RIO-TYPE INEQUALITY FOR THE EXPECTATION OF
PRODUCTS OF RANDOM VARIABLES
B. L. S. PRAKASA RAO
Received 26 July 2004
We develop an inequality for the expectation of a product of n random variables generalizing the recent work of Dedecker and Doukhan (2003) and the earlier results of Rio
(1993).
1. Introduction
Let (Ω,Ᏺ,P) be a probability space and let (X,Y ) be a bivariate random vector defined
on it. Suppose that E(X 2 ) < ∞ and E(Y 2 ) < ∞. Hoeffding proved that
Cov(X,Y ) =
R
2
P(X ≤ x, Y ≤ y) − P(X ≤ x)P(Y ≤ y) dx d y.
(1.1)
In [5], Lehmann gave a simple proof of this identity and used it in his study of some
concepts of dependence. This identity was generalized to functions h(X) and g(Y ) with
E[h2 (X)] < ∞ and E[g 2 (Y )] < ∞ and with finite derivatives h (·) and g (·) by Newman
[6]. Multidimensional versions of these results were proved by Block and Fang [1], Yu
[13], and more recently by Prakasa Rao [7]. Related covariance identities for exponential
and other distributions are given by Prakasa Rao in [9, 10].
Suppose that ᏹ is a sub-σ-algebra of Ᏺ and Y is measurable with respect to ᏹ. Let
σ(X) be the sub-σ-algebra generated by the random variable X. Define
α(ᏹ,X) = sup P(A ∩ B) − P(A)P(B), A ∈ ᏹ, B ∈ σ(X) .
(1.2)
Define
z
QX (u) = inf x : P |X | > x ≤ u ,
GX (s) = inf z :
0
QX (t)dt ≥ s ,
HX,Y (s) = inf t : E |X |I[|Y |>t] ≤ s .
Copyright © 2005 Hindawi Publishing Corporation
Journal of Inequalities and Applications 2005:1 (2005) 7–14
DOI: 10.1155/JIA.2005.7
(1.3)
8
Rio-type inequality
Rio [11] proved that
Cov(X,Y ) ≤ 2
α(ᏹ,X)/2
QY (u)QX (u)du.
0
(1.4)
Related results are given in [12, page 9]. These results were generalized by Bradley [2]
for a strong-mixing process and by Prakasa Rao [8] for rth-order joint cumulant under
rth-order strong mixing. In a recent work, Dedecker and Doukhan [3] proved that
E(XY ) ≤
E(X |ᏹ)1
0
HX,Y (t)dt ≤
E(X |ᏹ)1
QY oGX (t)dt
0
(1.5)
and obtained an improved version of the above inequality. If Xi , 1 ≤ i ≤ n, are positivevalued random variables, it is easy to see that
E X1 X2 · · · Xn ≤
1
0
QX1 (u)QX2 (u) · · · QXn (u)du.
(1.6)
For a proof, see [12, Lemma 2.1, page 35].
We now obtain an improved version of the above inequality following the techniques
of Dedecker and Doukhan [3] and Block and Fang [1].
2. Main result
Let {Xi , 1 ≤ i ≤ n} be a sequence of nonnegative random variables defined on a probability space (Ω,Ᏺ,P). Then the random variable Xi can be represented in the form
Xi =
∞
0
I(xi ,∞) Xi dxi ,
(2.1)
where

1
if Xi > xi ,
I(xi ,∞) Xi = 
0 if Xi ≤ xi .
(2.2)
Hence
E X1 X2 · · · Xn = E X1 Πni=2
=
=
Rn+−1
Rn+−1
∞
0
I(xi ,∞) Xi dxi
E X1 Πni=2 I(xi ,∞) Xi dx2 · · · dxn
(2.3)
E X1 I[Xi >xi , 2≤i≤n] X2 ,...,Xn dx2 · · · dxn
by the Fubini’s theorem, where R+n−1 = {(x2 ,...,xn ) : xi ≥ 0,2 ≤ i ≤ n}. Observe that
E X1 I[Xi >xi , 2≤i≤n] X2 ,...,Xn
≤ min E[X1 ],E X1 I[Xi >xi , 2≤i≤n] X2 ,...,Xn
(2.4)
B. L. S. Prakasa Rao 9
and hence
E X1 X2 · · · Xn ≤
Rn+−1
EX1
0
χ(E[X1 I[Xi >xi , 2≤i≤n] (X2 ,...,Xn )]>u) (u)du dx2 · · · dxn .
