Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis
Volume 2007, Article ID 74191, 8 pages
doi:10.1155/2007/74191
Research Article
On the Lower Bound for the Number of Real Roots of
a Random Algebraic Equation
Takashi Uno
Received 27 February 2007; Accepted 30 October 2007
We estimate a lower bound for the number of real roots of a random alegebraic equation
whose random coeffcients are dependent normal random variables.
Copyright © 2007 Takashi Uno. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let Nn (R,ω) be the number of real roots of the random algebraic equation
Fn (x,ω) =
n
aν (ω)xν = 0,
(1.1)
ν =0
where the aν (ω), ν = 0,1,...,n, are random variables defined on a fixed probability space
(Ω,Ꮽ,Pr ) assuming real values only.
During the past 40–50 years, the majority of published researches on random algebraic
polynomials has concerned the estimation of Nn (R,ω). Works by Littlewood and Offord
[1], Samal [2], Evans [3], and Samal and Mishra [4–6] in the main concerned cases in
which the random coefficients aν (ω) are independent and identically distributed.
For dependent coefficients, Sambandham [7] considered the upper bound for Nn (R,ω)
in the case when the aν (ω), ν = 0,1,...,n, are normally distributed with mean zero and
joint density function
|M |1/2 (2π)
−(n+1)/2
exp − (1/2)a Ma ,
(1.2)
where M −1 is the moment matrix with σ i = 1, ρi j = ρ, 0 < ρ < 1, (i= j), i, j = 0,1,...,n
and a is the transpose of the column vector a. Also, Uno and Negishi [8] obtained the
same result as Sambandham in the case of the moment matrix with σ i = 1, ρi j = ρ|i− j | ,
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Journal of Applied Mathematics and Stochastic Analysis
(i= j), i, j = 0,1,...,n, where ρ j is a nonnegative decreasing sequence satisfying ρ1 < 1/2
and ∞
j =1 ρ j < ∞ in (1.2).
The lower bound for Nn (R,ω) in the case of dependent normally distributed coefficients was estimated by Renganathan and Sambandham [9] and Nayak and Mohanty
[10] under the same condition of Sambandham [7]. Uno [11] pointed out the defect in
the proofs of the above papers and obtained the result for the lower bound. Additionally,
Uno [12] estimated the strong result for this particular problem in the sense of Evans [3].
The term strong indicates that the estimation for the exceptional set is independent of the
degree n.
The object of this paper is to find the lower bound for Nn (R,ω) when the coefficients
are nonidentically distributed dependent normal random variables. We remark that this
result is the general form of Uno [11] and that the exceptional set is dependent on the
degree n. In this paper, we suppose that the aν (ω), ν = 0,1,...,n, have mean zero, and the
moment with
⎧
⎪
⎪
⎪
⎨1
ρi j = ⎪ρ|i− j |
⎪
⎪
⎩0
(i = j),
1 ≤ |i − j | ≤ m ,
|i − j | > m , i, j = 0,1,...,n,
(1.3)
for a positive integer m, where 0 ≤ ρ j < 1, j = 1,2,...,m in (1.2). That is to say we assume
the aν (ω) s to be m-dependent stationary Gaussian random variables. With Yoshihara
([13, page 29]), we see that this assumption is equivalent to the following two statements
for a stationary Gaussian sequence:
(i) {aν } is ∗ -mixing;
(ii) {aν } is φ-mixing.
Throughout the paper, we suppose n is sufficiently large. We will follow the line of
proof of Samal and Mishra [5].
Theorem 1.1. Let
fn (x,ω) =
n
aν (ω)bν xν = 0
(1.4)
ν =0
be a random algebraic equation of degree n, where the aν (ω)’s are dependent normally distributed with mean zero, and the moment matrix given by (1.3) and the bν , ν = 0,1,...,n,
be positive numbers such that lim n→∞ (kn /tn ) is finite, where kn = max 0≤ν≤n bν and tn =
min 0≤ν≤n bν .
Then for n > n0 , the number of real roots of most of the equations fn (x,ω) = 0 is at least
εn logn outside a set of measure at most
μ
k
+ n
εn logn
tn
β
exp −
μ β
,
εn
β > 0,
(1.5)
provided εn tends to zero, but εn log n tends to infinity as n tends to infinity, and μ and μ are
positive constants.
Takashi Uno 3
2. Proof of theorem
Let {λn } be any sequence tending to infinity as n tends to infinity and M is the integer
defined by
M=
α2 λ2n
kn
tn
2 + 1,
(2.1)
where α is a positive constant and [x] denotes the greatest integer not exceeding x. Let k
be the integer determined by
M 2k ≤ n < M 2k+2 .
