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Hung and Giang Journal of Inequalities and Applications 2014, 2014:291
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RESEARCH
Open Access
On bounds in Poisson approximation for
integer-valued independent random
variables
Tran Loc Hung* and Le Truong Giang
*
Correspondence:
[email protected]
University of Finance and Marketing,
Ho Chi Minh City, Vietnam
Abstract
The main aim of this note is to establish some bounds in Poisson approximation for
row-wise arrays of independent integer-valued random variables via the Trotter-Renyi
distance. Some results related to random sums of independent integer-valued
random variables are also investigated.
MSC: 60F05; 60G50; 41A36
Keywords: Poisson approximation; random sums; Le Cam’s inequality; Trotter’s
operator; Renyi’s operator; probability distance; integer-valued random variable
1 Introduction
Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent integervalued random variables with success probabilities P(Xn,j = ) = pn,j ; P(Xn,j = ) =  – pn,j –
qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Set Sn = nj= Xn,j and λn =
E(Sn ) = nj= pn,j . Suppose that limn→∞ λn = λ ( < λ < +∞). We will denote by Zλ the
Poisson random variable with mean λ. It has long been known that in the case of all qn,j =
 (j = , , . . . , n; n = , , . . .), the partial sum Sn is said to be a Poisson-binomial random
variable, and the probability distributions of Sn , n = , , . . . , are usually approximated by
the distribution of Zλ . Specially, under the assumptions that limn→∞ max≤j≤n pn,j = , the
well-known Poisson approximation theorem states that
d
→ Zλ
Sn −
as n → ∞.
()
d
→ means the convergence in distribution. It is to
Here, and from now on, the notation −
be noticed that, for the information on the quality of the Poisson approximation, Le Cam
() [] established the remarkable inequality
∞
n
P(Sn = k) – P(Zλ = k) ≤ 
pn,j .
()
j=
k=
It is to be noticed that another inequality in Poisson approximation is usually expressed
in terms of the total variation distance dTV (Sn , Zλ )
dTV (Sn , Zλ ) ≤
n
pn,j ,
()
j=
© 2014 Hung and Giang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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where for the distributions P and Q on Z+ = {, , , . . .}, the total variation distance between P and Q will be defined as follows:
dTV (P, Q) :=
 P(x) – Q(x).
 x∈Z
()
+
(For other surveys, see [–], and [].)
In recent years many powerful tools for establishing the Le Cam’s inequality for a wide
class of discrete independent random variables have been demonstrated, like the coupling
technique, the Stein-Chen method, the semi-group method, the operator method, etc.
Results of this nature may be found in [–], and [].
The main aim of this paper is to establish the bounds of the Le Cam-style inequalities for
independent discrete integer-valued random variables using the Trotter-Renyi distance
based on Trotter-Renyi operator (see [, ], for more details). It is to be noticed that
during the last several decades the Trotter-operator method has been used in many areas
of probability theory and related fields. For a deeper discussion of Trotter’s operator we
refer the reader to [–], and [].
The results obtained in this paper are extensions of known results in [, , –], and
[]. The present paper is also a continuation of [].
This paper is organized as follows. The second section deals with the definition and
properties of Trotter-Renyi distance, based on Trotter’s operator and Renyi’s operator. Section  gives some results on Le Cam’s inequalities, based on the Trotter-Renyi distance, for
independent integer-valued distributed random variables. The random versions of these
results are also given in this section.
2 Preliminaries
In the sequel we shall recall some properties of Trotter-Renyi operator, which has been
used for a long time in various studies of classical limit theorems for sums of independent random variables (see [–, , ], and [], for the complete bibliography). Based
on Renyi’s definition ([], Chapter , Section , p.), we redefine the Trotter-Renyi
operator as follows.
Definition . The operator AX associated with a discrete random variable X is called
the Trotter-Renyi operator, defined by
∞
f (x + k)P(X = k),
(AX f )(x) = E f (X + x) =
∀f ∈ K, ∀x ∈ Z+ ,
()
k=
where by K is denoted the class of all real-valued bounded functions f on the set of all
non-negative integers Z+ := {, , , . . .}. The norm of the function f ∈ K is defined by
f = supx∈Z+ |f (x)|.
It is to be noticed that Renyi’s operator defined in [] actually is a discrete form of
Trotter’s operator (we refer the readers to [, , –], and [], for a more general and
detailed discussion of Trotter’s operator).
We shall need in the sequence the following main properties of Trotter-Renyi operator,
for all functions f , g ∈ K and for α ∈ R:
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.
.
.
.
.
Page 3 of 11
AX (f + g) = AX (f ) + AX (g).
AX (αf ) = αAX (f ).
AX (f ) ≤ f .
AX (f ) + AY (f ) ≤ AX (f ) + AY (f ).
Suppose that AX , AY are operators associated with two independent random
variables X and Y . Then, for all f ∈ K,
AX+Y (f ) = AX AY (f ) = AY AX (f ).
In fact, for all x ∈ Z+
AX+Y f (x) =
∞
f (x + l)P(X + Y = l) =
l=
∞
f (x + k + r)P(Y = k)P(X = r)
r,k=
= AX AY f (x) = AX AY f (x).
.
.
Suppose that AX , AX , . . . , AXn are the operators associated with the independent
random variables X , X , . . . , Xn . Then, for all f ∈ K, ASn (f ) = AX AX · · · AXn (f ) is the
operator associated with the partial sum Sn = X + X + · · · + Xn .
Suppose that AX , AX , . . . , AXn and AY , AY , . . . , AYn are operators associated with
independent random variables X , X , . . . , Xn and Y , Y , . . . , Yn . Moreover, assume
that all Xi and Yj are independent for i, j = , , . . . , n. Then, for every f ∈ K,
n
AX (f ) – AY (f ).
An X (f ) – An Y (f ) ≤
k
k
k= k
k= k
()
k=
Clearly,
AX AX · · · AXn – AY AY · · · AYn
=
n
AX AX · · · AXk– (AXk – AYk )AYk+ · · · AYn .
k=
Accordingly,
n
An X (f ) – An Y (f ) ≤
AX . . . AX (AX – AY )AY · · · AY (f )
n

