Research Article On the Lower Bound for the Number of Real Roots of a Random Algebraic Equation

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1 Appied Mathematics and Stochastic Anaysis Voume 007, Artice ID 74191, 8 pages doi: /007/74191 Research Artice On the Lower Bound for the Number of Rea Roots of a Random Agebraic Equation Takashi Uno Received 7 February 007; Accepted 30 October 007 We estimate a ower bound for the number of rea roots of a random aegebraic equation whose random coeffcients are dependent norma random variabes. Copyright 007 Takashi Uno. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. 1. Introduction Let N n (R,ω be the number of rea roots of the random agebraic equation F n (x,ω = n a ν (ωx ν = 0, (1.1 ν=0 where the a ν (ω, ν = 0,1,...,n, are random variabes defined on a fixed probabiity space (Ω,,Pr assuming rea vaues ony. During the past years, the majority of pubished researches on random agebraic poynomias has concerned the estimation of N n (R,ω.WorksbyLittewoodandOfford [1], Sama [], Evans [3], and Sama and Mishra [4 6] in the main concerned cases in which the random coefficients a ν (ω are independent and identicay distributed. For dependent coefficients, Sambandham [7] considered the upper bound for N n (R,ω inthecasewhenthea ν (ω, ν = 0,1,...,n, are normay distributed with mean zero and joint density function M 1/ (π (n+1/ exp ( (1/a Ma, (1. where M 1 is the moment matrix with σ i = 1, ρ ij = ρ, 0<ρ<1, (i j, i, j = 0,1,...,n and a is the transpose of the coumn vector a. Aso, Uno and Negishi [8] obtainedthe same resut as Sambandham in the case of the moment matrix with σ i = 1, ρ ij = ρ i j,

2 Appied Mathematics and Stochastic Anaysis (i j, i, j = 0,1,...,n, whereρ j is a nonnegative decreasing sequence satisfying ρ 1 < 1/ and j=1 ρ j < in (1.. The ower bound for N n (R,ω in the case of dependent normay 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 resut for the ower bound. Additionay, Uno [1] estimated the strong resut for this particuar probem in the sense of Evans [3]. The term strong indicates that the estimation for the exceptiona set is independent of the degree n. TheobjectofthispaperistofindtheowerboundforN n (R,ω when the coefficients are nonidenticay distributed dependent norma random variabes. We remark that this resut is the genera form of Uno [11] and that the exceptiona set is dependent on the degree n. In this paper, we suppose that the a ν (ω, ν = 0,1,...,n,havemeanzero,andthe moment with 1 (i = j, ( ρ ij = ρ i j 1 i j m, (1.3 ( 0 i j >m, i, j = 0,1,...,n, for a positive integer m,where0 ρ j < 1, j = 1,,...,m in (1.. That is to say we assume the a ν (ω s to be m-dependent stationary Gaussian random variabes. With Yoshihara ([13, page 9], we see that this assumption is equivaent to the foowing two statements for a stationary Gaussian sequence: (i {a ν } is -mixing; (ii {a ν } is φ-mixing. Throughout the paper, we suppose n is sufficienty arge. We wi foow the ine of proof of Sama and Mishra [5]. Theorem 1.1. Let f n (x,ω = n a ν (ωb ν x ν = 0 (1.4 ν=0 be a random agebraic equation of degree n, where the a ν (ω s are dependent normay distributed with mean zero, and the moment matrix given by (1.3 andtheb ν, ν = 0,1,...,n, be positive numbers such that im n (k n /t n is finite, where k n = max 0 ν n b ν and t n = min 0 ν n b ν. Then for n>n 0, the number of rea roots of most of the equations f n (x,ω = 0 is at east ε n ogn outside a set of measure at most ( μ β ε n ogn + kn exp( μ β, β>0, (1.5 t n ε n provided ε n tends to zero, but ε n ogn tends to infinity as n tends to infinity, and μ and μ are positive constants.

3 Takashi Uno 3.Proofoftheorem Let {λ n } be any sequence tending to infinity as n tends to infinity and M is the integer defined by [ M = α λ n ( kn t n ] + 1, (.1 where α is a positive constant and [x] denotes the greatest integer not exceeding x. Letk be the integer determined by We wi consider f n (x,ω at the points for = [k/] + 1,[k/] +,...,k. Let f n ( x,ω = 1 M k n<m k+. (. ( x = 1 1 1/ M (.3 ( a ν (ωb ν x ν + + a ν (ωb ν x ν = U (ω+r (ω, (say, (.4 3 where ν ranges from M 1 +1toM +1 in 1,from0toM 1 in and from M to n in 3. The foowing emmas are necessary for the proof of the theorem. We wi use the fact that each a ν (ω has margina frequency function (π 1/ exp( u /. Lemma.1. For α 1 > 0, where Proof. First, we have σ = +1 i=m i=m 1 +1 σ >α 1 t n M, (.5 bi x i + b i x i M i=m 1 +1 j=i+1 M >tn x i i=m 1 +1 > ( B A where A and B are positive constants such that A>1and0<B<1. ρ j i. (.6 t nm, (.7

4 4 Appied Mathematics and Stochastic Anaysis Second, we get M i=m 1 +1 j=i+1 = t n ρ j i >tn x (M x { m m ρ i x i i=1 1 M x i+j ρ j i i=m 1 +1 j=i+1 i=1 ρ i x (M M 1 i } ( B A ρ 0 t nm, (.8 where ρ 0 = m j=1 ρ j and A and B are positive constants satisfying A > 1and0<B < 1. So we get where α 1 is a positive constant, as required. Lemma.. Let where σ α 1t nm, (.9 ({ } Pr ω; a ν (ωb ν x ν >λ n σ < σ 1 = bi x i + i=0 Proof. We get ({ } Pr ω; a ν (ωb ν x ν >λ n σ = M 1 1 i=0 1 j=i+1 e λ n/, (.10 π λ n ρ j i. (.11 e u / e du < λ n/. (.1 π λ n π λ n Lemma.3. Let ({ } Pr ω; a ν (ωb ν x ν >λ n σ < 3 e λ n/, (.13 π λ n where σ = n i=m bi x i + The proof is simiar to that of Lemma.. Lemma.4. For a fixed, n 1 i=m j=i+1 Pr ({ ω; R (ω <σ } > 1 π n ρ j i. (.14 1 λ n e λ n/. (.15

5 Takashi Uno 5 Proof. ByLemmas. and.3,weget,foragiven, R (ω <λ n ( σ + σ (.16 outside a set of measure at most (/π 1/ λ 1 n exp( λ n/. Again, we have M 1 1 i=0 1 j=i+1 1 i=0 m ρ j i kn Hence we get, for a positive constant α, b i x i k nm 1, M 1 (i 1 ρ i x j+i i=1 j=1 ρ 0 k nm 1. (.17 σ α k nm 1. (.18 Simiary, we have σ α 3k nm 1 (.19 for a positive constant α 3. Therefore, we obtain, outside the exceptiona set, R (ω ( ( <λ n α + α 3 kn M (1/ α + α < 3 k n /M λ n σ 1/ <σ, (.0 α 1 t n bylemma.1 and (.1. Let us define random events E p, F p by It can be easiy seen that E p = { ω; U 3p (ω σ 3p,U 3p+1 (ω < σ 3p+1 }, F p = { ω; U 3p (ω < σ 3p,U 3p+1 (ω σ 3p+1 }. (.1 Pr ( E p F p = δ p (say >δ, (. where δ>0isacertain constant. Let η p be a random variabe such that 1 one p F p, η p = 0 esewhere. (.3 Then we get E ( η p = δ p, V ( η p = δ p δ p. (.4 Let q be the tota number of pairs (U 3p,U 3p+1 for which [ k ] +1 3p<3p +1 k, (.5

6 6 Appied Mathematics and Stochastic Anaysis q must be at east equa to [k/3] [([k/] + 1/3] 1. Take η = η p, (.6 where the summation is taken over a the q pairs. Appying Tschebyscheff inequaity, we have, for 0 <ε<δ, Pr ({ η E(η } V(η δ qε q ε p q ε 1 qε, (.7 since for n sufficienty arge, Cov(η i,η j = 0(i j. But [ ] [ k [k/] + 1 q 3 3 ] 1 k ( (k/ = (k 14 μ 1k, (.8 where μ 1 is a positive constant. Therefore, outside a set of measure at most μ /k, that is, or η E(η <qε, (.9 η E(η > qε (.30 η>e(η qε = δ p qε > q(δ ε μ 3 k, (.31 where μ and μ 3 are positive constants. Thus we have proved that outside a set of measure at most μ /k, either U 3p σ 3p and U 3p+1 < σ 3p+1,orU 3p < σ 3p and U 3p+1 σ 3p+1 for at east μ 3 k vaues of. Define 0 if R 3p <σ 3p, R 3p+1 <σ 3p+1, ζ p = (.3 1 esewhere. Let ξ p = η p η p ζ p.ifξ p = 1, there is a root of the poynomia in the interva (x 3p,x 3p+1. Hence the number of rea roots in the interva (x [k/]+1,x k must exceed ξ p,wherethe summationis takenovera the q pairs. Now, by using Lemma.4, wehave E ( ηp ζ p = E ( ( ( η p ζ p E ζ p = Pr ζ p = 1 { ( ( } Pr R 3p σ3p +Pr R 3p+1 σ3p+1 (.33 <μ 4 (k +1 1 λ n e λ n/, where μ 4 is a constant. Hence we have, for β>0, ({ ηp } Pr ζ p >μ 4 (k +1λ β 1 n e λ n/ < λ n E ( η p ζ p μ 4 (k +1λn β 1 e < 1 λ n/ λ β n. (.34

7 Takashi Uno 7 So we get ηp ζ p μ 4 (k +1λ β 1 n e λ n/, (.35 except for a set of measure at most 1/λ β n. Therefore, we have, outside a set of measure at most μ /k +1/λ β n, N n > ξ p >μ 3 k μ 4 (k +1λ β 1 n e λ n/ k ( μ 3 ε 1, (.36 where 0 <ε 1 <μ 3 (since μ 4 λn β 1 exp( λ n/ tends to zero as n tends to infinity. But it foows from (.1and(.that ( kn μ 5 λ n t M μ 6 n ( kn t n λ n, μ 7 ogn og (( μ k 8 ogn k n /t n λn og ((, k n /t n λn (.37 where μ i, i = 5,6,7,8, are constants. Hence we get outside the exceptiona set N n > where μ 9 is a constant. Taking λ n = (t n /k n exp(μ 9 /ε n, we obtain μ 9 ogn og (( k n /t n λn, (.38 N n >ε n ogn (.39 outside a set of measure at most ( μ β ε n ogn + kn exp( μ β, (.40 t n ε n where μ and μ are constants. This competes the proof of the theorem. Acknowedgment The author wishes to thank the referee for his/her vauabe comments. References [1] J.E.LittewoodandA.C.Offord, On the number of rea roots of a random agebraic equation II, Proceedings of the Cambridge Phiosophica Society, vo. 35, pp , [] G. Sama, On the number of rea roots of a random agebraic equation, Proceedings of the Cambridge Phiosophica Society, vo. 58, pp , 196. [3] E. A. Evans, On the number of rea roots of a random agebraic equation, Proceedings of the London Mathematica Society. Third Series, vo. 15, no. 3, pp , [4] G. Sama and M. N. Mishra, On the ower bound of the number of rea roots of a random agebraic equation with infinite variance, Proceedings of the American Mathematica Society, vo. 33, pp , 197.

8 8 Appied Mathematics and Stochastic Anaysis [5] G. Sama and M. N. Mishra, On the ower bound of the number of rea roots of a random agebraic equation with infinite variance. II, Proceedings of the American Mathematica Society, vo. 36, pp , 197. [6] G. Sama and M. N. Mishra, On the ower bound of the number of rea roots of a random agebraic equation with infinite variance. III, Proceedings of the American Mathematica Society, vo. 39, no. 1, pp , [7] M. Sambandham, On the upper bound of the number of rea roots of a random agebraic equation, The the Indian Mathematica Society. New Series, vo. 4, no. 1 4, pp. 15 6, [8] T. Uno and H. Negishi, On the upper bound of the number of rea roots of a random agebraic equation, The the Indian Mathematica Society. New Series, vo. 6, no. 1 4, pp. 15 4, [9] N. Renganathan and M. Sambandham, On the ower bounds of the number of rea roots of a random agebraic equation, Indian Pure and Appied Mathematics,vo.13,no.,pp , 198. [10] N. N. Nayak and S. P. Mohanty, On the ower bound of the number of rea zeros of a random agebraic poynomia, The the Indian Mathematica Society. New Series, vo. 49, no. 1-, pp. 7 15, [11] T. Uno, On the ower bound of the number of rea roots of a random agebraic equation, Statistics & Probabiity Letters, vo. 30, no., pp , [1] T. Uno, Strong resut for rea zeros of random agebraic poynomias, Appied Mathematics and Stochastic Anaysis, vo. 14, no. 4, pp , 001. [13] K. Yoshihara, Weaky dependent stochastic sequences and their appications. Vo. I. Summation theory for weaky dependent sequences, Sanseido, Tokyo, Japan, 199. Takashi Uno: Facuty of Urban Science, Meijo University, Nijigaoka, Kani, Gifu , Japan Emai address: uno@urban.meijo-u.ac.jp

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