Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance
© Kesemen et al.; licensee Springer 2013
Received: 23 November 2012
Accepted: 12 March 2013
Published: 29 March 2013
In this paper, a semi-Markovian random walk with a discrete interference of chance is considered. In this study, it is assumed that the sequence of random variables , , which describes the discrete interference of chance, forms an ergodic Markov chain with the Weibull stationary distribution. Under this assumption, the ergodic theorem for the process is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process and the limit form of the ergodic distribution is derived.
MSC:60G50, 60K15, 60F99.
Keywordssemi-Markovian random walk discrete interference of chance ergodic distribution weak convergence asymptotic expansion ladder variables
Many interesting problems of stochastic finance, mathematical biology, reliability, queuing, stochastic inventory and mathematical insurance can be expressed by means of random walk processes. Some important studies on this topic exist in literature (see, for example, Aliyev et al. [1–3]; Alsmeyer ; Borovkov ; Khaniyev et al. [8, 9]; Lotov ; Rogozin ; Skorohod and Slobodenyuk ; Spitzer etc.).
Note that in the studies of Khaniyev et al.  and Aliyev et al. [1, 2], the random variables , , which describe the discrete interference of chance, have exponential, gamma and triangular distribution, respectively, and stationary moments of the ergodic distribution of a semi-Markovian random walk process have been investigated. Moreover, Aliyev et al.  and Khaniyev and Atalay  investigated a weak convergence theorem for the ergodic distribution of the renewal-reward process when the random variables , , have gamma and triangular distribution, respectively. In this study, unlike Aliyev et al. [1–3] and Khaniyev et al. [8, 9], we assume that the random variables , , which describe the discrete interference of chance, are independent and identically distributed random variables with the Weibull distribution, and the weak convergence theorem is proved for the ergodic distribution of a semi-Markovian random walk process, and the limit distribution is derived for the ergodic distribution of the considered process.
This process might be useful in the following situation.
Consider a stochastic model, which can be used in the field of insurance. This model can be described as follows.
Suppose that the amount of initial capital of an insurance company is equal to . Assume that the premiums and claims arrive at the insurance company randomly at the times , , here , , are the random time intervals between two successive claims and premiums. Level of total capital of the company fluctuates in accordance with , . The random variable expresses difference of claims and premiums, which can take both positive and negative values. The amount of the total capital of the insurance company continues its variation until a random time which is the time at which the capital level first falls below zero. When the above conditions take place, the amount of the company’s capital increases immediately to the level , which is a random variable having a certain distribution in the interval . Thus, the first period is completed. Then the insurance company keeps working in a way similar to the previous period with a new initial capital and so on.
Denote the stochastic process expressed this model mathematically by . Thus, the amount of capital of the insurance company at each time t is represented by the process . The process is known to be as a semi-Markovian random walk process with a discrete interference of chance.
We now proceed to a mathematical construction of the process .
2 Mathematical construction of the process
and is stipulated.
Let , , ; , , and define as .
if , ; .
Note that the process describes the amount of the total capital of an insurance company at any time .
The main purpose of this study is to prove the weak convergence theorem for the ergodic distribution of the process , as . For this aim, we first discuss the ergodicity of the process .
3 The ergodicity of the process
State the following proposition on the ergodicity of the process .
Proposition 3.1 (Ergodic theorem )
Proof The process belongs to a wide class of processes which is called ‘The class of semi-Markov processes with a discrete interference of chance’ in literature. General ergodic theorem of type ‘Smith’s key renewal theorem’ exists in literature for this class (see, Gihman and Skorohod , p.243). It is not difficult to show that the assumptions of the general ergodic theorem are satisfied under the conditions of Proposition 3.1. Therefore, the ergodicity of the process is derived by using this general ergodic theorem. □
Now we define the characteristic function of the ergodic distribution of the process as follows: , .
where ; ; .
4 Weak convergence theorem for the ergodic distribution of the process
In this section, we use the ladder variables of the random walk , , with the initial state . Let , .
where is n-fold convolution of the distribution function .
Our aim is to prove the weak convergence theorem for the ergodic distribution as . For the Weibull distribution, it is known that . In this study, α will be fixed. Therefore while , the parameter λ should converge to zero. Hence, we need to give the following lemma first.
Since M is finite and is an arbitrary positive number, the proof of Lemma 4.1 is completed. □
We can give the following lemma, which has a similar proof to that of Lemma 4.1.
For the investigation of the asymptotic behavior of the ergodic distribution of the process as , we define the auxiliary process as and investigate the asymptotic behavior of its ergodic distribution function. It is easily seen that the process is a linear transform of . Therefore, from Proposition 3.1, it is immediately follows that the process is also ergodic under the conditions of Proposition 3.1. Let us denote the characteristic function of the ergodic distribution of by and formulate the following statement.
where is Euler’s gamma function.
Note that is the limit distribution of a ‘residual waiting time’ (see, Feller , p.368).
where is the characteristic function of the Weibull distribution with parameter .
Now we can investigate the asymptotic behavior of as .
The conditions and ensure that is finite.
where , , .
It means that the family of characteristic functions converges to a limit characteristic function . Here is the characteristic function of the Weibull distribution with parameter .
This completes the proof of Theorem 4.1. □
Note that is the limit distribution for the ‘residual waiting time’ generating by the sequence (see, Feller , p.386).
In this paper, a semi-Markovian random walk with a discrete interference of chance is considered and the ergodic theorem for this process is discussed under some conditions. Finally, the weak convergence theorem is proved for the ergodic distribution of the process , and the limit form of the ergodic distribution is established. Note that is a limit distribution of a residual waiting time of a renewal process generated by the random variables having the Weibull distribution with parameters . In the terms of insurance, here we derived the explicit form of the limit distribution of the amount of the capital of an insurance company which is working for a long time as .
Dedicated to Professor Hari M Srivastava.
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