Georgian Mathematical Journal
Volume 7 (2000), Number 2, 379-386
EXPECTED NUMBER OF REAL ZEROS OF LÉVY AND
HARMONIZABLE STABLE RANDOM POLYNOMIALS
S. REZAKHAH AND A. R. SOLTANI
Abstract. Assuming (A0 , A1 , . . . , An ) is a jointly symmetric α-stable random vector, 0 < α ≤ 2, a general formula for the expected number of real
n
P
zeros of the random algebraic polynomial
Ai xi was obtained by Rezai=0
khah in 1997. Rezakhah’s formula is applied to the case where (A0 , . . . , An )
is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived
n
P
Ai xi .
for the expected number of real zeros of
i=0
2000 Mathematics Subject Classification: Primary 60H25; secondary
60G52, 60E07
Key words and phrases: Random algebraic polynomial, number of real
zeros, expected density, Lévy stable noise, harmonizable processes, Lévy and
harmonizable coefficients
§
1. Introduction
Suppose (A0 , A1 , . . . , An ) is a symmetric α-stable random vector, 0 < α ≤ 2,
i.e., every linear combination of the components of the vector is an α-stable
random variable. Then (A0 , A1 , . . . , An ) admits the following spectral representation (see [1]–[3]).
Theorem 1.1. Let (A0 , A1 , . . . , An ) be a symmetric stable random vector of
index α, 0 < α ≤ 2. Then there is a symmetric α-stable random measure of
independent increments Φ on (R, B) and a set of functions h1 , . . . , hn in Lα (µ)
such that
Z
h1 (s)dΦ(s), . . . ,
R
Z
!
hn (s)dΦ(s)
R
(1.1)
has the same joint distribution as (A0 , A1 , . . . , An ), i.e.,
(
E exp it
n
X
j=0
λ j Aj
)!
= exp
(


n
X
α

− 
λj hj 

j=0
Lα (µ)
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / 
)
|t|α , t ∈ R,
(1.2)
380
S. REZAKHAH AND A. R. SOLTANI
for every choice of scalars λ0 , . . . , λn , where µ is called the control measure of
the random vector and is given by

kΦ(a, b)kα
α,
1 ≤ α ≤ 2,
µ(b) = 
kΦ(a, b)kα , 0 < α ≤ 1.
(1.3)
The norm in (1.3) is the Schilder norm. For any symmetric stable random
variable X, kXkα the Schilder norm of X is defined by


‘‘1/α

 − log EeiX
,
‘


− log EeiX ,
kXkα =
1 ≤ α ≤ 2,
(1.4)
0 < α ≤ 1.
The spectral representation in Theorem 1.1 is not unique. For the latest
results on the uniqueness see [4]. Although (1.1) may not be a version of
(A0 , A1 , . . . , An ), it can be used for the models considered in this article, but
due to the following lemma this has no effect on the expected number of real
n
P
zeros of the algebraic polynomial
Ai z i if the coefficients Ai are replaced by
the corresponding terms in (1.1).
i=0
Lemma 1.1. Suppose the random vectors (A0 , . . . , An ) and (B0 , . . . , Bn )
have the same joint distribution. Then the algebraic polynomials
n
P
i=0
n
P
i=0
Ai z i , and
Bi z i possess the same expected number of real zeros on any interval.
Proof. Due to the Kac–Rice formula (see [5]), the expected number of real zeros
of the polynomial Qn (z) ≡
ENn (a, b) =
n
P
i=0
Zb
Ai z i on an interval (a, b), say ENn (a, b), satisfies
fn (z)dz,
a
fn (z) =
+∞
Z
−∞
|η|p(z, 0, η)dη,
where fn (z) is called the expected density of real zeros, and p(z, x, η) is a continuous version of the joint probability density function of Qn (z) = x and
Q0n (z) = η . Thus the result follows as p(z, x, η) is the same for
n
P
i=0
Bi z i .
n
P
i=0
Ai z i , and
Let Z be an α-stable random measure with independent increments whose
control measure is m, the Lebesgue measure on [0, ∞). Define Z(t) = Z[0, t].
Then Z(t), t ≥ 0, is the Lévy stable noise of the index α. Note that
E(exp{isZ(t)}) = exp{−t|s|α }. If Aj = Z(tj ), j = 0, . . . , n, for 0 ≤ t0 <
· · · < tn < ∞, then in (1.2) µ is the Lebesgue measure and hj (λ) = I[0,tj ] (λ),
where I[a,b] (·) stands for the indicator function. In that case