k–
k
k
k+
k
k
k=
k=
k=
n
AY · · · AY (AX – AY )(f )
≤
n
k+
k
k
k=
≤
n
AX (f ) – AY (f ).
k
k
k=
.
AnX (f ) – AnY (f ) ≤ nAX (f ) – AY (f ).
Lemma . The equation AX f (x) = AY f (x) for f ∈ K, x ∈ Z+ shows that X and Y are identically distributed random variables.
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Let AX , AX , . . . , AXn , . . . be a sequence of Trotter-Renyi’s operators associated with the
independent discrete random variables X , X , . . . , Xn , . . . , and assume that AX is a TrotterRenyi operator associated with the discrete random variable X. The following lemma
states one of the most important properties of the Trotter-Renyi operator.
Lemma . A sufficient condition for a sequence of random variables X , X , . . . , Xn , . . . to
converge in distribution to a random variable X is that
lim AXn (f ) – AX (f ) = ,
n→∞
for all f ∈ K.
Proof Since limn→∞ AXn (f ) – AX (f ) = , for all f ∈ K, we conclude that
∞
lim f (x + k) P(Xn = k) – P(X = k) = ,
n→∞
for all f ∈ K and for all x ∈ Z+ .
k=
Taking
f (x) =
⎧
⎨,
if  ≤ x ≤ t,
⎩, if x > t,
then we recover
t
P(Xn = k) – P(X = k) = .
lim n→∞
k=
d
It follows that P(Xn ≤ t) – P(X ≤ t) →  as n → +∞. We infer that Xn −
→ X as n → +∞.
Before stating the definition of the Trotter-Renyi distance we firstly need the definition
of a probability metric. Let (, A, P) be a probability space and let Z(, A) be a space of
real-valued A-measurable random variables X : → R.
Definition . A functional d(X, Y ) : Z(, A) × Z(, A) → [, ∞) is said to be a probability metric in Z(, A) if it possesses for the random variables X, Y , Z ∈ Z(, A) the following
properties (see [, ] and [] for more details):
. P(X = Y ) =  ⇒ d(X, Y ) = ;
. d(X, Y ) = d(Y , X);
. d(X, Y ) ≤ d(X, Z) + d(Z, Y ).
We now return to the definition of a probability distance based on the Trotter-Renyi
operator (see [, ], and []).
Definition . The Trotter-Renyi distance dTR (X, Y ; f ) of two random variables X and Y
with respect to the function f ∈ K is defined by
dTR (X, Y ; f ) := AX f – AY f = sup Ef (X + x) – Ef (Y + x).
x∈Z+
()
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Based on the properties of the Trotter-Renyi operator, some properties of the TrotterRenyi distance are summarized in the following (see [, , , ], and [] for more
details) and we shall omit the proofs.
. It is easy to see that dTR (X, Y ; f ) is a probability metric, i.e. for the random variables
X, Y , and Z the following properties are possessed:
(a) For every f ∈ K, the distance dTR (X, Y ; f ) =  if P(X = Y ) = .
(b) dTR (X, Y ; f ) = dTR (Y , X; f ) for every f ∈ K.
(c) dTR (X, Y ; f ) ≤ dTR (X, Z; f ) + dTR (Z, Y ; f ) for every f ∈ K.
. If dTR (X, Y ; f ) =  for every f ∈ K, then FX ≡ FY .
. Let {Xn , n ≥ } be a sequence of random variables and let X be a random variable.
The condition
lim dTR (Xn , X; f ) = ,
n→+∞
for all f ∈ K,
d
.
→ X as n → ∞.
implies that Xn −
Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent random variables (in
each group). Then, for every f ∈ K,
dTR
n
Xj ,
j=
n
≤
Yj ; f
j=
n
dTR (Xj , Yj ; f ).
()
j=
Moreover, if the random variables are identically (in each group), then we have
dTR
n
Xj ,
j=
.
≤ ndTR (X , Y ; f ).
Yj ; f
j=
Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent random variables (in each
group). Let {Nn , n ≥ } be a sequence of positive integer-valued random variables
that are independent of X , X , . . . , Xn and Y , Y , . . . , Yn . Then, for every f ∈ K,
dTR
N
n
Xj ,
j=
.
n
Nn
≤
Yj ; f
j=
∞
k=
P(Nn = k)
k
dTR (Xj , Yj ; f ).
()
j=
Suppose that X , X , . . . , Xn ; Y , Y , . . . , Yn are independent identically distributed
random variables (in each group). Let {Nn , n ≥ } be a sequence of positive
integer-valued random variables that are independent of X , X , . . . , Xn and
Y , Y , . . . , Yn . Moreover, suppose that E(Nn ) < +∞, n ≥ . Then, for every f ∈ K, we
have
dTR
N
n
j=
Xj ,
Nn
Yj ; f
≤ E(Nn ) · dTR (X , Y ; f ).
j=
Finally, we emphasize that the Trotter-Renyi distance in () and the total variation distance in () have a close relationship if the function f is chosen as an indicator function of
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a set A ∈ Z+ , namely
⎧
⎨, if x ∈ A,
f (x) = χA (x) =
⎩, if x ∈/ A.
Then
dTR (X, Y , χA ) = dTV (X, Y ),
where we denote by dTV (X, Y ) the total variation distance between two integer-valued
random variables X and Y , defined as follows:
 P(X = k) – P(Y = k).
dTV (X, Y ) = sup P(X ∈ A) – P(Y ∈ A) =

