Journal
and Stochastic Analysis, 13:3
of Applied Mathematics
(2000), 261-267.
NEGATIVELY DEPENDENT BOUNDED RANDOM
VARIABLE PROBABILITY INEQUALITIES AND
THE STRONG LAW OF LARGE NUMBERS
M. AMINI and A. BOZORGNIA
Ferdowsi University
of Mathematical Sciences
Faculty
Mashhad, Iran
E-maih Bozorg@science2.um.ac.ir
(Received January, 1998; Revised January, 2000)
Let Xl,...,X n be negatively dependent uniformly bounded random
variables with d.f. F(x). In this paperwe obtain bounds for
probabiliis the
ties P(I Y=IXil >_nt) and P(l(pn-pl >e) where
is the pth quantile of F(x). Moreover, we
sample pth ^quantile and
show that
under mild^
is a strongly consistent estimator of
restrictions on F(x) in the neighborhood of
We also show that (pn
converges completely to
Key words: Probability Inequalities, Strong Law of Large Numbers,
the^
pn
pn
p
p.
p.
p
Complete Convergence.
AMS subject classifications: 60E15, 60F15.
1. Introduction
In many stochastic models, the assumption that random variables are independent is
not plausible. Increases in some random variables are often related to decreases in
other random variables so an assumption of negative dependence is more appropriate
than an assumption of independence. Lehmann [12] investigated various conceptions
of positive and negative dependence in the bivariate case. Strong definitions of bivariate positive and negative dependence were introduced by Esary and Proschen [7].
Also Esary, Proschen and Walkup [8] introduced a concept of association which implied a strong form of positive dependence. Their concept has been very useful in reliability theory and applications. Multivariate generalizations of conceptions of dependence were initiated by Harris [9] and Brindley and Thompson [4]. These were later
developed by Ebrahimi and Ghosh [6], Karlin [11], Block and Ting [2], and Block,
Savits and Shaked [1]. Furthermore, Matula [13] studied the almost sure convergence
of sums of negatively dependent (ND) random variables and Bozorgnia, Patterson
and Taylor [3] studied limit theorems for dependent random variables. In this paper
we study the asymptotic behavior of quantiles for negatively dependent random variPrinted in the U.S.A.
@2000 by
North Atlantic Science Publishing Company
261
M. AMINI and A. BOZORGNIA
262
ables.
Definition 1" The random variables
X1,...,X n
are pairwise negatively dependent
if
__ -
P(Xi <_ xi, X j <_ xj) <_ P(X <_ xi)P(X j <_ xj)
for all x i, x j
e R,
,
7 j. It can be shown that
(1)
(1)
is equivalent to
P(X > xi, X j > xj) <_ p(X > xi)P(X j > xj)
for all xj, x E
# j.
Defmition 2: The random variables
X1,... X n are said to be ND
P(NY= I(Xj <- xj))
and
P(N= I(Xj > xj))
H= 1P(XJ
if we have
xj),
(3)
IIY= 1P(Xj > xj),
(4)
for all Xl,...,x n R.
Conditions (3) and (4) are equivalent for n- 2. However, Ebrahimi and Ghosh [6]
show that these definitions do not agree for n >_ 3. An infinite sequence {Xn, n >_ 1}
is said to be ND if every finite subset {X1,...,Xn} is ND.
Definition 3: For parametric function g(0), a sequence of estimators {Tn, n >_ 1} is
strongly consistent if
Tn--og(O
Definition 4: The sequence
completely (denoted limn__,ocX n
a.e.
{Xn, n > 1}
of random variables converges to zero
0 completely) if for every > 0,
P[IXl >1
(5)
<c.
n--1
In the following example, we will show that the ND properties are not preserved
for absolute values and squares of random variables.
Example: Let (X, Y) have the following p.d.f:
f(- 1,
1)
f(1,0)
0, f(- 1,0)
f(0,0)
f(0,
1)
f(0,1)
f(1,1)
1/9,
f(-1,1)- f(1,-1)-2/9.
Then
X and Y
are
ND random variables since for each x, y G R
we have
F(x,y) <_ Fx(x)Fy(y ).
(ii)
X and V-y2 are not ND random variables because for -1 <x<0 and
0<v<l we have
Negatively Dependent Bounded Random Variable Probability Inequalities
263
F(x, v)- 1/9 > Fx(x)Fv(v -(3/9)(2/9).
U X 2 and V y2 are not ND random variables nor are
since0_<u<l, 0<v<l we have
(iii)
IX land YI
F(u, v)- 1/9 > Fu(u)Vv(v -(2/9)(3/9).
The following lemmas are used to obtain the main result in the next section.
Detailed proofs of these lemmas can be founded in the Bozorgnia, Patterson and
Taylor [3].
Lemma 1: Let {Xn, n > 1} be a sequence of ND random variables let {fn, n > 1}
be a sequence of Borel functions all of which are monotone increasing (or all are
monotone decreasing). Then {fn(Xn),n > 1} is a sequence of NO random variables.
Lemma 2: Let X1,...,X n be ND random variables and let tl,...,t n be all
nonnegative (or all nonpositive). Then
E[e
i=1
Ee tiXi
<
i=1
Lemma3: Let X be a r.v. such that
every real number h we have
E(X)-O
Ee hX <
Proof: For c- 1, see Chow
replaced by X/c.
e
and
XI
<_c<oc
a.e.
Then
for
h2c2.
