ON THE VARIANCE OF THE NUMBER OF REAL ZEROS OF A RANDOM TRIGONOMETRIC POLYNOMIAL

The asymptotic estimate of the expected number of real zeros of the polynomial T(θ)=g1cosθ


Introduction
Let n T(O) Tn(O w) E gj(w)cos j0, (1.1) 3--1 where gl(w), g2(w),..., gn(w) is a sequence of independent random variables defined on a probability space (fl, A, Pr), each normally distributed with mean zero and variance one.Much has been written concerning N K(O 2r), the number of crossings of a fixed level K by T(0), in the interval (0,2r).From the work of Dunnage [2] we know that, for all sufficiently large n, the mathematical expectation of N0(0 2r) _--N(0, 2) is asymptotic to 2n/v/-.In [3] and [5] we show that this asymptotic number of crossings remains invariant for any K-K n such that K2/n----O as n--.cx.However, less information is known about the variance of N(0, 2r).The only attempt so far is in [4], where an (fairly large) upper bound is obtained.Indeed this could be justified since the problem with finding the variance consists of different levels of difficulties compared with finding the mean.The degree of difficulty With this challenging problem is reflected in the delicate work of Maslova [8] and Sambandham et al. [7] who have obtained the variance of N for the case of random algebraic polynomial, Printed in the U.S.A. ()1997 by North Atlantic Science Publishing Company 57 E : ogjx3; a case involving analysis that is usually easier to handle.Qualls [9] also studied the variance of the number of real roots of a random trigonometric n jO+ polynomial.However he studied a different type of polynomial j 0ajcos hi sin j0 which has the property of being stationary and for which a special theorem has been developed by Cramfir and Leadbetter [1].
Here we look at the random trigonometric polynomial (1.1) as a non-stationary random process.First we are seeking to generalize Cramfir and Leadbetter's [1] works concerning factorial moments which are mainly for the stationary case.To evaluate the variance specially, and some other applications generally, it is important to con- sider the covariance of the number of real zeros of ((t) in any two disjoint intervals.To this end, let (t) be a (non-stationary) real-valued separable normal process possessing continuous sample paths, with probability one, such that for any 01 02 the joint normal process ((01) ((02) '( 01) and ('(02) is non-singular.Let (a,b) and (c,d) be any disjoint intervals on which ((t)is defined.The following theorem and the formula for the mean number of zero crossings [1, page 85] obtain the covariance of N(a, b) and N(c, d).
Threm 1: For any two disjoint intervals, (a,b) and (c,d) on which the process (. is defined, we have where for a 01 b and c O2 d, POl,O2(Zl, Z2, x,y) denoles lhe four dimensional of A modification of the proof of Theorem 1 will yield the following theorem which, in reality, is only a corollary of Theorem 1.
Theorem 2: For POl,02(Zl, Z2,x,y) defined as in Theorem 1 we have b b By applying Theorem 2 to the random trigonometric polynomial (1.1) we will be able to find an upper limit for the variance of its number of zeros.This becomes possible by using a surprising and nontrivial result due to Wilkins [12] which reduces the error term involved for EN(0,2r) to O(1).We conclude by proving the following.
Theorem 3: If the coefficients gj(w), j-1,2,...,n in (1.1) be a sequence of independent random variables defined on probability space (,A, Pr), each normally distributed with mean zero and variance one, then for all sufficiently large n the variance of the number of real zeros of T(O) satisfies war {N(0, 7r)} 0(n3/2).

