Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance

Journal of Inequalities and Applications Limit distribution for a semi-Markovian random walk with Weibull distributed interference of chance Tülay Kesemen 0 Rovshan Aliyev 1 2 Tahir Khaniyev 1 3 0 Faculty of Science, Department of Mathematics, Karadeniz Technical University , Trabzon, 61080 , Turkey 1 Institute of Cybernetics of Azerbaijan National Academy of Sciences , F. Agayev Str. 9, AZ 1141, Baku , Azerbaijan 2 Department of Probability Theory and Mathematical Statistics, Baku State University , Z. Khalilov Str. 23, AZ 1148, Baku , Azerbaijan 3 Department of Industrial Engineering, TOBB University of Economics and Technology , Sögütözü, Ankara 06560 , Turkey In this paper, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered. In this study, it is assumed that the sequence of random variables {ζn}, n = 1, 2, . . . , 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 X(t) is discussed. Then the weak convergence theorem is proved for the ergodic distribution of the process X(t) and the limit form of the ergodic distribution is derived. MSC: 60G50; 60K15; 60F99 semi-Markovian random walk; discrete interference of chance; ergodic distribution; weak convergence; asymptotic expansion; ladder variables 1 Introduction 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. [–]; Alsmeyer []; Borovkov []; Khaniyev et al. [, ]; Lotov []; Rogozin []; Skorohod and Slobodenyuk []; Spitzer [] etc.). Note that in the studies of Khaniyev et al. [] and Aliyev et al. [, ], the random variables 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 like Aliyev et al. [–] and Khaniyev et al. [, ], we assume that the random variables 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 This process might be useful in the following situation. The model 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 z ∈ (, ∞). Assume that the premiums and claims arrive at the insurance company randomly at the times Tn = in= ξi, n ≥ , here ξi, i ≥ , are the random time intervals between two successive claims and premiums. Level of total capital of the company fluctuates in accordance with {–ηn}, n ≥ . The random variable ηn 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 τ 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 ζ, 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 ζ and so on. Denote the stochastic process expressed this model mathematically by X(t). Thus, the amount of capital of the insurance company at each time t is represented by the process X(t). The process X(t) 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 X(t). 2 Mathematical construction of the process X(t) Let (ξn, ηn, ζn), n ≥  be a sequence of independent and identically distributed vector of random variables, defined on any probability space ( , , P), such that ξn takes only positive values, ηn takes positive and negative values as well as positive ones, ζn has the Weibull distribution with parameters (α, λ), α > , λ > . Suppose that ξ, η, ζ are mutually independent random variables and the distribution functions of them are known, i.e., π (z) ≡ P{ζ ≤ z} =  – exp –(λz)α , z ≥ , α > , λ > . Define the renewal sequence {Tn} and the random walk {Sn} as follows: Tn = Sn = T = S = , n = , , . . . , and define a sequence of integer-valued random variables {Nn} as follows: N = , N ≡ N (z) = inf{n ≥  : z – Sn < }, z ≥ ; Nn+ = i (...truncated)