Exponential inequality for associated random variables

Statistics & Probability Letters 42 (1999) 423 – 431

Exponential inequality for associated random variables D.A. Ioannidesa , G.G. Roussasb; ∗;1 b University

a University

of Macedonia, Thessaloniki, Greece of California, Division of Statistics, One Shields Avenue, Davis, CA 95616-8705, USA Received April 1998; received in revised form August 1998

Abstract Under mild conditions, a Bernstein–Hoeding-type inequality is established for covariance invariant positively associated random variables. A condition is given for almost sure convergence, and the associated rate of convergence is speci ed c 1999 Elsevier Science B.V. All rights reserved in terms of the underlying covariance function. Keywords: Exponential inequality; Associated random variables; Covariance function

1. Introduction The concept of positively associated random variables (r.v.’s) (associated, in the original terminology) was introduced in the statistical literature by Esary et al. in 1967. The same concept has been used in statistical mechanics under the name of FKG-inequalities, taken from the initials of the authors Fortuin et al. (1971). It seems that Esary’s et al. (1967) motivation for introducing the notion of association stemmed from its signi cance in systems reliability. An early account of it can be found in the book Barlow and Proschan (1975), in Sections 2 and 3, pp. 29 –39, and Section 4, pp. 142–152. Extensive discussions on association can also be found in recent books and monographs; see, for example, Shaked and Tong (1992), Shaked and Shanthikumar (1994), Szekli (1995), and Yoshihara (1997). Also, relevant are the recent papers Cohen and Sackrowitz (1992, 1994), and Sarkar and Chang (1997). Early contributions in the statistical mechanics framework were papers by Harris (1960), Lebowitz (1972), Simon (1973), Preston (1974), and Kemperman (1977). Newman (1980) was the rst to establish a central limit theorem for positively associated r.v.’s. Other relevant works by the same author, alone or jointly, are Newman (1984, 1990) and Newman and Wright (1981, 1982). There is a signi cant number of contributions from the probability and statistical viewpoint. Some of them are those represented by the papers Cox and Grimmett (1984), Birkel (1988a,b, 1989) Bagai and Prakasa Rao (1991, 1995), Roussas (1991, 1993, 1994, 1995, 1996), Yu (1993) and Cai and Roussas (1997a,b, 1998a,b). Birkel (1988a) was the rst to provide moment bounds for positively associated r.v.’s; exponential inequalities are not available as yet, to the best of ∗ 1

Corresponding author. Tel.: +1-530-752-8142; fax: +1-530-752-7099; e-mail: [email protected]. This research was supported in part by a research grant from the University of California, Davis.

c 1999 Elsevier Science B.V. All rights reserved 0167-7152/99/$ – see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 8 ) 0 0 2 4 0 - 5

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our knowledge. The reason may be that the relevant moment inequalities go the “wrong” way for associated r.v.’s. It is the purpose of this paper to ll in this void. The signi cance of exponential inequalities toward several probability and statistical applications is well known, and needs no elaboration here. Such applications may be discussed in a forthcoming paper. The remaining sections of the paper are organized as follows. In the next section, the necessary notation and terminology are introduced before the main result, inequality (2.7), can be formulated. In addition to the basic assumption of positive association, the conditions required are that the underlying r.v.’s be bounded, covariance invariant (however, see also Remark 2.2), and that the covariance involved decays at a rate subject to a bound (see condition (2.6)). The proof of the theorem rests heavily on Lemma 3.1, which is formulated and proved in Section 3. Actually, all preliminary results needed are taken care of in the same section, as is the proof of the theorem. The optimal associated rate of convergence depends on the underlying covariance function, and is explicitly calculated by way of formulas (3.26) and (3.23) for a given covariance function. Section 4 is devoted to the discussion of two classes of covariances falling into scope of Theorem 2.1. To avoid unnecessary repetitions, it is stated at the outset that all limits are taken as n → ∞. 2. Notation, assumptions, and formulation of main results At the outset, the de nition of positively associated r.v.’s is given. Deÿnition 2.1. For a nite index set I , the r.v.’s {Xi ; i ∈ I } are said to be positively associated (PA for short), if for any real-valued coordinatewise increasing functions G and H de ned on R I , Cov(G(Xi ; i ∈ I ); H (Xj ; j ∈ I ))¿0;