(2.2)
We will consider fn (x,ω) at the points
xl = 1 −
1
M 2l
1/2
(2.3)
for l = [k/2] + 1,[k/2] + 2,...,k.
Let
fn xl ,ω =
1
aν (ω)bν xlν +
2
+
3
aν (ω)bν xlν = Ul (ω) + Rl (ω),
(say),
(2.4)
where νranges from M 2l−1 + 1 to M 2l+1 in 1 , from 0 to M 2l−1 in 2 and from M 2l+1 + 1
to n in 3 .
The following lemmas are necessary for the proof of the theorem. We will use the fact
that each aν (ω) has marginal frequency function (2π)−1/2 exp(−u2 /2).
Lemma 2.1. For α1 > 0,
σ l > α1 tn M l ,
(2.5)
where
σ 2l =
2l+1
M
i=M 2l−1 +1
bi2 xl2i + 2
2l+1
M 2l+1
−1 M
i+ j
bi b j xl ρ j −i .
(2.6)
i=M 2l−1 +1 j =i+1
Proof. First, we have
2l+1
M
bi2 xl2i
2l
−
1
i=M
+1
2l
> tn2
M
xl2i >
i=M 2l−1 +1
B 2 2l
t M ,
A n
where A and B are positive constants such that A > 1 and 0 < B < 1.
(2.7)
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Journal of Applied Mathematics and Stochastic Analysis
Second, we get
2l+1
M 2l+1
−1 M
i+ j
bi b j xl ρ j −i > tn2
i=M 2l−1 +1 j =i+1
2l −1
M
2l
M
i+ j
xl ρ j −i
i=M 2l−1 +1 j =i+1
2(M 2l−1 +1) m
m
B
2(M 2l −M 2l−1 )−i
i
2 xl
= tn
ρ i xl − ρ i xl
ρ0 tn2 M 2l ,
≥
2
A
1 − xl
i =1
i=1
where ρ0 =
So we get
m
j =1 ρ j
(2.8)
and A and B are positive constants satisfying A > 1 and 0 < B < 1.
σ 2l ≥ α21 tn2 M 2l ,
(2.9)
where α1 is a positive constant, as required.
Lemma 2.2. Let
Pr
2
2 e−λn /2
ν
ω; aν (ω)bν xl > λn σ l <
,
π λn
2
(2.10)
where
2
σ l =
2l−1
M
i =0
bi2 xl2i + 2
−1
2l−1
M 2l
−1 M
i =0
i+ j
bi b j xl ρ j −i .
(2.11)
j =i+1
Proof. We get
Pr
2
∞
2
2 e−λn /2
ν
−u2 /2
ω; aν (ω)bν xl > λn σ l =
e
du <
.
π λn
π λn
2
(2.12)
Lemma 2.3. Let
Pr
2
2 e−λn /2
ν
ω; aν (ω)bν xl > λn σ l <
,
π λn
3
(2.13)
where
n
2
σ l =
i=M 2l+1 +1
bi2 xl2i + 2
n
−1
n
i=M 2l+1 +1
j =i+1
i+ j
bi b j xl ρ j −i .
(2.14)
2 1 −λ2n /2
e
.
π λn
(2.15)
The proof is similar to that of Lemma 2.2.
Lemma 2.4. For a fixed l,
Pr
ω; Rl (ω) < σ l > 1 − 2
Takashi Uno 5
Proof. By Lemmas 2.2 and 2.3, we get, for a given l,
Rl (ω) < λn σ l + σ
l
(2.16)
outside a set of measure at most 2(2/π)1/2 λ−n 1 exp(−λ2n /2). Again, we have
2l−1
M
i=0
−1
M 2l
−1 M 2l−1
i=0
bi2 xl2i ≤ 2kn2 M 2l−1 ,
m
2
i+ j
bi b j xl ρ j −i
j =i+1
≤ kn
ρi
i =1
−(i−1)
M 2l−1
j =1
(2.17)
2 j+i−2
xl
≤ ρ0 kn2 M 2l−1 .
Hence we get, for a positive constant α2 ,
2
σ l ≤ α22 kn2 M 2l−1 .
(2.18)
Similarly, we have
2
σ l ≤ α23 kn2 M 2l−1
(2.19)
for a positive constant α3 . Therefore, we obtain, outside the exceptional set,
Rl (ω) < λn α2 + α3 kn M l−(1/2) <
α2 + α3 kn
λn σ l M 1/2 < σ l ,
α1 tn
(2.20)
by Lemma 2.1 and (2.1).
