I Definitions of Functions
Let be an index set. For each , let be a finite set. For each , let
be a random variable taking values in
. For , . In particular for , we write . Letbe a set of all probability distributions on
. For , we write its disribution as . For , we often omit its subscript to simply write . For , let denote the probability distribution of , which is the marginral distribution of . We adopt similar notations for other variables or sets. For , we consider a function having the following form:(1)  
(2) 
In (1), , , are given nonnegative functions and , are given real valued coefficients. In (2), the quantities and are given positive coefficients. Furthermore, and are given subsets of . We define
(3) 
In this paper we assume that the function satisfy the following property.
Assumption 1

For any , is nonnegative and bounded, i.e., there exists a positive such that for any .

is a continuous function of .
Let be a given subset of . The following two optimization problems
(4) 
frequently appear in the analysis of capacity or rate regions in the field of multiterminal information theory. In this paper we give one example of and , which is related to the source coding with side information at the deconder posed and investigated Wyner and Ziv [6]. This example is shown below.
Example 1
Let , , , and be four random variables, respectively taking values in the finite sets , , , and . We consider the case where . Let a probability distribution of . For , we define
(5) 
where are distortion measures. In this example we have the following:
(6) 
Let
In this example we denote the quantity by , which has the following form:
The quantity
yields the following hyperplane expression of WynerZiv rate distortion region
:In the above example because of the two Markov chains
and , the computation of becomes a nonconvex optimization problem, which is very hard to solve in its present form. As we can see from this example, the computations of and are in general highly challenging. To solve those problems, alternative optimization problems having one parameter on some relaxed condition of are introduced. Let be some suitable onto mapping satisfying We set On the above , we assume the following:Assumption 2

Let denote a feasible region on those relaxed optimization problems. On the feasible region , we assume that for any , its support set includes the support set of .

For any and for any , we have
where and are positive constants and the quantities and are subsets of satistying the following:
For and , define
We consider the following two optimization problems:
(7) 
Those optimization problems appear in recent results that the author [1][4], Tyagi and Watanabe [5] obtained on the proofs of the strong converse theorems for multiterminal source or channel networks.
Example 2
We consider the case of Example 1. Define by The feasible region is given by
For and for , we have
(8) 
From (8), we have that and . We denote the quantity by , which has the following form:
According to Tyagi and Watanabe [5], a single letter characterization of the rate distortion region using the function plays an important role in the proof of the the strong converse theorem for WynerZiv source coding problem.
Ii Main Results
Our aim in this paper is to evaluate the differences between and and between and . It is obvious that we have
(9) 
for any . In fact, restricting the feasible region in the definitions of or to , we obtain the bounds in (9). We first describe explicit upper bounds of and by standard analytical arguments. This result is given by the following proposition.
Proposition 1
For any positive , we have
(10)  
(11) 
where we set
Proof of this proposition is given in Appendix A. We set
For and , define
Furthermore, define
(12) 
For , define
Furthermore, set
Note that the quantity depends on and the quantity depends on . Our main result is given in the following proposition.
Proposition 2
For any satisfying , we have
(13) 
where is a suitable positive constant depending on . Furthermore, for any satisfying , we have
(14) 
where is a suitable positive constant depending on .
Proof of this proposition will be given in the next section. We can see from the above proposition that the two bound (13) and (14) in Propostion 2, respectively, provide significant improvements from the bounds (10) and (11) in Proposition 1.
We next consider an application of Propostion 2 to the case discussed in Examples 1 and 2. As stated in Examples 1 and 2, and . Set
Here we note that and depend on the value of . Hence we write and when we wish to express that those are the functions of . Applying Proposition 2 to the example of WynerZiv source coding problem, we have the following result.
Proposition 3
For any and any satisfying , we have
Specifically, for any satisfying , we have
where
Let and for fixed source block length , let be the rate distortion region consisting of a pair of compression rate and distortion level such that the decoder fails to obtain the sources within distortion level with a probability not exceeding . Formal definition of is found in [2]. The above theorem together with the result of Tyagi and Watanabe [5] yields a new strong converse outer bound. To describe this result for , we set
According to Tyagi and Watanabe [5], we have the following theorem.
Theorem 1 (Tyagi and Watanabe [5])
For any ,
(15) 
Theorem 2
For any satisfying , we have
(16) 
where
(17) 
In (17), we choose For this choice of , the quantity becomes the following:
The quantity indicates a gap of the outer bound of from . This gap is tighter than the similar gap given by
where is some positive constant not depending on . The above was obtained by the author [2] in a different method based on the theory of information spectrums [7].
Iii Proof of the Main Result
For , and for , define
We can show that the functions we have definded so far satisfy several properties shown below.
Property 1

For fixed positive , a sufficient condition for to exist is

For , define a probability distribution by
For , define a probability distribution by
Then, we have
(18) (19) (20) (21) Specifically, we have
(22) (23) For fixed , a sufficient condition for the three times derivative of to exist is Furthermore, a sufficient condition for the three times derivative of to exist is .

Let be some positive constant depending on . Then, for any , we have
(24)
Property 2

For fixed positive , a sufficient condition for to exist is

For fixed positive , a sufficient condition for the three times derivative of to exist is

Let be some positive constant depending on . Then, for any , we have
(28)
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