Elementary Uncertain Renewal Reward Theorem and Its Strict Proof ()
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Subject Areas: Mathematical Analysis
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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
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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
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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
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(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
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*Corresponding author.