[5]. For general c, apply
the c-1 result with X
2. An Extension of the Theorem of Hoeffding for ND Random Variables
In this section we extended the theorem of Hoeffding (Theorem 1 below) and then
obtain the strong law of large numbers for ND uniformly bounded random variables.
Theorem 1: (Hoeffding [10]) .Let X1,...,X n be independent random variables satisfying P[a <_ X <_ b]- 1 for each where a < b, and let S n
l(Xk- EXk).
Then for any t > O and c b-a
=
P[S n >_ nt] <_ exp
c2.
j.
Theorem 2: Let X1,...,X n be ND random variables satisfying
for each where a<b, and let S n- =I(Xk-EXk). Then
P[a <_ X <_ b]- 1
for any t>O and
c-b-a,
p[s _> ] _< txp 4: j.
Proofi We define
YI
c a.e.
Yk- Xk-EXk
for k- 1,...,n. Then we have EY k -0 and
we have
Hence, by Lemmas 2 and 3,
M. AMINI and A. BOZORGNIA
264
p(Sn >_ nt) <_ exp(- nth)Ee hSn
_< exp[- nth]
H EehYk <- exp[- nth + nh2c2].
k-1
for h- t__._2"
exp[- nt2]
2
The right side of this inequality attains its minimum value
Thus, for each t > 0,
P[S n>_nt]<_exp
Corollary 1: Under the assumptions
4c 2
P(
Sn
> nt)
V!
j.
of Theorem 1, for
P[IS.I >-nt]<-2exp
every t
>0
].
4c 2
P[S n >_ nt] + P[ S n >_ nt]
nt
4c 2
P[S n >_ nt] + P[S’n >_ nt] <_ 2exp
whereS2nE=lZkandZ k- -Yk, k-1,...,n.
Theorem
Theorem 3: Under the
assumptions
1
--a
n
2c
4c
of
2]
j
2, for every a > 1/2 we have
n
Z (Xk- EXk)-*O
completely.
k=l
Proof: By Theorem 2 and Corollary 1, for each
ZP(ISnl >na)<2exp
n--1
n--1
>0
we have
2n2a
l
4c 2
j’ <
]
Hence, for a- 1 we obtain the strong law of large numbers for negatively dependent uniformly bounded random variables.
3. Asymptotic Behavior of Quantiles for ND Random Variables
pn
The following two theorems and one corollary given conditions under which
is
contained in a suitably small neighborhood of
with probability one for all sufficiently large n.
Theorem 4: Let Xl,...,X n be ND random variables with d.f. F(x). Let
O < p < 1. Suppose that
is the unique solution x of F(x- <_ p <_ F(x). Then for
every > 0 and n we have
p
p
P(
"p.- p
> ) <_ 2exp(- n62/4)
(6)
Negatively Dependent Bounded Random Variable Probability Inequalities
where
min{F(p + ) p, p F(p
Proof: For every^ > 0 we have
P[
pn- p
265
pn is the sample pth quantile.
+ p] + P[pn < p- ]
)}
and
>]
P[pn >
Pip > Fn(p + )] + PIp < Fn( p
We define
V
I[x > p+ e] and U
I[x < p_e] for
Since X,...,X n are ND random variables, by Lemma 1
ND random variables. Hence, by Theorem 2 23 have
P(P > Fn(p + ))
E (Yi- E(Yi)
P
1,2,...,n.
U,...,U n and V1,..., V n are
> nl) -< exp
i=1
1 F(p + )- p. Similarly we have
exp ..’4...
P[p < Fn(v-)] P
(U i-EUi) > n52
where 5 p- F(p- ). Define 5e min{51, 52}. Thus we have (6).
where
Corolly 2: Let X1,...,X n be ND random variables with
the sample pth quantile. Then
pn----p
completely as
d.f. F(x)
H
and let
pn be
n--,cx.
Proof: By Theorem 4 we have
n1( n52 )
P[l’p -p[ >1< 2
n
n=l
pn---p
Hence
completely.
By the Borel-Cantelli lemma, we have
exp
4
<"
Vl
pn---p
a.e.
Thus
pn
is a strongly
consistent estimator of (p.
Theorem 5: Let X1,...,X n be ND random variables with d.f. F(X). Let
with F’(p) f(p) > O. Then for
0 < p < 1. Suppose that F is differentiable at
some fl > O and O < a 1
p
<_-
(2 + fl)lna(n)
Proof:
Since F is continuous at
p with
F’(p)> 0,
p is a unique solution of
F(x-) < p < F(x) and F(p)- p. Thus, we may apply Theorem 4. Writing
(2 + 3)lnC(n)
we have
r(p + n)
P
nf((p) + (n) >
(2 /3)inC’(n)
M. AMINI and A. BOZORGNIA
266
where n is sufficiently large. Similarly
p
(2
F(p n) nf(p) + (n) > +
fl)21n(n)
Thus for all sufficiently large n,
)21n2a(n) (2 + )21n(n)
n(2 n /4 > (2 +4n2a1
4
where
6%
Hence, for
min{F(p + en)
P, P
F(p en)}"
a constant c we have
n--1
n=l
n (+/2)
2
which completes the proof.
Let X1,...,X n be independent random variables with d.f. F(x). Then in this case,
all the above theorems and corollaries are true. In particular, Theorems 4 and 5 are
extensions of Theorem 2.3.1 and Lemma B, respectively, pages 96 of Settling [14].
Acknowledgement
We wish to express our appreciation and gratitude to the referees for their suggestions
which improved the exposition.
References
[1]
[2]
[3]
[4]
[5]
[8]
[9]
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