A⊆Z+
k∈Z+
For a deeper discussion of the total variation distance, we refer the reader to [–], and
[].
3 Main results
Let {AXn,j , j = , , . . . , n; n = , , . . .} be a sequence of operators associated with the integervalued random variables Xn,j , j = , , . . . , n; n = , , . . . , and let {AZpn,j , j = , , . . . , n; n =
, , . . .} be a sequence of operators associated with the Poisson random variables with parameters pn,j , j = , , . . . , n; n = , , . . . . Since Zλn is a Poisson random variable with positive
d parameter λn = nj= pn,j , we can write Zλn = nj= Zpn,j , where Zpn, , Zpn, , . . . , Zpn,n are independent Poisson random variables with positive parameters pn, , pn, , . . . , pn,n , and the
d
notation = denotes coincidence of distributions.
Theorem . Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent, integer-valued random variables with probabilities P(Xn,j = ) = pn,j , P(Xn,j =
) =  – pn,j – qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Let us write
Sn = nj= Xn,j and λn = nj= pn,j . We will denote by Zλn the Poisson random variable with
parameter λn . Then, for all functions f ∈ K,
dTR (Sn , Zλn ; f ) ≤ f n
pn,j + qn,j .
j=
Proof Applying (), we have
dTR (Sn , Zλn , f ) ≤
n
dTR (Xn,j , Zpn,j ; f ) =
j=
n
AX (f ) – AZ (f ).
pn,j
n,j
k=
Moreover, for all f ∈ K, for all x ∈ Z+ and r ∈ {, , . . . , n} we conclude that
AXnj f (x) – AZpn,j f (x) =
∞
f (x + r) P(Xnj = r) – P(Zλpn,j = r)
r=
=
∞
r=
e–pn,j prn,j
f (x + r) P(Xnj = r) –
r!
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= f (x)  – pn,j – qn,j – e–pn,j
+ f (x + ) pn,j – pn,j e–pn,j
∞
e–pn,j prn,j
.
+
f (x + r) P(Xn,j = r) –
r!
r=
Therefore, for all functions f ∈ K , and for all x ∈ Z+ , we have
AX f (x) – AZ f (x)
pn,j
n,j
= f (x)  – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j
∞
e–pn,j prn,j +
f (x + r) P(Xn,j = r) –
r!
r=
= f (x)  – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞
e–pn,j prn,j f (x + r) P(Xn,j = r) –
+
r!
r=
≤ f (x)  – pn,j – qn,j – e–pn,j + f (x + ) pn,j – pn,j e–pn,j ∞
∞
e–pn,j prn,j f (x + r)P(Xn,j = r) + f (x + r)
+
r! r=
≤ e–pn,j
r=
+ pn,j + qn,j –  sup f (x) + pn,j – pn,j e–pn,j sup f (x)
x∈Z+
x∈Z+
∞
∞ –p
e n,j prn,j + sup f (x)
P(Xn,k = r) + sup f (x)
x∈Z+
r! x∈Z+
r=
r=
= sup f (x) e–pn,j + pn,j + qn,j –  + pn,j – pn,j e–pn,j + qn,j +  – e–pn,j – pn,j e–pn,j
x∈Z+
= f pn,j – pn,j e–pn,j + qn,j
≤ f pn,j + qn,j .
One infers that
∀f ∈ K,
AX (f ) – AZ (f ) ≤ f p + qn,j .
pn,j
n,j
n,j
Therefore, applying (), we can assert that
dTR (Sn , Zλn ; f ) ≤ f n