(2.5)
Here χA (·) denotes the indicator function of the set A. Let
gX1 x2 ,...,xn = E X1 I[Xi >xi , 2≤i≤n] X2 ,...,Xn ].
(2.6)
Then
E X1 X2 · · · Xn ≤
=
Rn+−1
EX1
0
χ[gX1 (x2 ,...,xn )>u] (u)du dx2 · · · dxn
E(X1 ) [(x2 ,...,xn ):gX1 (x2 ,...,xn )>u]
0
(2.7)
1dx2 · · · dxn du.
Let
HX1 ,X2 ,...,Xn (u) = λ x2 ,...,xn : gX1 x2 ,...,xn > u ,
(2.8)
where λ is the Lebesgue measure on the space R+n−1 . Hence
E X1 X2 · · · Xn ≤
E(X1 )
HX1 ,X2 ,...,Xn (u)du.
0
(2.9)
Observe that
gX1 x2 ,...,xn = E X1 I[Xi >xi , 2≤i≤n] X2 ,...,Xn
≤
E[I[X >x , 2≤i≤n] (X2 ,...,Xn )]
i
i
QX1 (u)du
0
(2.10)
from the Fréchet’s inequality [4]. Here QX1 (·) is the generalized inverse of the function
TX1 (x) = P(X1 > x) as defined earlier. Let
MX1 (y) =
y
0
QX1 (t)dt.
(2.11)
Observe that MX1 (·) is nondecreasing in y. Let GX1 (u) = inf {z : MX1 (z) ≥ u} as defined
earlier. Let
TX2 ,...,Xn x2 ,...,xn = P Xi > xi , 2 ≤ i ≤ n .
(2.12)
Note that
gX1 x2 ,...,xn ≤ MX1 E I[Xi >xi , 2≤i≤n] X2 ,...,Xn
gX1 x2 ,...,xn > u =⇒ MX1 E I[Xi >xi , 2≤i≤n] X2 ,...,Xn
,
>u
=⇒ E I[Xi >xi , 2≤i≤n] X2 ,...,Xn > GX1 (u)
=⇒ P Xi > xi , 2 ≤ i ≤ n > GX1 (u).
(2.13)
10
Rio-type inequality
Hence the set
x2 ,...,xn ∈ R+n−1 : gX1 x2 ,...,xn > u
(2.14)
is contained in the set
x2 ,...,xn ∈ R+n−1 : P Xi > xi , 2 ≤ i ≤ n > GX1 (u) .
(2.15)
In particular, it follows that the Lebesgue measure of the former set is less than or equal
to that of the latter. Let
QX∗2 ,...,Xn GX1 (u)
(2.16)
denote the Lebesgue measure of the set (2.15).
Then
HX1 ,X2 ,...,Xn (u) ≤ QX∗2 ,...,Xn GX1 (u)
(2.17)
for all 0 ≤ u ≤ 1. Hence
E X1 X2 · · · Xn ≤
E(X1 )
0
QX∗2 ,...,Xn GX1 (u) du.
(2.18)
We have proved the following inequality.
Theorem 2.1. Let Xi , 1 ≤ i ≤ n, be nonnegative random variables defined on a probability
space (Ω,Ᏺ,P). Then
E X1 X2 · · · Xn ≤
E(X1 )
0
HX1 ,X2 ,...,Xn (u)du ≤
E(X1 )
0
QX∗2 ,...,Xn oGX1 (u)du,
(2.19)
where the functions H,Q∗ , and G are as defined earlier.
3. Applications
We now suppose that the random variables {Xi , 1 ≤ i ≤ n} are arbitrary but with
EX1 X2 · · · Xn < ∞.
(3.1)
Define
gX1 x2 ,...,xn = E X1 I[|Xi |>xi , 2≤i≤n] X2 ,...,Xn ,
HX1 ,X2 ,...,Xn (u) = λ x2 ,...,xn : gX1 x2 ,...,xn ≤ u ,
TX2 ,...,Xn x2 ,...,xn
= P Xi > xi , 2 ≤ i ≤ n ,
(3.2)
and define MX1 (·),QX1 (·),QX∗2 ,...,Xn , and GX1 accordingly. The following theorem follows
by arguments analogous to those given in Section 2.
