maxima below the level zero, or any level u such that u/n k → 0 as n → ∞, is negligible compared with the number of maxima in all.We will show in the proof of Theorem 1.3 that k can be any positive constant.
In Section 3, we will prove the following two theorems.
Theorem 1.2.If the coefficients of T (x) in (1.3) are independent, normally distributed random variables with mean zero and variance one, then for all sufficiently large n, and u ≡ u n defined below, the mathematical expectation of the number of maxima below the level u satisfies , where u ≥ √ n.
(1.4) Theorem 1.3.For the same polynomial T (x) defined above, and for all sufficiently large n, and u ≡ u n defined below, then The second result in Theorem 1.2 is not surprising when we consider the results, outlined earlier, from Farahmand [4].From Kac [8] and Das [3] we know that for the intervals (−∞, −1) and (1, ∞) the expected number of both zero crossings and turning points are asymptotically equal.This suggests that asymptotically all the maxima are above the level zero; this partially explains the first result in Theorem 1.3.

Local maxima of a normal process. Farahmand proved in Theorem 1.1 of [5] that
where p t (x,y,z) denotes the 3-dimensional normal density function for ξ(t), ξ (t), and ξ (t).The 3-dimensional density function is based on the covariance matrix in which and the determinant of the covariance matrix, denoted | |, is We define p t (x, 0,z), not in terms of the variables above, but with the more convenient notation Combining (2.1) and (2.6) we find that By splitting the exponential element in (2.7) into two parts, one of which isolates x, we have (2.8) It is possible now to rewrite the portion of (2.8) referring to x, in terms of v; that is, Therefore, (2.10) At this point we interchange the order of the integration in (2.10) with respect to the variables v and z, thus (2.11) We can integrate with respect to z in (2.11) to find that (2.12) After a little algebraic manipulation we can separate the exponential in the second integral of (2.12) into two parts, that is (2.13) Using (2.13) and integration by substitution, together with our knowledge of the probability distribution function Φ(x) we can rewrite (2.12) as ( This completes the proof of the corollary.

Proof of theorems for the random algebraic polynomial.
In order to find EM u (α, β) for the random algebraic polynomial (1.3), we principally use the result in Corollary 1.1.We also use a result due to Das [3].
Applying Corollary 1.1 to the polynomial (1.3), we find where and Since the coefficients of T (x) in (1.3) are independent, normally distributed random variables with mean zero and variance one, it is a trivial task to show that By repeated differentiation of the geometric sum we obtain Looking at the properties of the random coefficients in (1.3) it is clear that EM u (−b, −a) = EM u (a, b).Obviously, we need only find the leading behaviour for EM u (0, 1) and EM u (1, ∞) to prove the two theorems.
To this end we look initially at the interval 0 < x < 1− , where n ≡ = n −α , and α is any positive value, satisfying The positive constant k is critical to these theorems and we will return to define a set of qualifying k values later.From (3.6), and since for all n sufficiently large,  for all sufficiently large n, all terms inside O{ } in (3.8) and (3.9) will tend to zero if k is any number greater than 2. Using (3.1) and (3.9) we have that Obviously, (3.12) Concentrating on the integral common to both sides of the composite inequality (3.12) and using the fact that Φ{(1 − x 6 ) 1/2 u} is a decreasing function on the interval For some δ ≡ δ n , such that 0 < δ n < 1, and lim n→∞ δ n = 0, we can see that Using partial fraction expansions in (3.14) we can quickly show that if f is any function of n such that it tends to infinity as n tends to infinity.The condition on u in (3.17) can be greatly simplified, leading to a more meaningful result, by increasing the level u marginally, to yield At this juncture we turn our attention to the integral of I 2 (x) and we employ many of the techniques previously used.Using (3.1) and (3.9), As in the integral of I 1 (x), the integral common to both sides of the double inequality (3.20), is critical to our proof.Since x 2 (1−x 2 ) 1/2 has a maximum at Simply by integrating the terms in (3.21) we find ).We will use a result due to Das [3] to complete the proof of the theorem.From (3.18) and (3.24) it is clear that This is the same asymptotic value that Das [3] obtained for the number of maxima on the interval (0, 1).Therefore, EM u (1 − , 1) = o(log n), irrespective of the level u and the theorem is proved.In Theorem 1.3, we want to evaluate EM u (1, ∞) asymptotically.It becomes apparent later that we need only find EM u (1 + ε, ∞) (where ε is defined below), and it proves convenient to make the substitution y = 1/x.If we let 1+ε = (1− ) −1 , where = n −α as before.It will be necessary to define a set of qualifying k values for (3.7).We make the substitution in (3.6) to find

.28)
It can easily be shown that Φ{ } in (3.28) is an increasing function on the interval (0, 1 − ).In much the same way as we proved Theorem 1.2, we see that (3.29) As in Theorem 1.2, we concentrate on the integral in the interval and use the fact that Φ{ } is an increasing function (3.30) Hence, using (3.29), (3.30), and a little simple integration it is clear that if f is any function which tends to infinity as n tends to infinity.The only significant part to prove now is the integral of I 2 (x) on the interval ((1 + ) −1 , ∞).From (3.1) and (3.27) we can show

. 8 )
Using(3.8)  and the definitions we have for a, b, c, d, e, and | | it is a reasonably simple task to show that