Neural, Parallel, and Scientific Computations 24 (2016) 97-106
EXPECTED NUMBER OF REAL ROOTS OF CERTAIN GAUSSIAN
RANDOM TRIGONOMETRIC POLYNOMIALS
SOUDABEH SHEMEHSAVAR∗ AND KAMBIZ FARAHMAND
School of Mathematics, Statistics and Computer Science
University of Tehran, Tehran, Iran
Department of Mathematics, University of Ulster at Jordanstown, UK
Pn
ABSTRACT. Let Dn (θ) = k=0 (Ak cos kθ + Bk sin kθ) be a random trigonometric polynomial
where the coefficients A0 , A1 , . . . , An , and B0 , B1 , . . . , Bn , form sequences of Gaussian random vari-
ables. Moreover, we assume that the increments ∆1k = Ak −Ak−1 , ∆2k = Bk −Bk−1 , k = 0, 1, 2, . . . , n,
are independent, with conventional notation of A−1 = B−1 = 0. The coefficients A0 , A1 , . . . , An ,
and B0 , B1 , . . . , Bn , can be considered as n consecutive observations of a Brownian motion. In this
paper we provide the asymptotic behavior of the expected number of real roots of Dn (θ) = 0 as order
√
2 √2n
.
3
Also by the symmetric property assumption of coefficients, i.e., Ak ≡ An−k , Bk ≡ Bn−k , we
2n
.
show that the expected number of real roots is of order √
3
Key words: Random Trigonometric Polynomials, Brownian motion, Symmetric Property.
1. PRELIMINARIES
There are two different forms of random trigonometric polynomials previously
studied.
n
X
Ak cos(kθ)
Tn (θ) =
k=0
and
(1.1)
Dn (θ) =
n
X
(Ak cos kθ + Bk sin kθ),
k=0
Dunnage [2] first studied the classical random trigonometric polynomials T (θ). He
showed that in the case of identically and normally distributed coefficients A0 , A1 , . . . ,
An with mean zero and variances 1, the expected number of real roots in the interval
2n
(0, 2π), outside of an exceptional set of measure zero, is √
+ O{n11/3 (log n2/3 )}, when
3
n is large. In Farahmand [3, 4, 5], it is shown the asymptotic formula for the expected
number of K-level crossings remain valid when the level K increases. The work of
Sambandham and Renganathan [13] and Farahmand [6] among other obtained this
result for different assumption on the distribution of the coefficients. For various
e-mail:[email protected]
Received February 20, 2016
∗
c
1061-5369 $15.00 Dynamic
Publishers, Inc.
98
S. SHEMEHSAVAR AND K. FARAHMAND
aspects on random polynomials see Bharucha-Reid and Sambandham [1], which includes a comprehensive reference. Farahmand and Sambandham [8] study a case of
coefficients with different mean and variances, which shows an interesting results for
the expected number of level crossing in the interval (0, 2π). Farahmand and T. Li [9]
obtained asymptotic behavior for the expected number of real roots of two different
forms of random trigonometric polynomials Tn (θ) and Dn (θ), where the coefficients
of polynomials are normally distributed random variables with different means and
variances. Also They studied a case of reciprocal random polynomials for Tn (θ) and
Dn (θ). We consider the classical forms of random trigonometric polynomials Dn (θ)
where the coefficients A0 , A1 , . . . , An and B0 , B1 , . . . , Bn be a mean zero Gaussian
(1)
(2)
random sequence in which the increments ∆k = Ak − Ak−1 and ∆k = Bk − Bk−1 ,
k = 0, 1, 2, . . . , are independent, A−1 = 0, B−1 = 0. The sequence A0 , A1 , . . .
and B0 , B1 , . . . may be considered as successive Brownian points, i.e., Ak = W1 (tk ),
Bk = W2 (tk ),k = 0, 1, . . . , n, where t0 < t1 < · · · and {Wi (tk ), t ≥ 0}, i = 1, 2, are the
(i)
standard Brownian motion. In this physical interpretation, Var(∆k ) is the distance
between successive times tk−1 , tk . We note that
(1)
(1)
(1)
Ak = ∆0 + ∆1 + · · · + ∆k ,
(2)
(2)
(2)
Bk = ∆0 + ∆1 + · · · + ∆k
(i)
k = 0, 1, . . . , n,
(i)
where ∆k ∼ N(0, σi2 ), k = 0, 1, . . . , n, i = 1, 2, and ∆k are independent. Thus
" n
#
!
!
n
n
X
X
X
(1)
(2)
cos jθ ∆k +
sin jθ ∆k
Dn (θ) =
j=k
k=0
=
n X
j=k
(2)
(1)
ak1 (θ)∆k + bk1 (θ)∆k
k=0
and
′
Dn (θ) =
n
X
k=0
=
"
−
n X
n
X
j sin jθ
j=k
(1)
!
(1)
∆k
j=k
(2)
ck1 (θ)∆k + dk1 (θ)∆k
k=0
+
n
X
j cos jθ
!
(2)
∆k
#
where
ak1 (θ) =
n
X
cos jθ =
sin(2n + 1) 2θ − sin(2k − 1) θ2
,
2 sin( 2θ )
sin jθ =
cos(2k − 1) θ2 − cos(2n + 1) θ2
,
2 sin( 2θ )
j sin jθ =
sin(2n + 1) θ2 − sin(2k − 1) θ2
2 sin( 2θ )
j=k
bk1 (θ) =
n
X
j=k
ck1 (θ) = −
n
X
j=k
!′
,
GAUSSIAN TRIGONOMETRIC POLYNOMIALS
(1.2)
dk1 (θ) =
n
X
j cos jθ =
j=k
cos(2k − 1) 2θ − cos(2n + 1) θ2
2 sin( 2θ )
99
!
Now given Dn (θ) in (1.1) with a symmetric property of coefficients, i.e., Ak ≡ An−k
and Bk ≡ Bn−k for k = 0, 1, . . . , n, we can write Qn (θ) for odd n’s as follows:
n−1
(1.3)
Qn (θ) =
2
X
k=0
[Ak (cos kθ + cos (n − k)θ) + Bk (sin kθ + sin (n − k)θ)]
The polynomials will have one additional term for even n’s and we will not discuss
this case here.
n−1
Qn (θ) =
2
X
k=0
n−1
=
2
X
k=0
n−1
[Ak (cos kθ + cos (n − k)θ) + Bk (sin kθ + sin (n − k)θ)]