(2.1)

provided EG 2 (Xi ; i ∈ I ) ¡ ∞; EH 2 (Xj ; j ∈ I ) ¡ ∞. If I is not nite, the r.v.’s {Xi ; i ∈ I } are said to be PA, if any nite subcollection is a set of PA r.v.’s. Remark 2.1. Along with PA, the concept of negatively associated r.v.’s was introduced by Joag-Dev and Proschan (1983). Accordingly, for a nite index set I , the r.v.’s {Xi ; i ∈ I } are said to be negatively associated (NA for short), if for any nonempty and disjoint subsets A and B of I , and any coordinatewise increasing functions G and H with G : R A → R and H : R B → R and EG 2 (Xi ; i ∈ A) ¡ ∞; EH 2 (Xj ; j ∈ B) ¡ ∞; Cov(G(Xi ; i ∈ A); H (Xj ; j ∈ B))60:

(2.2)

If I is not nite, the r.v.’s {Xi ; i ∈ I } are said to be NA, if any nite subcollection is a set of NA r.v.’s. Negatively associated r.v.’s will not occupy our interest in this paper. Suces only to say that an exponential inequality, as the one discussed here is not hard to obtain; actually, this was done in Roussas (1996). For the formulation of the assumptions to be made in this paper, the introduction of some notation is required. This notation is closely related to the way the proofs are carried out. Namely, for positive integers 16p = p(n) ¡ n and p → ∞, divide the set {1; 2; : : : ; n} into successive groups each containing p elements. Let r = r(n) be the largest integer with: 0 ¡ r ¡ n;

r → ∞;

and

2pr6n;

(2.3)

which implies that n=2pr → 1. Thus, the set {1; 2; : : : ; n} is split into 2r groups, each consisting of p elements; the remaining n − 2pr ¡ p elements constitute a set which may be empty. The assumptions under which the main result in this paper is obtained are gathered together below for easy reference.

D.A. Ioannides, G.G. Roussas / Statistics & Probability Letters 42 (1999) 423 – 431

425

Assumptions. (A1) The basic assumption is that the r.v.’s Xi ; i¿1, are PA. (A2) The r.v.’s are bounded, |Xi |6M=2; i¿1, and covariance invariant, Cov(Xi ; Xi+k )=Cov(X1 ; Xk+1 ); i¿1; k¿1. For later reference, set C(k) = Cov(X1 ; Xk+1 );

k¿1:

(2.4)

(A3) Without loss of generality, it is assumed that C(k) is nonincreasing as k → ∞. Remark 2.2. The results in this paper hold, if, in assumption (A2), the covariance invariance is dropped, and C(k) is rede ned by C(k) = sup{Cov(Xi ; Xi+k ); i¿1}; k¿1: The nondecreasing property of C(k), assumed in (A3), is retained. (See also Remark 3.1 below.) This suggestion was made by an anonymous referee, and is thankfully acknowledged here. De ne S n and n by



n

Sn =

1X (Xi − EXi ); n

n =

i=1

M 2 2

1=2 

log n r

1=2 ;

(2.5)

where M is as in assumption (A1), r is as in (2.3), and  is an arbitrary constant ¿ 1 (see discussion just prior to relation (3.20)). Then the exponential inequality to be established is the following. Theorem 2.1. Let S n and n be deÿned by (2.5). Then, under assumptions (A1) – (A3), and the proviso   4(M + 1)   1=2 1=2 (r log n) ; (2.6) C(p)6exp − 3M 2 it holds P(|S n |¿n )6C0 exp (−crn2 );

c = 2=9M 2 ;

(2.7)

for all suciently large n, n¿n0 , say, where C0 is a constant ( for example, C0 = 12). Furthermore, S n → 0 a.s. at the rate 1=n . The optimal speci cation of r is given by (3.25) or (3.26), and the respective 1=n is given by (3.23). 3. Preliminary results Pn Set Yi = Xi − EXi , so that |Yi |6M; Yi ; i¿1, are PA, Sn = i=1 Yi and S n = Sn =n. With p and r as in the previous section, de ne the r.v.’s Ui ; Vi ; i = 1; : : : ; r and Wn by Ui = Y2(i−1)p+1 + · · · + Y(2i−1)p ; Wn = Y2pr+1 + · · · + Yn ; and

r

Un =

1X Ui ; n

Vi = Y(2i−1)p+1 + · · · + Y2ip ;

r

Vn =

i=1

1X Vi ; n

Wn =

i=1

Wn ; n

(3.1) (3.2)

(3.3)

so that S n = U n + V n + W n: We may now formulate the next lemma.

