Elementary Uncertain Renewal Reward Theorem and Its Strict Proof

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**Subject Areas:** **Mathematical Analysis**

1. Introduction

In probability theory, renewal process and renewal reward process are two important uncertain processes in which interarrival times and rewards are regarded as random variables.

Note that probability theory is applicable only when the obtained probability is close enough to the real frequency. Otherwise, some counterintuitive results will happen [1] . But in real life, we are often lack of observed data or historical data to estimate the probability distributions of interarrival times and reward, so we have to invite some domain experts to evaluate their belief degree of the interarrival times and reward. Since human tends to overweight unlikely events (Kahneman and Tversky [2] ), the belief degree may have a much larger than the real frequency. Thus probability theory fails to model the renewal process and renewal reward process in this situation. In order to resolve these problems, an uncertainty theory is founded by Liu [3] and refined by Liu [4] based on normality, duality, subadditivity and product axioms. Nowadays, uncertainty theory has been applied to uncertain programming [5] [6] , uncertain process [7] - [10] etc. [11] [12] , uncertainty theory. In the framework of uncertainty theory, Liu [13] first assumed the interarrival times and reward of an renewal process as uncertain variables, and proposed an uncertain renewal process. Then Liu [4] also proposed an uncertain renewal reward process which interarrival times and rewards were both regarded as uncertain variables and gave the an elementary renewal reward theorem. At present, there is a lack of strict proof for the elementary theorem. Therefore, the paper will give its strict proof with two lemmas by some techniques.

2. Preliminary

Definition 1. (Liu [3] ) Let be a -algebra on nonempty set. A set function is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality); for the universal set;

Axiom 2. (Duality) for any event;

Axiom 3. ( Subadditivity) For every countable sequence of events, we have

In this case, the triple is called an uncertainty space.

In [14] , Liu further presented the following axiom:

Axiom 4. (Product Axiom) Let be uncertainty spaces for. Then the product uncertain measure is an uncertain measure satisfying

where are arbitrarily chosen events from for, respectively.

Definition 2. (Liu [3] ) An uncertain variable is a measurable function from an uncertainty space to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event.

Definition 3. (Liu [3] ) The uncertainty distribution of an uncertain variable is defined by for any real number x.

Definition 4. (Liu [4] ) An uncertainty distribution is said to be regular if its inverse function exists and is unique for each.

Definition 5. (Liu [14] ) The uncertain variables are said to be independent if

for any Borel sets of real numbers.

Definition 6. (Liu [3] (2007)) The expected value of uncertain variable is defined by

provided that at least one of the two integrals is finite.

Theorem 1. (Liu [4] ) Let be an uncertain variable with uncertainty distribution. If the expected value exists, then

Theorem 2. (Liu [14] ) Let be independent uncertain variables with uncertainty distributions, respectively. If is strictly increasing with respect to and strictly decreasing with respect to then is an uncertain variable with uncertainty distribution

and inverse uncertainty distribution

In particular, if have a common uncertainty distribution, then have a uncertainty distribution

Definition 7. (Liu [3] ) Let be a sequence of uncertain variables with uncertainty distributions respectively, then is said to converge in distribution to if at every continuous point x of.

3. Uncertain Renewal Reward Process

Definition 8. (Liu [13] ) Let T be a index set and let be an uncertainty space. An uncertain process is a measurable function from to the set of real numbers, i.e., for any and any Borel set B of real numbers, the set is an event.

Definition 9. (Liu [13] ) Let be independent and identical distribution(iid) positive uncertain variables. Define and for. Then the uncertain process is called a renewal process.

Note that event is same with event.

For an uncertain renewal process, Liu [4] proved that converges in mean to, i.e.,

Definition 10. (Liu [4] ) Let be iid uncertain interarrival times, and let be uncertain rewards. It is also assume that are independent. Then

is called a renewal reward process, where is the renewal process.

Theorem 3. (Liu [4] ) Let be a renewal reward process with uncertain interarrival times and uncertain rewards. If those interarrival times and rewards have uncertainty distributions and, then has an uncertainty distribution

Here we set and when.

Liu gave an elementary uncertain renewal reward theorem in the book [4] (see latter Theorem 4). But, it is not strict to proof of the theorem. Therefore, in the following we strict prove it by two lemmas.

Lemma 1. If and are nonnegative continuous strict increasing functions on, and then

(i) for given, there exists such that

(ii)

Proof. Proof of (i) is easy. In following we prove (ii). Note that we have the following facts:

For given there exists such that

and for any integer such that

Thus, when,

Also,

and function at is continuous, thus

i.e.,

Lemma 2. If conditions of Lemma 1 are satisfied, and

converge, then

consistent convergent on about t.

Proof. It follows from process of proof of Lemma 1 that, for any

Therefore, for any

also,

is convergent, then

is consistent convergent on about t.

Theorem 4. (Elementary uncertain renewal reward theorem, Liu [4] ) Let be a renewal reward process with uncertain interarrival times and uncertain rewards If exists, then If those interarrival times and rewards have regular uncertainty distribution and satisfy the following conditions and then

Proof. Firstly, note that uncertainty distribution of is

and

Since the uncertainty distribution of is

and the uncertainty distribution of is

using Lemma 2 we have

2. Conclusion

This paper provides a strict proof of elementary uncertain renewal reward theorem by some technics.

Acknowledgements

This work was supported by National Natural Science Foundation of China Grants No. 61273044 and No. 11471152.

NOTES

^{*}Corresponding author.

References

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