X


cj Aj 

α
=
X




cj I[0,tj ] (λ)

Lα (m)
.
(1.5)
EXPECTED NUMBER OF REAL ZEROS
381
If in (1.2) µ is a finite measure on [0, 2π) and hj = eiλtj , j = 0, . . . n, −∞ <
t0 < t1 < · · · < tn < ∞, then A0 , . . . , An can be considered as consecutive
observations of a harmonizable stable process at epochs t0 , . . . , tn . Note that in
that case
X




cj Aj 




X
iλtj 

cj e 

(1.6)
Lα (µ)
for every choice of scalars cj , where µ is assumed to be a finite symmetric
measure about the origin, so that the coefficients A0 , . . . , An are real symmetric
α-stable random variables.
Let Nn (a, b, α) be the number of real roots of a random polynomial
=
α
Qn (x) =
n
X
Aj x j ,
j=0
−∞ < x < ∞,
(1.7)
in an interval (a, b), where the coefficients A0 , . . . , An satisfy (1.5) or (1.6) with
µ having a finite support. Under (1.5) the coefficients A0 , . . . , An is a sequence
with independent increments generating a linear space whose dimension increases with n, while under (1.6) µ has a finite support, the process {An } is
deterministic (for every n, (A0 , . . . , An ) is in a linear space of dimension not
greater than m, where m stands for the cardinality of the support of µ). This
is a new feature in studying random polynomials. Note that in neither case the
coefficients are independent. In this article we derive explicit formulas for the
expected density of real zeros of Qn (x) for both cases.
For A0 , . . . , An satisfying (1.1), by applying the method of Logan and Shepp
[5], Rezakhah [6] derived a general formula for the expected density of real zeros
fn (z, α), namely he showed that
∞
1 Z
fn (z, α) = 2
log gn (u, z, α) du dz,
π αz
(1.8)
−∞
where
gn (u, z, α) =
α



φz (·)u + ψz (·) α
L (µ)
Œ

α
Œ
<1−1/α>
1/α

Œu + 

φ
(·)ψ
(·)
 α
z
z
Œ
L
and
n
P
j=0
φz (λ) =  P
 n
z j hj (λ)
ψz (λ) = 

j=0
n
P
jz j hj (λ)
.

z k hk (·) α
L (µ)
L (µ)
k=0
k=0
<q>
q−1
For complex z and real q, z
= |z| z. To apply (1.8) to a certain model and
to derive a more explicit formula for the expected density one has to evaluate
gn (u, z, α) in (1.8), which is not an easy task in general. In Section 2 we consider
the case where the coefficients of the random polynomial are the consecutive
observations from the Lévy stable noise and in Section 3 we treat a certain
harmonizable case.