 n−1


 n−1
2
2
X
X

 (cos jθ + cos (n − j)θ) ∆(1)
 (sin jθ + sin (n − j)θ) ∆(2)
k +
k
j=k
j=k
2 X
(1)
(2)
ak2 (θ)∆k + bk2 (θ)∆k ,
=
k=0
n−1
Q′n (θ) =
2
X
k=0

−

n−1
2
X
j=k
(1)
(j sin jθ + (n − j) sin (n − j)θ) ∆k


 n−1
2
X
(2)
+  (j cos jθ + (n − j) cos (n − j)θ) ∆k 
j=k
n−1
2 X
(1)
(2)
=
ck2 (θ)∆k + dk2 (θ)∆k
k=0
where by using this results
n−1
ak2 (θ) =
2
X
j=k
(cos jθ + cos (n − j)θ) =
sin(2n − 2k + 1) 2θ − sin(2k − 1) θ2
,
2 sin θ2
n−1
cos(2k − 1) 2θ − cos(2n − 2k + 1) θ2
bk2 (θ) =
,
(sin jθ + sin (n − j)θ) =
2 sin θ2
j=k
n−1
ck2 (θ) = −
(1.4)
2
X
j=k
2
X
(j sin jθ+(n−j) sin (n − j)θ) =
sin(2n − 2k + 1) 2θ − sin(2k − 1) θ2
2 sin 2θ
!′
n−1
2
X
(j cos jθ + (n − j) cos (n − j)θ) =
dk2(θ) =
j=k
cos(2k − 1) θ2 − cos(2n − 2k + 1) 2θ
2 sin 2θ
,
!′
100
S. SHEMEHSAVAR AND K. FARAHMAND
2. Kac-Rice Formula
Let N(0, 2π) be denotes the number of real roots of the random trigonometric
polynomials in the interval (0, 2π) and E(N(0, 2π)) be its expected value. To deal
with the asymptotic behavior of the expected number of real roots of Dn (θ) = 0 and
Qn (θ) = 0, we refer to Kac-Rice formula [10, 11], which is defined as
Z 2π
∆
dθ,
(2.1)
E(N(0, 2π)) =
πA2
0
where ∆2 = A2 B 2 − C 2 . For Dn (θ) given in (1.1) we have
A2D = Var(Dn (θ)) =
n
X
(a2k1 (θ)σ12 + b2k1 (θ)σ22 ),
k=0
2
BD
=
Var(Dn′ (θ))
=
n
X
(c2k1 (θ)σ12 + d2k1 (θ)σ22 ),
k=0
(2.2)
CD =
Cov(Dn (θ), Dn′ (θ))
=
n
X
(ak1 (θ)ck1 (θ)σ12 + bk1 (θ)dk1 (θ)σ22 ),
k=0
where ak1 (θ), bk1 (θ), ck1 (θ) and dk1 (θ) are defined in (1.2). For Qn (θ) given in (1.3)
we have
n−1
A2Q
= Var(Qn (θ)) =
2
X
(a2k2 (θ)σ12 + b2k2 (θ)σ22 ),
k=0
n−1
2
BQ
= Var(Q′n (θ)) =
2
X
(c2k2 (θ)σ12 + d2k2 (θ)σ22 ),
k=0
n−1
(2.3)
CQ = Cov(Qn (θ), Q′n (θ)) =
2
X
(ak2 (θ)ck2 (θ)σ12 + bk2 (θ)dk2 (θ)σ22 ),
k=0
where ak2 (θ), bk2 (θ), ck2 (θ) and dk2 (θ) are defined in (1.4).
As in algebraic case the above identities are not well behaved around 0, π and
2π. Therefore we first consider the intervals (ε, π − ε), (π + ε, 2π − ε), where ε is any
positive constant, smaller than π and arbitrary at this point to be chosen later. It
should be positive and small enough to facilitate handling the roots in the intervals
(ε, π − ε), (π + ε, 2π − ε) and for roots inside this two intervals, we use (2.1). For
the real roots lying in the intervals (0, ε), (π − ε, π + ε) and (2π − ε, 2π), which it so
happens, are negligible, we will use a different method based on the Jensen’s theorem.
We now define some functions to make the estimations, define S(θ) = sin(2n +
