Available online www.ejaet.com
European Journal of Advances in Engineering and Technology, 2016, 3(2): 15-20
Research Article
ISSN: 2394 - 658X
Expected Number of Zeros of a Random Trigonometric Polynomial
PK Mishra and Dipty Rani Dhal
Department of Mathematics,
College of Engineering & Technology, Biju Patnaik University of Technology (BPUT), Odisha, India
[email protected]
_____________________________________________________________________________________________
ABSTRACT
n
The asymptotic estimates of the expected number of real zeros of the polynomial T (θ ) ≡T 0(θ ,ϖ ) = ∑ g j (ϖ ) cos jθ i.e.
j =1
T (θ ) = g1cosθ + g 2cos 2θ + ......g n cos nθ where gj(j=1,2,…..n) is a sequence of independent normally distributed
random variables is such a number. The expected number of zeros of the above polynomial with coefficients gj(w),
j=1,2,…..n in be a sequence of independent random variables defined on probability space (Ω, λ , Pr) , each normally
distributed with mean zero and variance one, then for all sufficiently large n the variance of the number of real
3/ 2
zeros of T(θ) is equal to var{N (0, π )} = O(n )
Key words: Level crossings, Trigonometric functions Independent, identically distributed random variables,
random algebraic polynomial, random algebraic equation, real roots, domain of attraction of the normal law,
slowly varying function
_____________________________________________________________________________________
INTRODUCTION
n
Let
T (θ ) ≡T 0(θ ,ϖ ) = ∑ g j (ϖ ) cos jθ
(1)
j =1
where g 1(ϖ ), g 2 (ϖ ),....... g n (ϖ ) is a sequence of independent random variables defined on a probability space
(Ω, λ , Pr) , each normally distributed with mean zero and variance one. Much has been written concerning
N K (0,2π ) , the number of crossings of a fixed level K by T(θ), in the interval (0,2 π ) . From the work of Dunnage
[2] we know that, for all sufficiently large n, the mathematical expectation of is a N 0(0,2π ) ≡ N (0,2π ) symptotic to
2n / 3 . In Farahmand [3] and [5] we show that this asymptotic number of crossings remains invariant for any
K ≡K n . such that K 2 /n → 0 as n → ∞ However, less information is known about the variance of N (0,2π ) . The only
attempt so far is ….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 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] with above
j
n
obtained the variance of N for the case of random algebraic polynomial ∑ j 0 g j x ; 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
polynomial. However, he studied a different type of polynomial ∑nj 0a j cos jθ +b j sin jθ which has the property of
being stationary and for which a special theorem has been developed by Cramer and Leadbetter [1].Here we look at
the random trigonometric polynomial (1) as a non-stationary random process. First we are seeking to generalize
Cramer and Leadbetter’s [1] works concerning fractional moments which are mainly for the stationary case. To
evaluate the variance specially, and some other applications generally it is important to consider the covariance of
the number of real zeros of ξ (t ) in any two disjoint intervals. To this end, let ξ (t ) be a (non-stationary) real valued
separable normal process possessing continuous sample paths, with probability one, such that for any θ 1≠θ 2 the joint
normal process ξ (θ 1), ξ (θ 2), ξ ' (θ 1) and ξ ' (θ 2) is non singular. Let (a,b) and (c,d) be any disjoint intervals on
15
Mishra and Dhal
Euro. J. Adv. Engg. Tech., 2016, 3(2):15-20
______________________________________________________________________________
which ξ (t ) is defined. The following theorem and the formula for the mean number of zero crossings Cramer and
Leadbetter’s [1] obtain the covariance of N(a,b) and N(c,d).
Theorem -1
For any two disjoint intervals, (a,b) and (c,d) on which the process ξ ' (θ 1) is defined, we have
db ∞ ∞
E {N ( a , b ) N (c , d )} = ∫ ∫ ∫ ∫ xy pθ 1θ 2 (0,0, x , y )dxdydθ 1dθ 2 where for a ≤θ 1≤ b and c ≤θ 2≤ d , pθ 1.θ 2( x1 , x 2 , x, y)
c a −∞ −∞
denotes the four dimensional density function of ξ ( θ 1 ), ξ ( θ 2 ), ξ ' ( θ 1 ), ξ ' ( θ 2 ).