(3.4)

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D.A. Ioannides, G.G. Roussas / Statistics & Probability Letters 42 (1999) 423 – 431

Lemma 3.1. Let U n be deÿned by (3.3), let n ¿ 0, and suppose assumptions (A1) – (A3) hold. Then, for a suitable constant C0 ( for example, C0 = 3), P(U n ¿n )6C0 exp (−2rn2 =M 2 ); provided C(p)6exp [− 4(M + 1)rn =M 2 ]. Proof. The r.v.’s U1 ; : : : ; Ur are PA and |Ui |6pM for all i. For some  ¿ 0 whose range will be speci ed below (see relation (3.12)), set h(x) = e(=n)x ; −pM ¡ x ¡ pM , so that h0 (x) = (=n)e(=n)x and for their sup-norms, it holds: ||h||∞ 6epM=n ; ||h0 ||∞ 6(=n)epM=n . With this h and A = {1; : : : ; r − 1}, B = {r}, so that #A + #B − 2 = r − 2, apply Lemma 3.1 (ii) in Birkel (1988a) to obtain r−1  2  Pr−1 X Cov(Ui ; Ur ); (06) Cov e(=n) i=1 Ui ; e(=n)Ur 6 2 eprM=n n i=1

so that

  Pr−1 EeU n = E e(=n) i=1 Ui · e(=n)Ur Pr−1

6 Ee(=n)

i=1

Ui

· e(=n)Ur +

r−1

2 prM=n X e · Cov(Ui ; Ur ): n2

(3.5)

i=1

But r−1 X

Cov(Ui ; Ur ) =

i=1

r−1 X

(2i−1)p

(2r−1)p

X

X

Cov(Yj ; Yk )

i=1 j=2(i−1)p+1 k=2(r−1)p+1

=

XX

Cov(Yj ; Yk )

j∈A k∈B

=

r−1 X X X

Cov(Yj ; Yk );

(3.6)

‘=1 j∈A‘ k∈B

where

      A = 1; : : : ; p; 2p + 1; : : : ; 3p; : : : ; 2(r − 2)p + 1; : : : ; (2r − 3)p {z } {z } |   | {z } | A1

A2

Ar−1

and B = {2(r − 2)p + 1; : : : ; (2r − 3)p}. By assumptions (A2) and (A3), XX Cov(Yj ; Yk ) 6 p Cov(Y1 ; Y2(r−1)p+1 ) + · · · + p Cov(Yp ; Y2(r−1)p+1 ) j∈A1 k∈B

6 p2 Cov(Yp ; Y2(r−1)p+1 )6p2 C((2r − 3)p): Likewise, XX

Cov(Yj ; Yk )6p2 C((2r − 5)p);

j∈A2 k∈B

and continuing like this, X X Cov(Yj ; Yk )6p2 C(p): j∈Ar−1 k∈B

D.A. Ioannides, G.G. Roussas / Statistics & Probability Letters 42 (1999) 423 – 431

Therefore, XX

427

Cov(Yj ; Yk ) 6 p2 [C(p) + · · · + C((2r − 5)p) + C((2r − 3)p)]

j∈A k∈B

6 p2 rC(p):

(3.7)

By means of (3.7) and (3.6), inequality (3.5) becomes Pr−1

EeU n 6Ee(=n)

i=1

Ui

· Ee(=n)Ur +

2 prM=n 2 e ·p rC(p): n2

Apply the inequality 1 + x6ex ; x ∈ R , for x = (=n)Ur and x = (=n) multiply the resulting inequalities in order to get Pr−1

16Ee(=n)

i=1

Ui

(3.8) Pr−1 i=1

Ui , take expectations, and

· Ee(=n)Ur :

(3.9)

From 2pr6n, we get p2 r=n2 61=4r. Therefore, by means of this result and (3.9), inequality (3.8) becomes   Pr−1 2 (3.10) EeU n 6Ee(=n) i=1 Ui · Ee(=n)Ur 1 + eM C(p) : 4r The inequality xe6ex ; x ∈ R , gives 2 =46e−2 e ¡ e , so that 1+

1 2 M e C(p)61 + e(1+M ) C(p); 4r r

(3.11)

and we wish to have e(1+M ) C(p)61 or, equivalently, 6 −

1 log C(p): M +1

(3.12)

On account of (3.11) and (3.12), inequality (3.10) yields   Pr−1 1 U n (=n) i=1 Ui (=n)Ur : · Ee 1+ Ee 6Ee r Repeating the process which led to (3.13) another r − 1 times, we obtain, under condition (3.12), r  r Y 1 EeU n 6 Ee(=n)Ui 1 + : r

(3.13)