1)θ/ sin θ, see from [5, page 74] which is continuous at θ = jπ and will occur frequently
GAUSSIAN TRIGONOMETRIC POLYNOMIALS
101
in follows. Since for θ ∈ (ε, π − ε) and θ ∈ (π + ε, 2π − ε), we have |S(θ)| < 1/ sin ε.
Hence, we can obtain
1
S(θ) = O
ε
Further
′
S (θ) = O
We can show
n
ε
,
′′
S (θ) = O
n2
ε
n
X
S( 2θ ) + 1
sin(2n + 1) 2θ
1
+ =
=O
cos kθ =
2
2
2 sin 2θ
k=0
(2.4)
1
,
ε
and
(2.5)
n
X
k=0
n
S′( θ )
k sin kθ = − 2 = O
,
4
ε
In similar way, we define P (θ) = cos θ −
Hence, we can obtain
n
X
k=0
S ′′ ( 2θ )
k cos kθ = −
=O
8
2
cos(2n+1)θ
,
2 sin θ
n2
ε
,
we also have |P (θ)| < 1/ sin ε.
1
P (θ) = O
ε
Further
′
P (θ) = O
We can show
n
ε
,
′′
P (θ) = O
n
X
cos θ2 − cos(2n + 1) 2θ
sin kθ =
=P
2 sin θ2
k=0
(2.6)
n2
ε
θ
1
=O
,
2
ε
and
(2.7)
n
X
n
P ′ ( 2θ )
,
=O
k cos kθ =
4
ε
k=0
n
X
P ′′ ( 2θ )
k sin kθ = −
=O
8
k=0
2
n2
ε
,
Now, using the above identities, we are able to evaluate the characteristics required
in using the Kac-Rice formula in (2.1).
3. Asymptotic Behavior of E(N(0, 2π))
This section includes two subsection. We evaluate the asymptotic behavior of the
expected number of real roots of Dn (θ) = 0 in the intervals (ε, π −ε), (π + ε, 2π −ε) in
subsection 3.1 and in the intervals (0, ε), (π − ε, π + ε), (2π − ε, 2π) in subsection 3.2.
102
S. SHEMEHSAVAR AND K. FARAHMAND
3.1. Results On the Intervals (ε, π − ε), (π + ε, 2π − ε). In this part, we obtain
our results by applying the Kac-Rice formula. The main contribution of this part for
the two different cases is stated separately in the following theorems.
Theorem 3.1. Let Dn (θ) be the random trigonometric polynomial given in (1.1) for
(1)
(1)
(1)
(2)
(2)
(2)
which Ak = ∆0 +∆1 +· · ·+∆k , Bk = ∆0 +∆1 +· · ·+∆k , k = 0, 1, . . . , n, where
(i)
∆k , k = 0, 1, . . . , n, i = 1, 2 are standard normal i.i.d random variables independent.
We prove that for all sufficiently large n, the expected number of real roots of the
equation Dn (θ) = 0, satisfies
√
2n
EN(ε, π − ε) = EN(π + ε, 2π − ε) ≃ √
3
Proof. In order to use the Kac-Rice formula, we first evaluate asymptotic value for
each variable needed by using the error terms obtained in (2.4)–(2.7). Since E(Ak ) = 0
and E(Bk ) = 0 we have
(3.1)
E(Dn′ (θ)) = 0,
E(Dn (θ)) = 0,
Now using (2.4)-(2.7) and (1.2) and using some trigonometric identities, we obtain
the variance of Dn (θ) and Dn′ (θ), as
A2D
n
X
= Var(Dn (θ)) =
(a2k1 (θ) + b2k1 (θ))
k=0
(3.2)
=
n
X
k=0
2
BD