pn,j + qn,j .
j=
This completes the proof.
Corollary . Under the assumptions of Theorem ., let r ∈ {, , . . . , n}, we have
n

P(Sn = r) – P(Zλ = r) ≤ 
pn,j + qn,k .
n
j=
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Remark . We consider Corollary . and assume that the following conditions are satisfied:
(i)
lim
n→∞
n
qn,j = ,
j=
lim max pn,j = ,
(ii)
n→∞ ≤k≤n
(iii)
lim λn = lim
n→∞
n
n→∞
pn,j = λ ( < λ < +∞).
j=
d
Then Sn → Zλ as n → ∞.
Theorem . Let {Xn,j , j = , , . . . , n; n = , , . . .} be a row-wise triangular array of independent, integer-valued random variables with probabilities P(Xn,j = ) = pn,j , P(Xn,j = ) =
 – pn,j – qn,j ; pn,j , qn,j ∈ (, ); pn,j + qn,j ∈ (, ); j = , , . . . , n; n = , , . . . . Moreover, we suppose that Nn , n = , , . . . are positive integer-valued random variables, independent of all
n
N n
Xn,j , j = , , . . . , n; n = , , . . . . Let us write SNn = N
j= Xn,j and λNn =
j= pn,j . We will denote by ZλNn the Poisson random variable with parameter λNn . Then, for all functions f ∈ K,
N
n

pNn ,j + qNn ,j .
dTR (SNn , ZλNn ; f ) ≤ f E
j=
Proof According to Theorem . and (), for all functions f ∈ K, and for all x ∈ Z+ , we
have
dTR (SNn , ZλNn ; f ) ≤
∞
P(Nn = m)dTR (Sm , Zλm ; f )
m=
≤
∞
P(Nn = m)f m=
= f ∞
pNn ,j + qNn ,j
j=
P(Nn = m)
m=
= f E
m
m
pNn ,j
+ qNn ,j
j=
N
n
pNn ,j
+ qNn ,j
.
j=
Therefore,
N
n

pNn ,j + qNn ,j .
dTR (SNn , ZλNn ; f ) ≤ f E
j=
The proof is complete.
Corollary . According to Theorem ., let r ∈ {, , . . . , n}, we have
N
n