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 pθ 1.θ 2( x1 , x 2 , x, y ) defined as in Theorem 1 we have
EN
2
d b ∞
(a , b ) = ∫ ∫ ∫
∞
∫ xy p θ 1 θ
c a −∞ −∞
2
(0 , 0 , x , y )dxdyd
θ 1d θ
2
By applying Theorem 2 to the random trigonometric polynomial (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 (0,2π ) to 0(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 (Ω, λ , Pr) , each normally distributed with mean zero and variance one, then for all sufficiently large n the
3/ 2
variance of the number of real zeros of T(θ) satisfies var{N (0, π )} = O(n )
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, Nu.
Indeed, the analysis for the number of down crossings would be similar and therefore, the result for the total number
of crossings will follow. In order to find E {N u ( a , b ) N u ( c , d ) } we require refining and extending the proof
presented by Cramer and Leadbetter [1]. However, our proof follows their method and in the following, we
highlight the generalization required to obtain our result. Let a k = ( b − a ) k 2 − m + a and similarly
b j = ( d − c ) l 2 − m + c for k, l = 0,1,2,.... 2 m - 1 and we define the random variable Xk,m and Xlm as
1
Xk , m = 
0
if ξ ( a k ) < 0 < ξ ( a k +1 )
1 ifξ (b l ) < 0 < ξ (b l +1 )
And Xlm = 
otherwise
otherwise
0
(2)
In the following we show that
2 m −1 2 m −1
Y m= ∑
l=0
∑ Xk , m , Xl , m
k =0
tends to N u ( a, b) N u (c, d ) as m → ∞ with probability one. See also Cramer and Leadbetter [1], we first note that i
E{N u (a, b) N u (c, d )} s finite and therefore {N u ( a, b) N u (c, d )} is finite with probability one. Let v and r be the
number of up crossings of ξ (t ) in (a,b) and (c,d), respectively, and write t1,t2,…..tv and t’1, t’2…t’r for the points of
upcrossings of zero by ξ (t ) , there can be found two sub intervals for each Is,m and Js’m such that ξ (t ) in one is
strictly positive and in the other, it is strictly negative. Thus it is apparent that Ym will count each of tsts’. That is,
ξ (t )
Y m≥ vr , for all sufficiently large m. On the other hand, if ξ ( a )ξ (b
k
k +1 ) < 0 and ξ (b t )ξ (b t +1 ) < 0 then
must have a zero in (a k ,a k +1) and (bl ,bl +1 ) and hence Y m ≤ vr and hence Y m→ N u ( a, b) N u (c, d ) as m → ∞ , with
probability one. Now from (2.1) we can see at once that
E (Y m) =
We write
2 m −1 2m −1
∑ ∑ Pr( X k, mX l, m = 1)
l =0 k =0
η k for the random variable 2 {ξ (a k +1 ) − ξ (a k )}
m
have Pr( X k, m = X l, m = 1)
16
=
2m −12m −1
∑ ∑Pr(X k, m =Xl, m =1)
l =0 k =0
and similarly
(3)
η '1 for 2 {ξ (b k +1 ) − ξ (b k )} , then we
m
Mishra and Dhal
Euro. J. Adv. Engg. Tech., 2016, 3(2):15-20
______________________________________________________________________________
−m
−m
= Pr(0 > ξ ( a k ) > 2 η k , and 0 > ξ (b l ) > 2 η l )
∞ ∞ 2− m x 2
=
−m y
∫∫ ∫
∫ Pm, k , l ( z 1 , z 2 , x, y)dz 1 dz 2 dxdy
00
0
0
(4)
For k,l, (z1,z2,x,y) denotes the four dimensional normal density function for η k and η ' k . A simple calculation
shows see Farahmand [6], that if θ 1 and θ 2 are the fixed interval (a,b) and (c,d), respectively and km and lm are such
that a k m <θ 1< a k m+1 and b l m <θ 2<b l m+1 for each m, then all members of the covariance matrix of pm,k,l (z1,z2,x,y)
will tend to the corresponding members of the covariance matrix of pθ 1 ,θ 2 (z1,z2,x,y). This co-variance matrix is,
m
m
indeed, nonsingular. Now let t = 2 z1 and r = 2 z 2 then from (3) and (4) we have
E (Y
m
) =
∞ ∞ x y
2 m −1 2 m −1
−2m
−m
t 2 − m r , x , y ) dtdrdxdy
∑ 2
∫ ∫ ∫ ∫ Pm , k , l ( 2
∑
l=0
k =0
b d ∞ ∞ x y
= ∫ ∫ ∫ ∫ ∫ ∫ Ψ
m
(5)
0 0 0 0
,θ 1 ,θ
(2
2
− m
t2
− m
r , x , y ) dtdrdxdyd
θ 1d θ
2
a c 0 0 0 0
in which Ψ m ,θ 1,θ 2
(t , r , x, y ) = Pm, k , l (t , r , x, y ) for a k <θ 1<a k +1 and b l <θ 2<b l +1 . It follows, similar to Cramer
and Leadbetter [1], that m → ∞
Ψ m ,θ 1,θ 2(2 − m t 2 −m r , x, y) → pθ 1θ 2(0,0, x, y) which together with dominated convergence proves Theorem 1.