z k hk (·)
,
n
P
Œα ,
Œ
Œ
Œ
(µ)
α

382
S. REZAKHAH AND A. R. SOLTANI
There is a rich literature on the theory of the expected number of real zeros of
random algebraic polynomials. This area of research was elaborated by the fundamental work of M. Kac [7]. The works of Logan and Shepp [5], [8], Ibragimov
and Maslova [9], Wilkins [10], and Farahmand [11], [12] are other fundamental
contributions to the subject. For various aspects on random polynomials see
Bharucha-Reid and Sambandham [13], Rezakhah [14], and the recent book of
Farahmand [15].
§
2. Lévy Stable Noise
Suppose the coefficients Aj , j = 0, . . . , n, of a random polynomial Qn (x) in
(1.7) are formed from a Lévy stable noise as defined in the Introduction. Then
the following theorem provides the expected number of real zeros of Qn (x).
Theorem 2.1. Suppose Z(t), t > 0, is the Lévy stable noise. Let Qn (x)
be a random algebraic polynomial given by (1.7) where Aj = Z(tj ) for given
0 < t0 < t1 < · · · < tn < ∞. Then the expected number of real zeros in (a, b) is
given by
Z∞
ENn (R) =
fn (z, α) =
1
π 2 αz
Z∞
−∞
n
P
log k=0
fn (z; α) dz,
|ak (z)u + bk (z)|α ∆tk
|A(z)u + B(z)|α
−∞
where ∆t0 = t0 , ∆tk = tk − tk−1 for k ≥ 1, and
ak (z) =
n
X
j
z , bk (z) =
n
X
j
jz ,
A(z) =
j=k
j=k
B(z) =
’X
n
n
X
α
k=0
a<α−1>
(z)bk (z)∆tk
k
k=0
“
(2.1)
du,
|ak (z)| ∆tk
1−α
A
!1/α
,
(2.2)
(z).
Proof. Note that under the assumption of the theorem and by (1.5), (1.8), since
h( j)(·) = I[0,,tj ] (·),
n
P
j=0
φz (λ) =  P
 n
z j I[0,tj ] =
j=0
Now let
A(z) :=


z k I[0,tk ] 
α
k=0
Let t−1 = 0; then
n
X
z j I[0,tj ] (λ)
n
X
j=0
zj
n
X
I[tk−1 ,tk ] =
k=0
k=0
j=k
n
n ’X
X
k=0


“
n ’X
n
X



j

I
z
[tk−1 ,tk ] 


α
=
.
“
z j I[tk−1 ,tk ] .
j=0
)1/α
( n Πn
X Œ X ŒŒα
Œ
.
z j ŒŒ ∆tk
Œ
k=0 j=k
EXPECTED NUMBER OF REAL ZEROS
383
Then
φz (λ) =
n  P
n
P
k=0
j=k
(
‘
z j I[tk−1 ,tk ] (λ)
,
A(z)
)
“
n ’X
n
X
1
j
ψz (λ) =
jz I[tk−1 ,tk ] (λ) .
A(z) k=0 j=k
Thus


“
n ’X
n

1  X

j
(u
j)z

−
I
kφz (·)u − ψz (·)kα =
[tk−1 ,tk ] (·)

A(z)  k=0 j=k
α
(
Œ
n
n ŒŒ X
X
Œα
1
jŒ
Œ
(u − j)z Œ ∆tk
=
A(z) k=0 Πj=k
)1/α
.
Now define ak (z), bk (z) and A(z) as in (2.2). So
A(z)