P(SN = r) – P(Zλ = r) ≤ E
pNn ,j + qNn ,j .
n
Nn
j=
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Theorem . Let {Xk,j } (k = , , . . . ; j = , , . . .) be a double array of independent integervalued random variables with probabilities P(Xk,j = ) = pk,j , P(Xk,j = ) =  – pk,j – qk,j ,
pn,k ∈ (, ); k = , , . . . ; j = , , . . . . Assume that for every k = , , . . . the random variables
Xk, , Xk, , . . . , are independent, and for every j = , , . . . the random variables X,j , X,j , . . . are
independent. Set Snm = nk= m
j= Xk,j . Let us denote by Zδn,m the Poisson random variable
p
with mean δn,m = nk= m
j= k,j . Then, for all f ∈ K,
dTR (Snm , Zδn,m , f ) ≤ f n m
pk,j + qk,j .
k= j=
Proof Applying the inequality in (), we have
dTR (Snm , Zδnm , f ) ≤
n
dTR (Skm , Zμk,m , f )
k=
≤
n m
dTR (Sk,j , Zλk,j , f ).
k= j=
According to Theorem ., for all functions f ∈ K, and for all x ∈ Z+ , we conclude that
dTR (Sk,j , Zλk,j , f ) ≤ f pk,j + qk,j .
Therefore,
dTR (Snm , Zδnm , f ) ≤ f n m
pk,j + qk,j .
k= j=
This completes the proof.
Theorem . Let {Xk,j , k = , , . . . ; j = , , . . .} be a double array of independent integervalued random variables with P(Xk,j = ) = pk,j ; P(Xk,j = ) =  – pk,j – qk,j ; pk,j , qk,j ∈ (, );
pk,j + qk,j ∈ (, ); k = , , . . . ; n = , , . . . . Assume that for every k = , , . . . the random variables Xk, , Xk, , . . . , are independent, and for every j = , , . . . the random vari ables X,j , X,j , . . . are independent. Set Snm = nk= m
j= Xk,j . Suppose that Nn , Mm are
non-negative integer-valued random variables independent of all Xn,m , n ≥ ; m ≥ .
Let us denote by ZδNn Mm the Poisson random variable with mean δNn Mm = E(SNn Mm ) =
N n Mm
j= pk,j . Then, for all functions f ∈ K,
k=
dTR (SNn Mm , ZδNn Mm , f ) ≤ f E
N M
n n
pk,j
+ qk,j
.
k= j=
Proof According to Definition ., we have
(ASNn Mm f )(x) := E f (SNn Mm + x)
=
∞
n=
P(Nn = n)
∞
m=
P(Mn = m)(ASnm f )(x)
Hung and Giang Journal of Inequalities and Applications 2014, 2014:291
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and
(AZδN
n Mm
f )(x) := E f (ZδNn Mm + x)
=
∞
P(Nn = n)
n=
∞
P(Mn = m)(AZδnm f )(x).
m=
Therefore, for all functions f ∈ K, and for all x ∈ Z+ , we have
AS
Nn Mm (f ) – AZδN
≤
∞
P(Nn = n)
n=
n Mm
∞
P(Mn = m)ASnm (f ) – AZδn,m (f )
m=
≤ f ∞
P(Nn = n)
n=
= f (f )
∞
∞
n m

pk,j + qk,j
P(Mn = m)
m=
k= j=
P(Nn = n)E
n=
n M
m
pk,j
+ qk,j
k= j=
N M
n m

pk,j + qk,j .
= f E
k= j=
Thus,
dTR (SNn Mm , ZδNn ,Mm , f ) ≤ f E
N M
n n
pk,j
+ qk,j
.
k= j=
The proof is straightforward.
Remark . In the case of all probabilities qn,j = , j = , , . . . , n; n = , , . . . the partial
sum Sn = nj= Xn,j will become a Poisson-binomial random variable, and one concludes
that the results of Theorems ., ., ., and . are extensions of results in [] (see []
for more details).
We conclude this paper with the following comments. The Trotter-Renyi distance
method is based on the Trotter-Renyi operator and it has a big application in the Poisson
approximation. Using this method it is possible to establish some bounds in the Poisson
approximation for sums (or random sums) of independent integer-valued random vectors.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly to this work. All authors drafted the manuscript, read and approved the
final version of the manuscript.
Acknowledgements
The authors wish to express their gratitude to the referees for valuable remarks and comments, improving the previous
version of this paper. The research was supported by the Vietnam National Foundation for Science and Technology
Development (NAFOSTED, Vietnam) under grant 101.01-2010.02.
Received: 23 January 2014 Accepted: 10 July 2014 Published: 18 August 2014
Hung and Giang Journal of Inequalities and Applications 2014, 2014:291
http://www.journalofinequalitiesandapplications.com/content/2014/1/291
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doi:10.1186/1029-242X-2014-291
Cite this article as: Hung and Giang: On bounds in Poisson approximation for integer-valued independent random
variables. Journal of Inequalities and Applications 2014 2014:291.
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