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 pθ 1,θ 2( z1, z 2 , x, y) , we
should omit those zeros in the squares of side 2-m obtained from equal points in the axes (and therefore to evaluate
EN(N-1)). To this and for any g=(g1,g2) lying in the unit square and c>0, let Ame denote the set of all points g in the
unit square that for all s belonging to the squares of side 2-m set containing g we have s 1 − s 2 > ε . Let λ m ε
denote the characteristic function of the set λ
. Finally, similar to the covariance case, let
mε
1 ifξ (a k ) < 0 < ξ (a k +1 )
X k ,m= 
otherwise
0
−m
for k=0,1,2,….2m-1, where a k = (b − a ) k 2 + a. Now let
M
2 m −1
∑
mε =
2 m −1
∑
k = 0 (l = 0 , l ≠ k )
X
k, m
X
l, m
λ mε ( 2 − m k ,2 − m l )
(6)
Similar to Cramer and Leadbetter [1] we show that Mme is a non decreasing function of m for any fixed e. It is
obvious that Mme is a non decreasing function of e for fixed to m, and then by two applications of monotone
convergence it would be justified to change the order of limits in lim
note that each term of the sums of M
mε
ε→0
lim
m→∞
lim
mε
. To this end, we
corresponds to a square of side 2 . For fixed ε > 0 , the typical term is
-m
one if both of the followings statements are satisfied; (i) every point s=(s1,s2) in the square is such that s1 − s 2 > ε
and (ii) Xk,m=Xl,m=1. When m is increase 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 m
by m+1 and each k or l by 2k and 2l, for ak+1 and al+1. Since Xk,m=Xl,m=1 we must, with probability one, have at least
one of these four terms equal one. Hence Mme is a non decreasing function of m.
In the following, we show that lim m → ∞ lim ε → 0 s = N u ( N u − 1).
We first note that if the typical term in the sum of M mε is nonzero it follows that s1 − s 2 > ε , since it is impossible
to have ξ (a k ) < 0 < ξ (a k +1 ) and ξ (a k +1 ) < 0 < ξ (a k + 2 ) . Therefore, the characteristic function appearing in the formula
for M mε
in
in (6) is one and hence
17
Mishra and Dhal
Euro. J. Adv. Engg. Tech., 2016, 3(2):15-20
______________________________________________________________________________
lim ε → 0 M
2 m −1
mε
= ∑
2 m −1
∑
k = 0 (l = 0, l ≠ k )
X
k,m
X l, m
(7)
is clearly in the form of Ym defined in Section 2 except that the summations in (7) cover all the k and l such that
k≠l
. Hence from (7), we can write lim m → ∞ lim ε → 0 M
Therefore the same pattern as for the covariance case yields
mε=
N u ( N u − 1)
∞∞
E [N u ( a , b ) {N u ( a , b ) − 1}] = lim ε → 0 ∫ ∫ ∫ ∫ xy , p θ 1 ,θ 2 (0,0, x , y ) dxdyd θ 1d θ 2
D (ε ) 0 0
(8)
where D(ε ) denotes the domain in the two dimensional space with coordinates θ 1,θ
2
such that a <θ 1,θ 2< b and
θ 1−θ 2 > ε . Now notice that for θ 1=θ 2= 0 the pθ 1.θ 2(0,0, x, y ) degenerates to just pθ (0, x) , the two dimensional
joint density function of ξ (θ ) .and ξ ' (θ ) . Hence from (8), we have
=
E [N u (a, b){N u (a, b) − 1}]
b b ∞∞
b ∞
a a 0 0
a 0
∫ ∫ ∫ ∫ xy , p θ 1 , θ 2 ( 0 , 0 , x , y ) dxdyd θ 1d θ 2 - ∫ ∫ x , p θ ( 0 , x ) dxd θ
(9)
b ∞
Now since ∫ ∫ x , p θ ( 0 , x ) dxd θ is EN u(a, b) the result of Theorem 2 follows.