Let us define random events E p , F p by
E p = ω; U3p (ω) ≥ σ 3p ,U3p+1 (ω) < −σ 3p+1 ,
F p = ω; U3p (ω) < −σ 3p ,U3p+1 (ω) ≥ σ 3p+1 .
(2.21)
It can be easily seen that
Pr E p ∪ F p = δ p
(say) > δ,
(2.22)
where δ > 0 is a certain constant. Let η p be a random variable such that
⎧
⎨1
ηp = ⎩
0
on E p ∪ F p ,
elsewhere.
(2.23)
Then we get
E ηp = δ p,
V η p = δ p − δ 2p .
(2.24)
Let q be the total number of pairs (U3p ,U3p+1 ) for which
k
+ 1 ≤ 3p < 3p + 1 ≤ k,
2
(2.25)
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Journal of Applied Mathematics and Stochastic Analysis
q must be at least equal to [k/3] − [([k/2] + 1)/3] − 1. Take
η=
ηp,
(2.26)
where the summation is taken over all the q pairs. Applying Tschebyscheff inequality, we
have, for 0 < ε < δ,
V (η)
Pr η − E(η) ≥ qε ≤ 2 2 ≤
q ε
δp
2
q ε2
≤
1
,
qε2
(2.27)
since for n sufficiently large, Cov(ηi ,η j ) = 0 (i= j). But
q≥
k
[k/2] + 1
k
(k/2) + 1
1
−
−1 ≥ −1−
− 1 = (k − 14) ≥ μ1 k,
3
3
3
3
6
(2.28)
where μ1 is a positive constant. Therefore, outside a set of measure at most μ2 /k,
η − E(η) < qε,
(2.29)
η − E(η) > −qε
(2.30)
that is,
or
η > E(η) − qε =
δ p − qε > q(δ − ε) ≥ μ3 k,
(2.31)
where μ2 and μ3 are positive constants. Thus we have proved that outside a set of measure
at most μ2 /k, either U3p ≥ σ 3p and U3p+1 < −σ 3p+1 , or U3p < −σ 3p and U3p+1 ≥ σ 3p+1 for
at least μ3 k values of l.
Define
⎧
⎨0
if R3p < σ 3p , R3p+1 < σ 3p+1 ,
ζp = ⎩
1 elsewhere.
(2.32)
Let ξ p = η p − η p ζ p . If ξ p = 1, there is a root of the polynomial in the interval (x3p ,x3p+1 ).
Hence the number of real roots in the interval (x[k/2]+1 ,xk ) must exceed ξ p , where the
summation is taken over all the q pairs. Now, by using Lemma 2.4, we have
E
ηpζ p =
E ηpζ p ≤
E ζp =
Pr ζ p = 1
Pr R3p ≥ σ 3p + Pr R3p+1 ≥ σ 3p+1
2
1
< μ4 (k + 1) e−λn /2 ,
λn
≤
(2.33)
where μ4 is a constant. Hence we have, for β > 0,
Pr
η p ζ p > μ4 (k + 1)λβn
1 −λ2n /2
e
λn
<
E
ηpζ p
2
μ4 (k + 1)λnβ−1 e−λn /2
<
1
.
λβn
(2.34)
Takashi Uno 7
So we get
η p ζ p ≤ μ4 (k + 1)λnβ−1 e−λn /2 ,
2
(2.35)
except for a set of measure at most 1/λβn . Therefore, we have, outside a set of measure at
most μ2 /k + 1/λβn ,
Nn >
ξ p > μ3 k − μ4 (k + 1)λnβ−1 e−λn /2 ≥ k μ3 − ε1 ,
2
(2.36)
where 0 < ε1 < μ3 (since μ4 λnβ−1 exp(−λ2n /2) tends to zero as n tends to infinity). But it
follows from (2.1) and (2.2) that
kn 2 2
k 2
λn ≤ M ≤ μ6 n λ2n ,
tn
tn
μ8 logn
μ7 log n
≤k≤
,
log kn /tn λn
log kn /tn λn
μ5
(2.37)
where μi , i = 5,6,7,8, are constants. Hence we get outside the exceptional set
Nn >
μ log n
9
,
log kn /tn λn
(2.38)
where μ9 is a constant.
Taking λn = (tn /kn )exp(μ9 /εn ), we obtain
Nn > εn logn
(2.39)
outside a set of measure at most
μ
k
+ n
εn logn
tn
β
exp −
μ β
,
εn
(2.40)
where μ and μ are constants. This completes the proof of the theorem.
Acknowledgment
The author wishes to thank the referee for his/her valuable comments.
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Takashi Uno: Faculty of Urban Science, Meijo University, 4-3-3 Nijigaoka, Kani,
Gifu 509-0261, Japan
Email address: uno@urban.meijo-u.ac.jp