‘
φz<1−1/α> (·)ψz1/α (·)
=
=
’X
n
k=0
’X
n
ak (z)I[tk−1 ,tk ]
“<1−1/α>’ X
n
<1−1/α>
ak
(z)I[tk−1 ,tk ]
k=0
=
=
bl (z)I[tl−1 ,tl ]
l=0
“’ X
n
“1/α
1/α
bl (z)I[tl−1 ,tl ]
l=0
n X
n
X
<1−1/α> 1/α
“
ak
bl I[tk−1 ,tk ] I[tl−1 ,tl ]
k=0 l=0
n
X
<1−1/α> 1/α
ak
bk I[tk−1 ,tk ] .
k=0
Therefore
α
A
(·)ψz1/α (·)kαα
(z)kφ<1−1/α>
z
=
=
n
X
k=0
n
X
<1−1/α>
|ak
1/α
(z)bk (z)|α ∆tk
(z)bk (z)∆tk .
a<α−1>
k
k=0
By using (1.8) we obtain
gn (u, z, α) =
n
P
k=0
|ak (z)u + bk (z)|α ∆tk
|A(z)u + B(z)|α
,
(2.3)
where B(z) is given in (2.2). Therefore the expected density of real zeros,
fn (z, α), is given by (2.1).
Remark . The integral in (2.1) can be evaluated for α = 2 (see [16]) as
fn (z, 2) =
1 (A2 (z)G2 (z) − F 2 (z))1/2
,
·
π
A2 (z)
384
S. REZAKHAH AND A. R. SOLTANI
where G(z) =
 P
‘1/2
n
,
b2 (z)σ 2
k
k=0
k
§
and F (z) =
n
P
k=0
ak (z)bk (z)σk2 .
3. Harmonizable Processes
In this section we assume that the coefficients Aj of a random polynomial
Qn (x) satisfy (1.6), where the measure µ is supported by the set {λ1 , . . . , λm ,
−λ1 , . . . , −λm }, λk ≥ 0, k = 1, . . . , m. This amounts to assuming that A0 , . . . , Am
is a certain deterministic harmonizable sequence of stable random variables. Let
Φk = Φ{λk } = Φ{−λk },
k = 1, . . . , m.
Then Φ1 , . . . , Φm is an independent sequence of symmetric α-stable random
variables, and (1.1) implies
Aj ≡ 2
m
X
Φk cos(λk j),
j = 0, . . . , n.
(3.1)
k=1
Note that for every n, (A0 , . . . , An ) belongs to the m-dimensional linear space
generated by Φ1 , . . . , Φm .
Theorem 3.1. Suppose the coefficients Aj are given by (3.1). Then the expected number of real zeros of Qn (x) in (a, b) is given by
ENn (a, b) =
Zb
fn (z; α, λ1 , . . . , λm ) dz,
(3.2)
a
where
∞
1 Z
fn (z; α, λ1 , . . . , λm ) = 2
log gn (u, z; α, λ1 , . . . , λm ) du,
π αz
−∞
T −1 (z)
gn (u, z; α, λ1 , . . . , λm ) = ŒŒ
where a(z, λ) =
n
P
j=0
m
P
k=1
|a(z, λk )u + b(z, λk )|α ck
−1
Œ
Œu + T (z)
z j eijλ , b(z, λ) =
T (z) =
n
P
j=0
m
X
k=1
Œα
Œ
Œ
c
k
α−1
Œ
|a(z,λk )|·|b(z,λk )|
m
P
|a(z,−λk )·b(z,−λk )|α
k=1
(3.3)
,
jz j eijλ and
|a(z, λk )|α ck .
Proof. Let us evaluate the Schilder norms of the corresponding functions in
(1.8). Note that the functions hj in (1.1) are given by hj (λ) = eijλ , j = 1, . . . , n.
Therefore
n
n
X
z j hj (λ) =


 n
α
X j


z hj (·)

α
j=0
z j eijλ := a(z, λ).
j=0
j=0
Hence
X
L
=2
(µ)
m
X
k=1
|a(z, λk )|α ck = 2T (z),
EXPECTED NUMBER OF REAL ZEROS
where ck = µ(λk ). Note that a(z, −λ) = ā(z, λ). Similarly



α
φz (·)u + ψz (·) α
L
where b(z, λk ) =
n
P
j=0
(µ)
Œ
385
Œα
m ŒX
n
Œ
1 X
Œ
j ijλk Œ
=
Œ
(u
+
j)z
e
Πck
Œ
T (z) k=1 Πj=0
=
m
1 X
|a(z, λk )u + b(z, λk )|α ck ,
T (z) k=1
jz j eijλk . Also note that

α


φz<1−1/α> (·)ψz1/α (·) α
L
(µ)
m Œ
Œ
1 X
Œα
Œ <1−1/α>
Œa
=
(z, λk )b<1/α> (z, λk )Œ ck
T (z) k=1
m
1 X
|a(z, −λk )b(z, −λk )|α ck
=
.
T (z) k=1 |a(z, λk )b(z, λk )|α−1
By applying these observations in (1.8) we arrive at (3.3).
Acknowledgement
This research was supported by Amirkabir University of Technology, the Office of the Research Council, Tehran, Iran. The authors would like to thank the
referee for reading carefully the manuscript and providing valuable comments.
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(Received 15.05.1999; revised 14.01.2000)
Authors’ addresses:
S. Rezakhah
Department of Mathematics
Amirkabir University of Technology
Hafez Avenue
Tehran, Iran
E-mail: [email protected]
A. R. Soltani
Department of Statistics
College of Sciences
Shiraz University
Shiraz 71454, Iran
E-mail: [email protected]