=
n
X
cos jθ
j=k
Var(Dn′ (θ))
!2
=
n
X
+
sin jθ
j=k
n
X
!2 
=
n
2 sin2
θ
2
1
+ O( ),
ε
(c2k1 (θ) + d2k1 (θ))
k=0
=
n
X
k=0
=
(3.3)
n
X
k=0
CD =

 −
n
X
j sin jθ
j=k
4n2 + 4k 2
+O
16 sin2 2θ
Cov(Dn (θ), Dn′ (θ))
=
n
X
!2
+
n
X
j=k
!2 
j cos jθ 
n2
n3
=
ε
3 sin2
θ
2
+O
n2
ε
,
(ak1 (θ)ck1 (θ) + bk1 (θ)dk1 (θ))
k=0
=
n
X
k=0
(3.4)
=O
"
n
ε
n
X
j=k
,
cos jθ
!
−
n
X
j=k
j sin jθ
!
+
n
X
j=k
sin jθ
!
n
X
j=k
j cos jθ
!#
GAUSSIAN TRIGONOMETRIC POLYNOMIALS
103
Then, finally from (3.2)-(3.4), we can obtain
2
(3.5)
∆ =
2
A2D BD
−
2
CD
n4
=
6 sin4
θ
2
+O
n3
ε
,
The results of (3.2) and (3.5) into the Kac-Rice formula (2.1), we have
√
2n
E(N(ε, π − ε)) = E(N(π + ε, 2π − ε)) ∼ √
3
The theorem is proved.
Theorem 3.2. Let Qn (θ) be the random trigonometric polynomial given in (1.3)
(1)
(1)
(1)
(2)
(2)
(2)
where Ak = ∆0 + ∆1 + · · · + ∆k , Bk = ∆0 + ∆1 + · · · + ∆k , k = 0, 1, . . . , n−1
,
2
(i)
where ∆k , k = 0, 1, . . . , n−1
, i = 1, 2, are standard normal i.i.d random variables.
2
We prove that for all sufficiently large n, the expected number of real roots of the
equation Qn (θ) = 0, satisfies
n
EN(ε, π − ε) = EN(π + ε, 2π − ε) ≃ √
3
Proof. We obtain our results by applying the Kac-Rice formula. Since E(Ak ) = 0 and
E(Bk ) = 0 we have
E(Qn (θ)) = 0, E(Q′n (θ)) = 0
Now from (1.4) and (2.4)–(2.7) and making some trigonometric identities, we obtain
n−1
A2Q = Var(Qn (θ)) =
2
X
(a2k2 (θ) + b2k2 (θ))
k=0