(3.14)

i=1

At this point, apply Lemma 1 in Devroye (1991), which states that, if EX = 0 and a6X 6b, then for every  ¿ 0; E exp (X )6 exp [2 (b − a)2 =8]. Take X = Ui , so that |Ui |6pM and b − a = 2pM , to obtain E exp ((=n)Ui )6 exp (2 p2 M 2 =2n2 ), and hence r Y

Ee(=n)Ui 6e

2

M 2 p2 r=2n2

6e

2

M 2 =8r

;

(3.15)

i=1

because 2 M 2  2 M 2 p2 r = 2 2n 8r



2pr n

2 6

2 M 2 : 8r

Since also (1 + (1=r))r 6C1 ( for example, C1 = 3), inequality (3.14) becomes Ee(=n)U n 6C1 e

2

M 2 =8r

;

subject to (3:12):

(3.16)

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D.A. Ioannides, G.G. Roussas / Statistics & Probability Letters 42 (1999) 423 – 431

Therefore, for n ¿ 0 and under (3.12), P(U n ¿n )6C1 e−n +(

2

M 2 =8r)

:

(3.17)

Minimizing, with respect to , the right-hand side in (3.17), we obtain 2

P(U n ¿n )6C1 e−2rn =M

2

for 0 = 4rn =M 2

subject to (3:12):

For 0 as in (3.18), the requirement (3.12) is equivalent to   4(M + 1) r  C(p)6 exp − n : M2

(3.18)

(3.19)

This completes the proof of the lemma. Remark 3.1. Under the modi cation mentioned in Remark 2.2, the proof of Lemma 3.1 isP somewhat shortened r−1 in that, in the derivations between relations (3.5) and (3.7), one may argue directly that i=1 Cov (Ui ; Ur )6 2 (r − 1)p C(p). This is so, as the indices of the Y ’s involved in Ui and Ur dier by at least p units. Remark 3.2. As stated in assumption (A3), the condition that the covariance function C(k) be nonincreasing is not, really, necessary, although it would not be easy to envision cases where it does not occur. This can be justi ed in the process of arriving at inequality (3.7) by way of (3.6). All one has to do is to produce some more re ned bounds for the covariances, but such a result does not seem worth the eort. For almost sure convergence purposes, we wish to have 2r n2 =M 2 = log n (for any arbitrary  ¿ 1), or equivalently, 1=2  1=2  log n M 2 : (3.20) n = 2 r For this choice of n , 0 becomes  1=2 8 (r log n)1=2 ; 0 = M2 and condition (3.19) yields   4(M + 1)   1=2 (r log n)1=2 : C(p)6exp − M 2

(3.21)

(3.22)

Summarizing these observations, we have Lemma 3.2. Under assumptions (A1) – (A3), and with n speciÿed by (3.20), it holds P(U n ¿n )6C1 exp (−2rn2 =M 2 ); provided C(p) satisÿes condition (3.22). Remark 3.3. It is obvious that V n , as de ned in (3.3), satis es the same inequalities as U n in Lemmas 3.1 and 3.2. The following observation is meant to explain that we may dispense with W n as de ned in (3.3). Lemma 3.3. Under assumptions (A1) – (A3) and with n deÿned by (3.20), P(|Wn |¿n )=0 for all suciently large n.

D.A. Ioannides, G.G. Roussas / Statistics & Probability Letters 42 (1999) 423 – 431

429

Proof. Wn consists of n−2pr terms and n−2pr ¡ p. Then |Wn | ¡ pM=n, so that P(|Wn |¿n )6P(M ¿nn =p). This last expression, however, is 0, for all suciently large n, because  1=2  2 1=2   M 2 n log n n nn 2 1=2 = (r log n)1=2 → ∞; = (2M ) p 2 p2 r 2pr this is so because n=2pr → 1. Proof of Theorem 2.1. It consists, essentially, in combining Lemma 3.2, Remark 3.3, and Lemma 3.3. The r.v.’s −Yi , i = 1; : : : ; n have the same properties as the r.v.’s Yi , i = 1; : : : ; n. Thus, always under the proviso stated in (3.22), P(|U n |¿n ) = P(U n ¿n ) + P(−U n ¿n )62C0 exp (−2rn2 =M 2 ); and similarly for P(|V n |¿n ). Therefore, P(|S n |¿3n ) 6 P(|U n |¿n ) + P(|V n |¿n ) + P(|W n |¿n ) 6 P(|U n |¿n ) + P(|V n |¿n )

(for n¿n0 ; say)