The Covariance of the Number of Crossings
To obtain the result for the covariance, we shall carry through the analysis for the number of upcrossings, N u.Indeed, the analysis for the number of downcrossings would be similar and, therefore, the result for the total number of crossings will follow.In order to find E{Nu(a,b)Nu(c,d)} we require to refine and extend the proof presented by Crarnr and Leadbetter [1, page 205].However, our proof follows their method and in the following, we highlight the generalization required to obtain our result.
Let a k (b a)k2 m + a and similarly b (d-c)12 m + c for k,1- 0, 1,2,...,2 rn-1 and we define the random variable Xk, m and Xl, rn &s and (2.1) In the following we show that with probability one.Let u and 7" be the number of upcrossings of (t) in (a,b) and (c, d), respectively, and write tl, t2,'", tu and t, t,..., t. for the points of upcrossings of zero by (t) in these two intervals, counted by Nu(a,b and Nu(c,d).Suppose Is, m and Js'm are the intervals of the form (ak, ak+l) and (bk, bk+l) which contains t s and t's, s 1,2,...,u and s'= 1,2,...,r, respectively.Then, by continuity of (t), there can be found two sub-intervals for each I, m and such that (t) in one is strictly positive and in the other, it is strictly negative.__hs it is apparent that Ym will count each of tst ,.That is, Ym > uv, for all sufficiently large m.On the other hand, if ((ak)(bk + 1I < 0 and ((b)((b-+ )< 0 then (t) must have a zero in (ak, ak + ) and (bl, b + 1) and hence Ym <-ur and hence Ym-.Nu(a,b)Nu(c,d) as meo, with probability one.Now from (2.1) we can see at once that 2m_1 2m_l k=O We write r/k for the random variable 2m[(ak+)--(ak)] and similarly r/ for + 1)-Pr(Xk, m Xl, m Pr{0 > (ak) > 2-mk and 0 > (bl) > 2- =JJJ J 0 0 0 0

The Variance of the Number of Real Zeros
It will be convenient to evaluate the EN(N-1) rather than the variance itself since N(N-1) can be expressed much more simply.The proof is similar to that established above for covariance, therefore we only point out the generalization required to obtain the result.
To avoid degeneration of the joint normal density, pol,0,(zl, z2, x, y), we should omit those zeros in the squares of side 2 m obtained from %qual points in the axes (and therefore to evaluate EN(N-1)).To this end for any g (gl,g2) lying in the unit square and e > 0, let Ara denote the set of all points g in the unit square such that for all s belonging to the squares of side 2-m containing g we have Sl s 2 > c.Let , denote the characteristic function of the set Ame. Finally, similar to the covariance case, let 1 if ((ak) < 0 < ((a k + 1) Xk, ra 0 otherwise for k=0,1,2,...,2m-l,wherea k=(b-a)k2-m+a.Now let 2 TM 1 2 TM 1 Mrne E E Xk, mXl, marne( 2 ink, 2 ral).
(3.1) k=O (t=O, tCk) Similar to [1, page 205] we show that Mrae is a nondecreasing function of rn for any fixed e.It is obvious that Mrac is a nondecreasing function of e for fixed m, and then by two applications of monotone convergence it would be justified to change the order of limits in lim_olimra__,Mra .To this end, we note that each term of the sums of Mine corresponds to a square of side 2-m.For fixed e > 0, the typical term is one if both of the following statements are satisfied: (i) every point s (sl, s2) in the square is such that Sl s21 > e and (ii) Xk, m Xl, m 1.When rn is increased by one unit, the square is divided into four subsquares, in each of which property (i) still holds.Correspondingly, the typical term of sum is divided into four terms, formed by replacing rn by m + 1 and each k or by 2k and 21, for a k + 1 and a + 1" Since Xk, m Xl, m 1 we must, with probability one, have at least one of these four terms equal one.Hence Mine is a nondecreasing function of rn.
In the following, we show that lim limM Nu(N u-1).
m---oo {--,0 m We first note that if the typical term in the sum of Mine is nonzero it follows that Sl-S2l >1, since it is impossible to have (ak)<0<(ak+l) and (ak + 1) < 0 < (a k + 2)" Therefore, the characteristic function appearing in the formula for Mm in (3.1) is one and hence 2) is clearly in the form of Ym defined in Section 2 except that the summations in (3.2) cover all the k and such that k 1. Hence from (3.2), we can write lim limM Nu(N u-1).where D() denotes the domain in the two dimensional space with coordinates 01,02 such that a<01, 02<b and 101-021 >e.Now notice that for 01-02-0 the POl,02(O,O,x,y) degenerates to just Po(O,x), the two dimensional joint density function of ((0) and '(0).Hence from (3.3 Now since f f Ix P0(0, x)dO is ENu(a b) the result of Theorem 2 follows.a 0