ON THE DISTRIBUTIONS OF A RENEWAL REWARD PROCESS AND IT ’ S ADDITIVE FUNCTIONAL

In this study, a renewal reward process with a discrete interference of chance ( ) t ( X ) is constructed and distribution of the process ) t ( X is investigated. Onedimensional distribution of the process X(t) is given by means of the probability characteristics of the renewal processes { } n T and { } n S . Moreover, one dimensional distribution function of the additive functional ( ) t Jf of the process ) t ( X is expressed by the probability characteristics of the initial sequences of the random variables { } n ξ and { } n η . Keywords– Renewal Reward Process, Additive Functional, Finite Dimensional Distribution, Discrete Interference of Chance.


INTRODUCTION
A number of very interesting problems of queuing, reliability, risk, sequential analysis and control of reserves theories, mathematical insurance, statistics, biology and physics are expressed by means of the renewal reward process with a discrete interference of chance.Numereous studies have been done about a renewal reward process with a discrete interference of chance because of their practical and theoretical importance.Jewell [4] generalized the study of the fluctuations of a reward process imbedded in a renewal process.Brown and Solomon [2] obtained second order asymptotic expansions for the first and second moments of the renewal reward process.
Khaniyev [10]  not difficult to see that there exist some connection between harmonic renewal measures and first passage times.For instance, Alsmeyer proved that if 0 ) ( E 1 > µ ≡ η then the following asymptotic expansion is true as η is a sequence of independent and identically distributed random variables and γ is Euler's constant.But the probability characteristics of additive functional of the renewal reward process aren't sufficient investigate in literature, which is very important for solving some problems of applied sciences. Therefore, in this study, a renewal reward process with a discrete interference of chance ( ) t ( X ) and it's additive functional ( ( )

CONSTRUCTION OF THE PROCESS AND IT'S ADDITIVE FUNCTIONAL
, be a sequence of independent and identically distributed triples of random variables, defined on any probability space ( and a sequence of integer valued random variables n N as: We can now construct desired stochastic process ) t ( X as follows: The process ) t ( X is called a renewal reward process with a discrete interference of chance.The following graph is one of the trajectories of the process ( ) Picture 1.One of the trajectories of the process → is a bounded measurable function.For each 0 t ≥ , define ( ) is called an additive functional of the process X(t).
The main purpose of this study is to investigate the distributions of the renewal reward process ) t ( X and it's additive functional ( )

3.ONE DIMENSIONAL DISTRIBUTION FUNCTIONS OF THE PROCESS X(t)
Let us denote by ( ) The main purpose of this study is to express one dimensional distrubition function In order to becoming more precise now we need further notation first: are independent, then the Laplace transform of the distrubition functions Q(t,x,z) of the process X(t) can be expressed by the probability characteristics of the initial random variables, as follows: Proof.Using total probability formula, the one dimensional distrubition function Q(t,x,z) of the process X(t) can be written as follows: Let the first term of the equality (3.3) be G(t,x,z).Let's rewrite the second term of the equality (3.3) as follows: Therefore, the equality (3.3) can be written as follows: If both sides of the the equality (3.4) are multiplied by ( ) and integrated respect to parameter z from s to ∞ + , we get: Then, we can write: Using the formulas (3.6) and (3.7), we can write: The above result show that ( ) Let's now calculate ( ) Therefore, we finally get: This completes the proof of the Theorem 3.1.

ONE DIMENSIONAL DISTRIBUTION FUNCTION OF THE ADDITIVE FUNCTIONAL ( ( ( ( ) ) ) ) t J f
Let us denote by , the one dimensional distrubition function of the additive functional ( ) . In order to becoming more precise now we need further notation first.
( ) For each the bounded measurable function ( ) , we define the following double transforms: ,  M , Theorem 4.1.Suppose that 1 ξ and 1 η are independent random variables.Then the double transform of the one dimensional distribution function of the additive functional ( ) can be written as follows: Proof.Let's try to write obtained the integral equation of renewal reward type for Denote the first term of the ( . Let's rewritte the second term of the equation (4.3) as follows: where Substituting the (4.4) in the (4.3) we can obtain the integral equation as follows: We can rewrite the equality (4.5) as follows: Averaging this equality respect to z, we can obtained: Then, we can write: Using the formulas (4.7) and (4.8) we can obtain as follows: The above result show that the demanded function This completes the proof of the Theorem 4.1.
Note: From Theorem 3.1 and Theorem 4.1 we can derive many usefull and valuable information about the process ) t ( X or additive functional ( ) t J f .For this aim let's consider the following example.Conclusion.Using similar methods, from Theorem 3.1 and Theorem 4.1 the different corollaries, very important for solving some problems of applied sciences,.canbe obtained.

AN EXAMPLE
a sequence of independent and identically distributed positive random variables.Khaniyev obtained analytic results about moments of the process.Then, in this study an asymtpotic behavior, as t → ∞ of the process is investigated.Morever, a number of authors have investigated the harmonic renewal measure R f are expressed by the probability characteristics of sequences of the random variables { } n ξ , { } n η as the formulas (4.1) and (4.2).

Example 5 . 1 .
Using Theorem 3.1, let obtain the general form of the moments of the process X(t) , when i ζ have the exponantial distribution with parameter 0 µ > .From Theorem 3.1, it is known that

Corollary 5 . 5 .
The asymmetry-symmetry coefficients of the process s X Corollary 5.3.Then, the variance of the process s X (t) can expressed as follows: