- Email: [email protected]
An inequality involving correlations
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 369–372 www.elsevier.com/locate/stapro
An inequality involving correlations A.R. Nematollahia,, A.R. Soltania,b a
Department of Statistics, College of Science, Shiraz University, Shiraz, 71454 Iran Department of Statistics and Operations Research, College of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
b
Received 3 May 2004; received in revised form 28 June 2005 Available online 15 September 2005
Abstract A useful inequality involving correlations between three random variables with mean zero and finite second moments is presented. The inequality is applied to show that the entries on the main diagonal of the spectral density of a periodically correlated Markov process, as derived in Nematollahi and Soltani [2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1) 127–140], are nonnegative. The inequality, when two variables with the same second moments are involved, is compared to the Cauchy–Schwartz inequality. r 2005 Elsevier B.V. All rights reserved. Keywords: Periodically correlated Markov processes; Second order processes; Cauchy–Schwartz inequality
1. Introduction Since the classical work on inequalities by Hardy et al. (1934), enormous efforts have been exerted to extend classical inequalities, resulting in discoveries of many types of inequalities with potential applications; see Tong (1984). In this paper we present a useful inequality involving covariances EXY , EYZ, where X ; Y and Z are mean zero random variables with finite second moments measured in the same units. The inequality, to be derived in Section 2, appears to be crucial in proving that f 0 ðlÞ, comprising the main diagonal of the spectral density of a periodically correlated Markov process of period 2, as derived in Nematollahi and Soltani (2000), is nonnegative. This application is presented in Section 3. Interestingly, when only X and Y are involved and EX 2 ¼ EY 2 , the inequality provides an upper bound for the jEXY j in terms of EX 2 and jEXY j itself which appears to be sharper than the one from the Cauchy–Schwartz inequality; see Remark 3.1.
2. Main result The following theorem is the main result of this article in which a new inequality involving correlations is obtained. Corresponding author.
E-mail addresses: [email protected] (A.R. Nematollahi), [email protected] (A.R. Soltani). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.032
ARTICLE IN PRESS A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
370
Theorem 2.1. Suppose X ; Y and Z are mean zero random variables with finite second order moments measured on the same units. If EjX j2 ¼ EjZj2 , then jEXY þ EYZj 1 ðEXY ÞðEYZÞ 1 þ p . (2.1) 2 EX 2 þ EY 2 EX 2 EY 2 Proof. Let qxy ¼ EXY =ðEX 2 EY 2 Þ1=2 and qyz ¼ EYZ=ðEY 2 EZ 2 Þ1=2 . It follows from the Cauchy–Schwartz inequality that ð1 q2xy Þð1 q2yz ÞX0, or jqxy þ qyz jp1 þ qxy qyz .
(2.2)
On the other hand for any two positive numbers u and v, always pffiffiffipffiffiffi u v 1 p . uþv 2 Substitutions u ¼ EX 2 and v ¼ EY 2 in (2.3) yield pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi EX 2 EY 2 1 p . 2 EX 2 þ EY 2
(2.3)
(2.4)
Derivations (2.2) and (2.4) provide that pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi EX 2 EY 2 jEXY þ EYZj jEXY þ EYZj pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi ¼ 2 2 EX 2 þ EY 2 EX þ EY EX 2 EY 2 1 p jqxy þ qyz j 2 1 p ð1 þ qxy qyz Þ 2 1 ðEXY ÞðEYZÞ 1þ ¼ ; 2 EX 2 EY 2 which proves the theorem.
&
3. Applications Consider a zero mean discrete time second order process X ¼ fX t ; t 2 Zg, Z the set of integers. The process is said to be periodically correlated (PC) with period T, if its covariance function Rðt; sÞ ¼ EX t X s satisfies Rðt; sÞ ¼ Rðt þ T; s þ TÞ
for every s; t 2 Z,
(3.1)
where T is the smallest positive integer. It is well known that every PC process X possesses a unique spectral density, whenever the density exists, given by the matrix hðlÞ ¼ ½hjk ðlÞj;k¼0;...;T1 ; 0plp2p, where 1 f ððl 2pjÞ=TÞ; T kj The functions f k satisfy Z 2p eitl f k ðlÞ dl, Bk ðtÞ ¼ hjk ðlÞ ¼
j; k ¼ 0; . . . ; T 1;
0plp2p.
0
and Bk ðtÞ are related to the correlations Rðn þ t; nÞ through Rðn þ t; nÞ ¼
T 1 X 0
Bk ðtÞe2pikn=T .
ARTICLE IN PRESS A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
371
The matrix hðlÞ is Hermitian and nonnegative definite. Moreover f 0 ðlÞX0; see Nematollahi and Soltani (2000). The spectral density hðlÞ for PC Markov processes was derived by the authors in (Nematollahi and Soltani, 2000). For T ¼ 2, f 0 appeared to be f 0 ðlÞ ¼
ðR0 ð0Þ þ R1 ð0ÞÞð1 ge2 Þ ½1 þ 2a cosðl=2Þ; 4pj1 geeil j2
l 2 ½0; 2pÞ,
(3.2)
where ge ¼ ½R0 ð1ÞR1 ð1Þ=½R0 ð0ÞR1 ð0Þ and a ¼ ½R0 ð1Þ þ R1 ð1Þ=½ðR0 ð0Þ þ R1 ð0ÞÞð1 þ geÞ. According to the theory it is expected f 0 ðlÞ is nonnegative. But it is not clear whether the expression for f 0 ðlÞ in (3.2) is nonnegative. The ratio appearing on the right of (3.2) will be nonnegative if je gjp1, which follows from the Cauchy–Schwartz inequality. Indeed, since for T ¼ 2, EX 22 ¼ EX 20 , it follows that R0 ð1Þ R1 ð1Þ jEX 1 X 0 EX 2 X 1 j ¼ je gj ¼ R0 ð0Þ R1 ð0Þ EX 20 EX 21 p
½EX 20 EX 21 EX 22 EX 21 1=2 EX 20 EX 21 ¼ ¼ 1. EX 20 EX 21 EX 20 EX 21
Thus it remains to show the term 1 þ 2a cosðl=2Þ is also nonnegative. We infer from Brockwell and Davis (1991, p. 121) that the term 1 þ 2a cosðl=2Þ is also nonnegative, for a given a, if and only if jajp12. We will use the inequality, derived in Section 2, to show that this indeed is the case. Replace X 0 for X, X 1 for Y and X 3 for Z in (2.1). Note that X 0 ; X 1 ; X 2 are mean zero random variables defined on the same probability space, and for T ¼ 2, EX 20 ¼ Rð0; 0Þ ¼ Rð2; 2Þ ¼ EX 22 : Hence the assumptions of Theorem 2.1 are fulfilled. Thus it follows from inequality (2.1) that ! jEX 1 X 0 þ EX 2 X 1 j 1 ðEX 1 X 0 ÞðEX 2 X 1 Þ 1þ , p 2 EX 20 þ EX 21 ðEX 20 ÞðEX 21 Þ or R0 ð1Þ þ R1 ð1Þ 1 R ð0Þ þ R ð0Þp 2 ð1 þ geÞ, 0
1
where Rt ðkÞ ¼ Rtþ2 ðkÞ; t; k 2 Z. Therefore jajp12, giving the result. Remark 3.1. Let in (2.1) EX 2 ¼ EY 2 , and EXY ¼ EYZ, then the inequality will become 1 E 2 XY jEXY jp EX 2 þ , 2 EX 2
(3.3)
which can be derived directly from the fact that for any a, 2 a 1 2 pffiffiffi p ffiffi ffi X0, EX 2EX 2 2 with a ¼ jEXY j. Since it is assumed that EX 2 ¼ EY 2 , by the Cauchy–Schwartz inequality jEXY jpEX 2 . Thus it follows that 1 E 2 XY 2 (3.4) jEXY j p EX þ p EX 2 . 2 EX 2 Therefore (3.3) provides sharper upper bound, but of course it depends on EXY itself. Let X and Y be discrete with masses 0.2, 0.3, 0.3 and 0.2 at ð1; 1Þ, ð1; 1Þ, ð1; 1Þ and ð1; 1Þ, respectively. Then EX 2 ¼ EY 2 ¼ 1; EXY ¼ 0:2. Therefore the terms in (3.4) for this example are 0.2, 0.52, 1, respectively; which shows the inequalities in (3.3) could be strict.
ARTICLE IN PRESS 372
A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
Acknowledgements The authors thank the referee for providing constructive comments. References Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Method. Springer, New York. Hardy, G.H., Littlewood, J.E., Polya, G., 1934. Inequalities. Cambridge University Press, Cambridge. Nematollahi, A.R., Soltani, A.R., 2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1), 127–140. Tong, Y.L., 1984. Inequalities in statistics and probability. Proceedings of the symposium held at the University of Nebraska, Lincoln, Nebraska, October 27–30, 1982, IMS Lecture Notes, Monograph Series, vol. 5. Institute of Mathematical Statistics, Hayward, CA.
Statistics & Probability Letters 76 (2006) 369–372 www.elsevier.com/locate/stapro
An inequality involving correlations A.R. Nematollahia,, A.R. Soltania,b a
Department of Statistics, College of Science, Shiraz University, Shiraz, 71454 Iran Department of Statistics and Operations Research, College of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
b
Received 3 May 2004; received in revised form 28 June 2005 Available online 15 September 2005
Abstract A useful inequality involving correlations between three random variables with mean zero and finite second moments is presented. The inequality is applied to show that the entries on the main diagonal of the spectral density of a periodically correlated Markov process, as derived in Nematollahi and Soltani [2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1) 127–140], are nonnegative. The inequality, when two variables with the same second moments are involved, is compared to the Cauchy–Schwartz inequality. r 2005 Elsevier B.V. All rights reserved. Keywords: Periodically correlated Markov processes; Second order processes; Cauchy–Schwartz inequality
1. Introduction Since the classical work on inequalities by Hardy et al. (1934), enormous efforts have been exerted to extend classical inequalities, resulting in discoveries of many types of inequalities with potential applications; see Tong (1984). In this paper we present a useful inequality involving covariances EXY , EYZ, where X ; Y and Z are mean zero random variables with finite second moments measured in the same units. The inequality, to be derived in Section 2, appears to be crucial in proving that f 0 ðlÞ, comprising the main diagonal of the spectral density of a periodically correlated Markov process of period 2, as derived in Nematollahi and Soltani (2000), is nonnegative. This application is presented in Section 3. Interestingly, when only X and Y are involved and EX 2 ¼ EY 2 , the inequality provides an upper bound for the jEXY j in terms of EX 2 and jEXY j itself which appears to be sharper than the one from the Cauchy–Schwartz inequality; see Remark 3.1.
2. Main result The following theorem is the main result of this article in which a new inequality involving correlations is obtained. Corresponding author.
E-mail addresses: [email protected] (A.R. Nematollahi), [email protected] (A.R. Soltani). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.032
ARTICLE IN PRESS A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
370
Theorem 2.1. Suppose X ; Y and Z are mean zero random variables with finite second order moments measured on the same units. If EjX j2 ¼ EjZj2 , then jEXY þ EYZj 1 ðEXY ÞðEYZÞ 1 þ p . (2.1) 2 EX 2 þ EY 2 EX 2 EY 2 Proof. Let qxy ¼ EXY =ðEX 2 EY 2 Þ1=2 and qyz ¼ EYZ=ðEY 2 EZ 2 Þ1=2 . It follows from the Cauchy–Schwartz inequality that ð1 q2xy Þð1 q2yz ÞX0, or jqxy þ qyz jp1 þ qxy qyz .
(2.2)
On the other hand for any two positive numbers u and v, always pffiffiffipffiffiffi u v 1 p . uþv 2 Substitutions u ¼ EX 2 and v ¼ EY 2 in (2.3) yield pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi EX 2 EY 2 1 p . 2 EX 2 þ EY 2
(2.3)
(2.4)
Derivations (2.2) and (2.4) provide that pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi EX 2 EY 2 jEXY þ EYZj jEXY þ EYZj pffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi ¼ 2 2 EX 2 þ EY 2 EX þ EY EX 2 EY 2 1 p jqxy þ qyz j 2 1 p ð1 þ qxy qyz Þ 2 1 ðEXY ÞðEYZÞ 1þ ¼ ; 2 EX 2 EY 2 which proves the theorem.
&
3. Applications Consider a zero mean discrete time second order process X ¼ fX t ; t 2 Zg, Z the set of integers. The process is said to be periodically correlated (PC) with period T, if its covariance function Rðt; sÞ ¼ EX t X s satisfies Rðt; sÞ ¼ Rðt þ T; s þ TÞ
for every s; t 2 Z,
(3.1)
where T is the smallest positive integer. It is well known that every PC process X possesses a unique spectral density, whenever the density exists, given by the matrix hðlÞ ¼ ½hjk ðlÞj;k¼0;...;T1 ; 0plp2p, where 1 f ððl 2pjÞ=TÞ; T kj The functions f k satisfy Z 2p eitl f k ðlÞ dl, Bk ðtÞ ¼ hjk ðlÞ ¼
j; k ¼ 0; . . . ; T 1;
0plp2p.
0
and Bk ðtÞ are related to the correlations Rðn þ t; nÞ through Rðn þ t; nÞ ¼
T 1 X 0
Bk ðtÞe2pikn=T .
ARTICLE IN PRESS A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
371
The matrix hðlÞ is Hermitian and nonnegative definite. Moreover f 0 ðlÞX0; see Nematollahi and Soltani (2000). The spectral density hðlÞ for PC Markov processes was derived by the authors in (Nematollahi and Soltani, 2000). For T ¼ 2, f 0 appeared to be f 0 ðlÞ ¼
ðR0 ð0Þ þ R1 ð0ÞÞð1 ge2 Þ ½1 þ 2a cosðl=2Þ; 4pj1 geeil j2
l 2 ½0; 2pÞ,
(3.2)
where ge ¼ ½R0 ð1ÞR1 ð1Þ=½R0 ð0ÞR1 ð0Þ and a ¼ ½R0 ð1Þ þ R1 ð1Þ=½ðR0 ð0Þ þ R1 ð0ÞÞð1 þ geÞ. According to the theory it is expected f 0 ðlÞ is nonnegative. But it is not clear whether the expression for f 0 ðlÞ in (3.2) is nonnegative. The ratio appearing on the right of (3.2) will be nonnegative if je gjp1, which follows from the Cauchy–Schwartz inequality. Indeed, since for T ¼ 2, EX 22 ¼ EX 20 , it follows that R0 ð1Þ R1 ð1Þ jEX 1 X 0 EX 2 X 1 j ¼ je gj ¼ R0 ð0Þ R1 ð0Þ EX 20 EX 21 p
½EX 20 EX 21 EX 22 EX 21 1=2 EX 20 EX 21 ¼ ¼ 1. EX 20 EX 21 EX 20 EX 21
Thus it remains to show the term 1 þ 2a cosðl=2Þ is also nonnegative. We infer from Brockwell and Davis (1991, p. 121) that the term 1 þ 2a cosðl=2Þ is also nonnegative, for a given a, if and only if jajp12. We will use the inequality, derived in Section 2, to show that this indeed is the case. Replace X 0 for X, X 1 for Y and X 3 for Z in (2.1). Note that X 0 ; X 1 ; X 2 are mean zero random variables defined on the same probability space, and for T ¼ 2, EX 20 ¼ Rð0; 0Þ ¼ Rð2; 2Þ ¼ EX 22 : Hence the assumptions of Theorem 2.1 are fulfilled. Thus it follows from inequality (2.1) that ! jEX 1 X 0 þ EX 2 X 1 j 1 ðEX 1 X 0 ÞðEX 2 X 1 Þ 1þ , p 2 EX 20 þ EX 21 ðEX 20 ÞðEX 21 Þ or R0 ð1Þ þ R1 ð1Þ 1 R ð0Þ þ R ð0Þp 2 ð1 þ geÞ, 0
1
where Rt ðkÞ ¼ Rtþ2 ðkÞ; t; k 2 Z. Therefore jajp12, giving the result. Remark 3.1. Let in (2.1) EX 2 ¼ EY 2 , and EXY ¼ EYZ, then the inequality will become 1 E 2 XY jEXY jp EX 2 þ , 2 EX 2
(3.3)
which can be derived directly from the fact that for any a, 2 a 1 2 pffiffiffi p ffiffi ffi X0, EX 2EX 2 2 with a ¼ jEXY j. Since it is assumed that EX 2 ¼ EY 2 , by the Cauchy–Schwartz inequality jEXY jpEX 2 . Thus it follows that 1 E 2 XY 2 (3.4) jEXY j p EX þ p EX 2 . 2 EX 2 Therefore (3.3) provides sharper upper bound, but of course it depends on EXY itself. Let X and Y be discrete with masses 0.2, 0.3, 0.3 and 0.2 at ð1; 1Þ, ð1; 1Þ, ð1; 1Þ and ð1; 1Þ, respectively. Then EX 2 ¼ EY 2 ¼ 1; EXY ¼ 0:2. Therefore the terms in (3.4) for this example are 0.2, 0.52, 1, respectively; which shows the inequalities in (3.3) could be strict.
ARTICLE IN PRESS 372
A.R. Nematollahi, A.R. Soltani / Statistics & Probability Letters 76 (2006) 369–372
Acknowledgements The authors thank the referee for providing constructive comments. References Brockwell, P.J., Davis, R.A., 1991. Time Series: Theory and Method. Springer, New York. Hardy, G.H., Littlewood, J.E., Polya, G., 1934. Inequalities. Cambridge University Press, Cambridge. Nematollahi, A.R., Soltani, A.R., 2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1), 127–140. Tong, Y.L., 1984. Inequalities in statistics and probability. Proceedings of the symposium held at the University of Nebraska, Lincoln, Nebraska, October 27–30, 1982, IMS Lecture Notes, Monograph Series, vol. 5. Institute of Mathematical Statistics, Hayward, CA.