a 0
RANDOM TRIGONOMETRIC POLYNOMIAL
To evaluate the variance of the number of real roots of(!) (1) in the interval ( 0, π ) we use Theorem 2 to consider the
interval
(ε ',π − ε ') . The variance for the intervals and (ε ' ,π − ε ')
theorem Rudin [10] and Titchmarsh, [11]. We chose ε ' = n
(ε ' , π
possible error term. First, for any θ 1 andθ 2 in
−1 / 2
are obtained using an application of Jenson’s
which as we will see later, yields the smallest
− ε ' ) such that
θ 1− θ
2
> ε
where
ε ' = n −1 / 2
, we
evaluate the joint density function of the random variable T (θ 1), T (θ 2), T ' (θ 1) and T ' (θ 2) . Since for any θ we
have
∑ cos j θ = [sin {( n + 1 / 2 )θ } / sin( θ / 2 ) − 1 ] / 2
n
j =1
and also since for the above choice of θ 1 and θ 2, θ 1+θ 2< 2(π − ε ' )
we can show
A ( θ 1 ,θ
2
) = cov
{T
( θ 1 ), T( θ
) } = ∑ cos
n
2
j =1
jθ
1
cosj θ
2
{
{θ θ
}
θ
θ }
 / 4
=  sin ( n + 1 / 2 )( 1 - 2 ) / sin ( 1 - 2 ) / 2
 + sin {( n + 1 / 2 )( θ 1 + θ 2 ) } / sin {( θ 1 + θ 2 ) / 2 } − 2 
= O ( 1 / ε ) + O (1 / ε ' )
(10)
Similarly, we can obtain the following two estimates
C (θ 1,θ 2 ) = cov {T (θ 1), T(θ 2 )} = − ∑ j sin jθ 1 cosj θ 2 = (ϑ / ϑθ 1){A θ 1,θ 2 } = O ( n / ε + ε −2 + n / ε '+ε ' −2 )
n
j =1
(11)
and
B(θ1,θ 2) = cov{T(θ1), T(θ 2)} = ∑ j sin jθ1 sinjθ 2 = (ϑ /ϑθ2){Cθ1,θ 2} = O(n2 / ε + n/ ε 2 + n2 / ε + n/ ε '2 +ε'3 )
n
j =1
(12)
Also in the lemma in Farahmand [3], we obtain
var( T (θ 10 )) = n / 2 + O ( ε ' − 1 ), var( T ' (θ 10 )) = n 3 / 6 + O ( n 2 / ε '+ n / ε ' 2 + ε ' − 3 )
and
cov{T (θ 1)T (θ 2)} = O ( n / ε '+ε '−2 )
These together with (10-12) give the covariance matrix for the joint density function
n / 2 + O(ε '−1 )

A(θ1,θ 2)
C(θ1,θ1)
C(θ 2,θ1)


−1
A(θ1,θ 2)
n / 2 + O(ε ' )
C(θ1,θ 2)
C(θ 2,θ 2) 

T (θ 1), T (θ 2), T ' (θ 1) and T ' (θ 2) as ∑ 

C(θ1,θ1)
C(θ1,θ 2)
n3 / 6 + O(n2 / ε ' )
B(θ1,θ 2)


3
2
C(θ 2,θ 2)
B(θ1,θ 2)
n / 6 + O(n / ε ' )
 C(θ 2,θ1)
18
(13)
Mishra and Dhal
Euro. J. Adv. Engg. Tech., 2016, 3(2):15-20
______________________________________________________________________________
This covariance matrix for all n ≥ 4,0 <θ 1,θ 2< π such that θ 1 ≠ θ
cofactor of the (ij)th element of
, then
∑
∑33 > 0, ∑44 > 0 and ∑34 = ∑43
2 −1
pθ 1.θ 2(0,0, x, y) = (4π ) ∑
(
+
Now let q = ∑ 33 / ∑
)
1/ 2
(
x and s = ∑ 44 / ∑
−1 / 2
+
[{
is positive definite. Hence
2
. From [1] we have
)
1/ 2
−1
2 −1
∫ ∫ x, y pθ 1.θ 2(0,0, x, y)dx dy = (4π ) ∑ 33
−∞ −∞
where ρ = (∑ 34 + ∑43 ) / 2(∑ 33 ∑44
)
and, if ∑ij is
]
}
exp− ∑33x + ∑44 y2 +(∑34+∑43 )xy / 2 ∑
2
(14)
y.