2  n−1
2 
n−1
2
2
X X
X




=
(cos
jθ
+
cos
(n
−
j)θ)
+
(sin jθ + sin (n − j)θ) 

n−1
2
k=0
(3.6)
=
n
4 sin2
j=k
θ
2
j=k
1
+ O( ),
ε
n−1
2
BQ
= Var(Q′n (θ)) =
2
X
(c2k2 (θ) + d2k2 (θ))
k=0

2
n−1
2
X
X
=
(j sin jθ + (n − j) sin (n − j)θ)
−
n−1
2
k=0
j=k
2 
 n−1
2
X

+  (j cos jθ + (n − j) cos (n − j)θ) 
j=k
n−1
(3.7)
=
2
X
4n2 − 8k 2 − 8nk
k=0
16 sin2
θ
2
n2
n3
+ O( ) =
ε
12 sin2
θ
2
n2
+ O( ),
ε
104
S. SHEMEHSAVAR AND K. FARAHMAND
ani
n−1
CQ =
Cov(Qn (θ), Q′n (θ))
=
2
X
(ak2 (θ)ck2 (θ) + bk2 (θ)dk2 (θ))
k=0
n−1
=
2
X
k=0
 n−1
  n−1

2
2
X
X
 (cos jθ + cos (n − j)θ) −
(j sin jθ + (n − j) sin (n − j)θ)
j=k
j=k
 n−1
  n−1