B. L. S. Prakasa Rao 11
Theorem 3.1. Let Xi , 1 ≤ i ≤ n, be arbitrary random variables defined on a probability
space (Ω,Ᏺ,P). Then
E X1 X2 · · · Xn ≤
E(|X1 |)
0
HX1 ,X2 ,...,Xn (u)du ≤
E(|X1 |)
0
QX∗2 ,...,Xn oGX1 (u)du,
(3.3)
QX2 oGX1 (u)du
(3.4)
where the functions H,Q∗ , and G are as defined above.
In particular, for n = 2, we have
E X1 X2 ≤
E(|X1 |)
0
HX1 ,X2 (u)du ≤
E(|X1 |)
0
since QX∗ = QX for any univariate random variable X. Furthermore,
GX1 −E(X1 ) (u) ≥ GX1
u
,
2
0≤u≤1
(3.5)
QX2 (u)QX1 (u)du.
(3.6)
(cf. [3]). Hence
E X1 X2 ≤
GX−1 (E(|X1 |)/2)
1
0
Therefore, for any two functions fi (·), i = 1,2, with fi (0) = 0 such that E| f1 (X1) f2 (X2)| <
∞, we obtain that
E f1 X1 f2 X2 ≤
G−f 1(X
1
1)
(E(| f1 (X1 )|)/2)
Q f2 (X2 ) (u)Q f1 (X1 ) (u)du.
0
(3.7)
Applying Theorem 3.1 for the random variables X1 − E(X1 ),X2 ,...,Xn , we get that
E X1 − E X1 X2 · · · Xn ≤
E(|X1 −E(X1 )|)
0
QX∗2 ,...,Xn oGX1 −E(X1 ) (u)du.
(3.8)
But
GX1 −E(X1 ) (u) ≥ GX1
u
,
2
u≥0
(3.9)
(cf. [3]). Hence
E X1 − E X1 X2 · · · Xn ≤
E(|X1 −E(X1 )|)/2
0
QX∗2 ,...,Xn oGX1 (u)du.
Observing that GX1 (·) is the inverse of the function MX1 (y) =
E X1 − E X1 X2 · · · Xn ≤
GX−1 (E(|X1 −E(X1 )|)/2)
Hence we have the following result.
1
0
y
0
(3.10)
QX1 (t)dt, it follows that
QX∗2 ,...,Xn (u)QX1 (u)du.
(3.11)
12
Rio-type inequality
Theorem 3.2. Let Xi , 1 ≤ i ≤ n, be arbitrary random variables defined on a probability
space (Ω,Ᏺ,P) with E|X1 | < ∞ and E|X1 X2 · · · Xn | < ∞. Then (3.11) holds.
Observe that QX∗ = QX for any univariate random variable X. Let n = 2 in Theorem 3.2.
Then QX∗2 = QX2 and the above result reduces to
GX−1 (E(|X1 −E(X1 )|)/2)
E X1 − E X1 X2 ≤
1
0
QX2 (u)QX1 (u)du.
(3.12)
QX2 −E(X2 ) (u)QX1 (u)du.
(3.13)
As a further consequence, we get that
E X1 − E X1 X2 − E X2 ≤
GX−1 (E(|X1 −E(X1 )|)/2)
1
0
Since
QX2 −E(X2 ) ≤ QX2 + EX2 ,
(3.14)
we obtain that
E X1 − E X1 X2 − E X2 ≤
GX−1 (E(|X1 −E(X1 )|)/2)
1
0
QX2 (u)QX1 (u)du + EX2 GX−1 (E(|X1 −E(X1 )|)/2)
1
0
QX1 (u)du.
(3.15)
Let
−1
α X1 ,X2 = max GX1
E X1 − E X1 2
−1
,GX2
E X2 − E X2 2
.
(3.16)
Then it follows that
E X1 − E X1 X2 − E X2 ≤
α(X1 ,X2 )
0
1 QX1 (u)QX2 (u)du +
EX1 2
α(X1 ,X2 )
0
QX1 (u)du + EX2 α(X1 ,X2 )
0
QX2 (u)du .