Then from (14) we can write
∞ ∞
>0
∑
∑ 44
−1
{
∞ ∞
}
2
2
∫ ∫ q, s exp − (q + s + 2 pqs) / 2 dq ds
−∞ −∞
(15)
and 0 ≤ p < 1 . The value of the integral in (15) can be obtained by a similar
2
1/ 2
2 1/ 2
2 1/ 2
method to Cramer and Leadbetter [1]. Let u = (1 − ρ ) q and v = (1 − ρ ) s then we have
∞∞
{
}
∞∞
{
}
I = ∫ ∫ exp − (q2 + s2 + 2 pqs) / 2 dq ds = (1- ρ 2 )-1 ∫ ∫ exp − (u2 + v2 + 2 puv) / 2(1 − ρ 2 ) du dv
00
{
00
}
ρ
= π (1 - ρ 2 ) -1/2 1 / 2 − (π ) − 1 ∫ (1 − x 2 ) − 1 / 2 dx = (1 - ρ 2 ) -1/2 arccos ρ = φ csc φ
where
0
ρ = cos φ . Use has been made of the fact that (see for example Cramer and Leadbetter [1])
{
∞∞
}
{
(16)
}
ρ
2
2
2
2 -1/2
−1
2 −1 / 2
∫ ∫ exp − (u + v + 2 puv) / 2(1 − ρ ) du dv = π (1 - ρ ) 1 / 2 − (π ) ∫ (1 − x ) dx
00
0
Therefore from (16) by differentiation we can obtain
{
∞ ∞
}
2
2
2
∫ ∫ qs exp − (q + s + 2 pqs) / 2 dq ds = -(dI)/(dρ ) = csc φ (1 - φcotφ )
0 −∞
(17)
We can easily show that
{
∞ ∞
}
2
2
2
∫ ∫ qs exp − ( q + s + 2 pqs ) / 2 dq ds = csc φ {1 + (π − φ cot φ )}
0 −∞
Which together with (17) evaluates the integral in (15) as
∞
∞
{
}
2
2
2
∫ ∫ qs exp − (q + s + 2 pqs) / 2 dq ds = 4csc φ {1 + (π / 2 − φ )cotφ }
−∞ −∞
(18)
Now from (13) we can show
∑
44
= n 5 / 24 + O ( n 4 / ε ' ) = ∑
and ∑
33
34
= O (n
4
/ ε ') = ∑
43
(19-20)
Also from (19) and (20) and with the above choice from (18) we can obtain
ρ = (∑34 + ∑33 ) / 2(∑34 ∑33
)1/ 2 = O(1/ nε ' ) → 0 as n → ∞
Therefore φ → π / 2 for all sufficiently large n and hence from (18), we can see
∞
∞
{
}
2
2
∫ ∫ qs exp − ( q + s + 2 pqs ) / 2 dq ds = 4 + O (1 / n ε ' )
−∞ −∞
(21)
Also from (10)-(11) we can write
−1
2
3
−1
2
∑ = {n / 2 + O(ε ' )} {n / 6 + O(ε + ε ' )}
Therefore from this (15) and (21) the integrand that appears in (8) is asymptotically independent of θ 1 and θ 2 and
since by the definition of D(e), the area of the integration is (π − 2ε ' ) ε (π − 2ε ' ) + ε = π + O (ε + ε ' ) we
have
E [N (ε ' , π − ε ' ){N (ε ' , π − ε ' ) − 1}] = n 2 / 3 + O ( n / ε '+ nε + nε ' )
(22)
We now denote the mathematical expectation of N2 in the interval (0,e). Similar to Farahmand [2-3] we apply
Jensen’s theorem on a random integral function of the complex variable z,
2
2
2
n
T ( z , w ) = ∑ g j ( w ) cos jz
j =1
Let N(r) denote the number of real zeros of T(z,w) in z<r. For any integer j from Farahmand [3], we have
Pr[N (ε ' ) > 3nε '+ j ] < (2 / n )ε '− j / 2 + exp(− j / 2 − n 2ε ' j / 2) < 3ε '− j / 2
19
(23)
Mishra and Dhal
Euro. J. Adv. Engg. Tech., 2016, 3(2):15-20
______________________________________________________________________________
Let n' = [3nε '] be the smallest integer greater than or equal to 3nε ' then since N (ε ' ) ≤ 2n is a non negative integer,
from (23) and by dominated convergence, for efficiently large n we have
EN 2 (ε ' ) = ∑ ( 2 j − 1) Pr( N (ε ' ) ≥ j )
j =0
=
n