2
2
X
X
+  (sin jθ + sin (n − j)θ)  (j cos jθ + (n − j) cos (n − j)θ)
j=k
(3.8)
=O
n
ε
j=k
,
Then, finally from (3.6)–(3.8) we can get
(3.9)
2
∆ =
2
A2Q BQ
−
CQ2
n4
=
48 sin4
θ
2
+O
n3
ε
,
The results of (3.6) and (3.9) into the Kac-Rice formula (2.1), we can obtain
n
E(N(ε, π − ε)) = E(N(π + ε, 2π − ε)) ∼ √
3
3.2. Results On the Intervals (0, ε), (π −ε, π +ε), (2π −ε, 2π). In this subsection,
we are going to show the expected number of real roots in the intervals (0, ε), (π −
ε, π + ε), (2π − ε, 2π) is negligible. The period of Dn (θ) is 2π, and so the number of
real roots in the interval (0, ε) and (2π − ε, 2π) is the same as the number in (−ε, ε),
the interval (π − ε, π + ε) can be treated in the same way to give the same result.
Here we deal only with Dn (θ), since the same method is applicable for the random
trigonometric polynomial, Qn (θ), and the results of Dn (θ) remain the same for Qn (θ).
We consider the function of the complex variable z,
Dn (z, ω) =
n
X
(Ak (ω) cos kz + Bk (ω) sin kz)
k=0
We seek an upper bound to the number of real roots in the segment of the real axis
joining the points ±ε, and this certainly does not exceed the number of real roots in
the circle |z| < ε.
Let N(r) ≡ N(r, ω) denote the number of real roots of Dn (z, ω) = 0 in |z| < ε.
We will modify the method based on the Jensen’s theorem [12], which has been used
by Dunnage [2], then By Jensen’s theorem,
Z 2ε
Z 2π
Z 2ε
Dn (2εeiθ , ω)
1
−1
−1
log |
r N(r)dr ≤
|dθ
r N(r)dr =
2π 0
Dn (0)
ε
0
GAUSSIAN TRIGONOMETRIC POLYNOMIALS
105
for which we have
2π
Dn (2εeiθ , ω)
|dθ,
Dn (0)
0
P P
Now since the distribution function of Dn (0, ω) = nk=0 nj=k ∆1k (ω) is
(2n3 + 9n2 + 13n + 6)
G(x) ∼ N 0,
6
1
N(ε) log 2 ≤
2π
(3.10)
Z
log |
We can see that for any positive υ,
(3.11)
P (−e−υ ≤ Dn (0, ω) ≤ e−υ )
s
Z e−υ
3
3t2
=
exp − 3
dt
π(2n3 + 9n2 + 13n + 6) −e−υ
2n + 9n2 + 13n + 6
√
2 3e−υ
,
<p
π(2n3 + 9n2 + 13n + 6)
Also we have
|Dn (2εeiθ )| = |
(3.12)
n
n
X
X
k=0
!
cos(2jεeiθ ) ∆1k +
j=k
2nε
≤ 2M(n + 1)(n + 2)e
,
n
n
X
X
k=0
j=k
!
sin(2jεeiθ ) ∆2k |
where M = Maxk (max |∆1k |, max |∆2k |). The distribution function of |∆1k | and |∆2k | is

 √1 R x e− t22 if x ≥ 0
2π 0
F (x) =
0
if x < 0
For any positive υ and all sufficiently large n, the probability M > neυ is
(3.13)
P (M > neυ ) ≤ nP (|∆11 | > neυ )
r
Z ∞
2
2
1
(neυ )2
− t2
e dt ≃
,
= n√
exp −υ −
π
2
2π neυ
Therefore from (3.12) and (3.13), except for sample functions in an ω-set of measure
not exceeding
r
2
(neυ )2
|Dn (2εeiθ )| < 3n(n + 1)(n + 2)e2nε+υ ,
exp −υ −
(3.14)
π
2
Hence from (3.11), (3.14) and since we obtain
(3.15)
|
Dn (2εeiθ , ω)
| ≤ 3n(n + 1)(n + 2)e2nε+2υ ,
Dn (0, ω)
Except for sample function in an ω-set of measure not exceeding
r
√
2
(neυ )2
2 3e−υ
p
exp −υ −
+
π
2
π(2n3 + 9n2 + 13n + 6)
106
S. SHEMEHSAVAR AND K. FARAHMAND
Therefore from (3.10) and (3.15) we can show that outside the exceptional set
(3.16)
N(ε) ≤
log 3 + log n log (n + 1) + log (n + 2) + 2nε + 2υ
,
log 2
ε = n−1/4 , it follows from (3.16) and for any sufficiently large n that
r
√
2 3e−υ
2
(neυ )2
+
,
(3.17) P (N(ε) > 3nε+2υ) ≤ p
exp −υ −
π
2
π(2n3 + 9n2 + 13n + 6)
Let n′ = [3n3/4 ] be the greatest integer less than equal to 3n3/4 , then from (3.17) and
for n large enough we obtain
X
X
X
P (N(ε) > j) +
P (N(ε) ≥ n′ + j)
EN(ε) =
P (N(ε) ≥ j) =
j>0
′
≤n +
(3.18)
1≤j≤n′
s
j≥1
X
12
e−j/2 +
π(2n3 + 9n2 + 13n + 6) j≥1
r
= O(n3/4 ),
j
(nej )2
2X
exp − −
π j≥1
2
2
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