6 4C1 exp (−2rn2 =M 2 ): Setting C0 = 4C1 and replacing n by n =3, we obtain, nally, P(|S n |¿n )6C0 exp (−crn2 );

c = 2=9M 2 ; n¿n0 ;

provided

  4(M + 1)   1=2 1=2 (r log n) : C(p)6exp − 3M 2 2

The speci cation of n by (3.20), leads to the convergence S n → 0 a.s. at the rate of 1=n . Regarding the latter part of the theorem, proceed as follows. For the value of n speci ed in (3.20), the rate of convergence is given by 1=2  1=2  2 r 1 = : (3.23) n M 2 log n Inequality (3.22) becomes, equivalently:  2 M [log C(p)]2 1 : r6 8 M + 1 log n

(3.24)

From (3.23), the maximum rate is attained for the maximum allowed value of r. This maximum value is obtained from (3.24) and is  2 M [log C(p)]2 1 : (3.25) r= 8 M + 1 log n This is so, because when r increases, p decreases, due to the fact that r and p are inverse proportional, and also to the assumption that C(k) is a decreasing function. Again, by the fact that n=2pr → 1, it follows that p = (1=2x n ) · (n=r), some 0 ¡ x n → 1, so that (3.25) becomes  2 M 1 n [log C(p)]2 1 ; p= · ; 0 ¡ x n → 1; (3.26) r= 8 M + 1 log n 2x n r and all one has to do is solve for r. Then the corresponding rate of convergence is obtained from (3.23).

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Remark 3.4. From (3.23), it follows that the optimal rate of convergence is obtained by taking r=n. However, such a choice is not allowed here. Consequently, the rate of convergence (n=log n)1=2 is unattainable in the present framework. 4. Examples In this section, two examples with typical covariance functions are discussed. In each case, the optimal choice of r, provided by formula (3.26), is given, as well as the corresponding best rate of almost sure convergence through formula (3.23). Example 4.1. Suppose the covariance function de ned in (2.4) is given by: C(k) = 0 k ; 0 ¡  ¡ 1; 0 ¿ 0. Then relation (3.26) is of the form: r = C1n

1 n n2 + C2n + C3n 2 log n r log n r log n

or r 3 = C1n

r2 nr n2 + C2n + C3n ; log n log n log n

(4.1)

and the last term on the right-hand side in (4.1) is of highest order. Therefore r 3 is of the order of n2 =log n, and r is of the order of (n2 =log n)1=3 . Then, by (3.23), it turns out that 1=n is of the order of [n=(log n)2 ]1=3 . Example 4.2. Here suppose the covariance C(k) is of the form C(k) = k0 k −Â , Â ¿ 0; k0 ¿ 0. Then relation (3.26) is of the form r = C1n

1 log r (log r)2 + C2n + C3n + C4n + C5n (log r) + C6n (log n); log n log n log n

(4.2)

and the last term on the right-hand side in (4.2) is of highest order. Thus, r is of order log n, and then, by means of (3.23), 1=n is a constant. In other words, in this case, we do have almost sure convergence but without rates. References Bagai, I., Prakasa Rao, B.L.S., 1991. Estimation of the survival function for stationary associated processes. Statist. Probab. Lett. 12, 385–391. Bagai, I., Prakasa Rao, B.L.S., 1995. Kernel-type density and failure rate estimation for associated sequences. Ann. Inst. Statist. Math. 47, 253–266. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. Birkel, T., 1988a. Moment bounds for associated sequences. Ann. Probab. 16, 1184 –1193. Birkel, T., 1988b. On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16, 1685–1698. Birkel, T., 1989. A note on the strong law of large numbers for positively dependent random variables. Statist. Probab. Lett. 7, 17–20. Cai, Z.W., Roussas, G.G., 1997a. Ecient estimation of a distribution function under quadrant dependence. Scand. J. Statist. 24, 1–14. Cai, Z.W., Roussas, G.G., 1997b. Smooth estimate of quantiles under association. Statist. Probab. Lett. 36, 275–287. Cai, Z.W., Roussas, G.G., 1998a. Berry-Esseen bounds for smooth estimator of a distribution function under association. J. Nonparametric Statist., to appear. Cai, Z.W., Roussas, G.G., 1998b. Kaplan–Meier estimator under association. J. Multivariate Anal., to appear. Cohen, A., Sackrowitz, H.B., 1992. Some remarks on a notion of positive dependence, association, and unbiased testing. In: Shaked, M., Tong, Y.L. (Eds.), Stochastic Inequalities. IMS, Lecture Notes-Monograph Series, vol. 22.

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