∑ ( 2 j − 1) Pr( N (ε ' ) ≥ j ) + ∑ ( 2 n '− 1 + 2 j ) Pr( N (ε ' ) ≥ n '+ j )
0 < j ≤ n'
n'
j =1
n
≤ ∑ ( 2 n '− 1) + 3 + ∑ ( 2 n '− 1 + 1 + 2 j )ε − j / 2 = n '+ O ( n ' )
j =1
= O (n 2ε '2 )
j =1
(24)
The interval (π − ε ' , π ) can also be treated in exactly the same way to give the same result. Now we can use delicate
result due to Wilkins [12] which states that EN (0, π ) = n / 2 + O(1). From this and (22 & 24) and since ε = ε ' , we obtain
var{N (0, π )} = E{N (0, ε ' ) + N (ε ' , π − ε ' ) + N (π − ε ' , π )} − {EN (0, π )}
{
}
= n 2 / 3 + O(n 2 ε ' 2 +n / ε '+n 2 ε ' ) − n / 3 + O(1)
O(n 2 ε ' 2 +n / ε '+n 2 ε ' )
2
2
2
(25)
Use has been made of the fact that EN (ε ' , π − ε ' ) ~ EN (0, ε ' ) = O(nε ' ), see Farahmand [5] and therefore
E[N(0, ε ' )N(ε ' , π − ε ' )] = nO(N(0, ε ' )) = O(n 2 , ε ' ) and also from (24) EN 2 (0, ε ' ) ~ EN 2 (π − ε ' , π ) = O(n2 , ε '2 ).
−1/ 2
Finally from (25) and since ε ' = n
CONCLUSION
After proving all theorem and lemmas we conclude that considering a polynomial (1) where the coefficients are a
sequence of independent random variables defined on probability space (Ω, λ , 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(θ) satisfies
var{N (0, π )} = O(n 3 / 2 )
Hence the theorem proved.
REFERENCES
[1] H Cramer and MR Leadbetter, Stationary and Related Stochastic Process, Wiley, New York, 1967.
[2] JEA Dunnage, The Number of Real Zeros of a Random Trigonometric Polynomial, Proceedings of London
Mathematical Society, 1966, 16, 53-84.
[3] K Farahmand, On the Average Number of Level Crossings of a Random Trigonometric Polynomials, The Annals
of Probability, 1990, 18, 1403-1409.
[4] K Farahmand, On the Variance of the Number of Real Roots of a Random Trigonometric Polynomial, Journal of
Applied Mathematics and Stochastic Analysis, 1990, 3, 253-262.
[5] K Farahmand, Level Crossings of a Random Trigonometric Polynomial, Proceedings of the American
Mathematical Society, 1991, 111, 551-557.
[6] K Farahmand, Local Maxima of a Random Trigonometric Polynomial, Journal of Theoretical Probability, 1994,
7, 175-185.
[7] V Thangaraj, M Sambandham and AT Bharucha-Reid, On the Variance of the Number of Real Zeros of Random
Algebraic Polynomials, Stochastic Analysis Applied, 1983, 1, 215-238.
[8] NB Maslova, On the Variance of the Number of Real Roots of Random Polynomial, Theory Probability Applied,
1974, 19, 35-52.
[9] C Qualls, On the Number of Zeros of a Stationary Gaussian Random Trigonometric Polynomial, Journal of
London Mathematical Society, 1970, 2, 216-220.
[10] W Rudin, Real and Complex Analysis, McGraw Hill, New York, 1974.
[11] EC Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1939.
[12] JE Wilkins, Mean Number of Real Zeros of a Random Trigonometric Polynomial, Proceedings of the American
Mathematical Society, 1991, 111, 851-863.
20