(3.17)
This inequality is different from the inequality in [12, page 9].
Let f1 and f2 be differentiable functions on R+ with fi (0) = 0. Let Xi , i = 1,2, be nonnegative random variables. Suppose that E[ fi2 (Xi )] < ∞, i = 1,2. It is easy to see that
fi Xi =
Then
E f1 X1 f2 X2
∞
0
fi Xi I(xi ,∞) Xi dxi .
∞ f2 X2 I(x2 ,∞) X2 dx2
= E f1 X1
0
E f1 X1 f2 X2 I(x2 ,∞) (X2 ) dx2
=
R+
(3.18)
(3.19)
B. L. S. Prakasa Rao 13
by the Fubini’s theorem. Observe that
E f1 X1 f2 X2 I[X2 >x2 ] X2
≤ min E f1 X1 f2 X2 ,E f1 X1 f2 X2 I[X2 >x2 ] X2
(3.20)
and hence
E f1 X1 f2 X2 ≤
E[| f1 (X1 ) f2 (X2 )|]
R+
χ(E[| f1 (X1 ) f2 (X2 )|I[X2 >x2 ] (X2 )]>u) (u)du dx2 .
0
(3.21)
Here χA (·) denotes the indicator function of the set A. Let
g f1 (X1 ), f2 (X2 ) x2 = E f1 X1 f2 X2 I[X2 >x2 ], X2 .
(3.22)
Then
E f1 X1 f2 X2 ≤
≤
R+
E[| f1 (X1 ) f2 (X2 )|]
χ([g f1 (X1 ), f (X2 ) (x2 )]>u) (u)du dx2
2
0
E[| f1 (X1 ) f2 (X2 )|]
0
[x2 :g f1 (X1 ), f (X2 ) (x2 )>u]
(3.23)
1dx2 du.
2
Let
H f1 (X1 ), f2 (X2 ) (u) = inf x2 : g f1 (X1 ), f2 (X2 ) (x2 ) ≤ u .
(3.24)
Then it follows that
E f1 X1 f2 X2 ≤
E[| f1 (X1 ) f2 (X2 )|]
0
H f1 (X1 ), f2 (X2 ) (u)du.
(3.25)
An analogous inequality holds by interchanging f1 (X1 ) and f2 (X2 ):
E f1 X1 f2 X2 ≤
E[| f1 (X1 ) f2 (X2 )|]
0
H f1 (X1 ), f2 (X2 ) (u)du.
(3.26)
References
[1]
[2]
[3]
[4]
[5]
[6]
H. W. Block and Z. B. Fang, A multivariate extension of Hoeffding’s lemma, Ann. Probab. 16
(1988), no. 4, 1803–1820.
R. C. Bradley, A covariance inequality under a two-part dependence assumption, Statist. Probab.
Lett. 30 (1996), no. 4, 287–293.
J. Dedecker and P. Doukhan, A new covariance inequality and applications, Stochastic Process.
Appl. 106 (2003), no. 1, 63–80.
M. Fréchet, Sur la distance de deux lois de probabilité, C. R. Acad. Sci. Paris 244 (1957), 689–692
(French).
E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist. 37 (1966), 1137–1153.
C. M. Newman, Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 74 (1980),
no. 2, 119–128.
14
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Rio-type inequality
B. L. S. Prakasa Rao, Hoeffding identity, multivariance and multicorrelation, Statistics 32 (1998),
no. 1, 13–29.
, Bounds for rth order joint cumulant under rth order strong mixing, Statist. Probab. Lett.
43 (1999), no. 4, 427–431.
, Covariance identities for exponential and related distributions, Statist. Probab. Lett. 42
(1999), no. 3, 305–311.
, Some covariance identities, inequalities and their applications: a review, Proc. Indian
Nat. Sci. Acad. Part A 66 (2000), no. 5, 537–543.
E. Rio, Covariance inequalities for strongly mixing processes, Ann. Inst. H. Poincaré Probab.
Statist. 29 (1993), no. 4, 587–597.
, Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Springer, Paris,
2000.
H. Yu, A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated
sequences, Probab. Theory Related Fields 95 (1993), no. 3, 357–370.
B. L. S. Prakasa Rao: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, New
Delhi 110 016, India
E-mail address: blsp@isid.ac.in