Tulay Kesemen

Asymptotic Expansions for the Moments of the Semi-Markovian Random Walk with Gamma Distributed Interference of Chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X(t) are obtained when the random variable ζ1, which describes a discrete interference of chance, has a gamma distribution with parameters (α, λ), α > 1, λ > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X(t), as λ → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace- Padé Method

In this study, the solutions of random partial differential equations are examined. The parameters and the initial conditions of the random component partial differential equations are investigated with Beta distribution. A few examples are given to illustrate the efficiency of the solutions obtained with the random Differential Transformation Method (rDTM). Functions for the expected values and the variances of the approximate analytical solutions of the random equations are obtained. Random Differential Transformation Method is applied to examine the solutions of these partial differential equations and MAPLE software is used for the finding the solutions and drawing the figures. Also the Laplace- Pade Method is used to improve the convergence of the solutions. The results for the random component partial differential equations with Beta distribution are analysed to investigate effects of this distribution on the results. Random characteristics of the equations are compared with the results of the deterministic partial differential equations. The efficiency of the method for the random component partial differential equations is investigated by comparing the formulas for the expected values and variances with results from the simulations of the random equations.

Deterministic stability and random behavior of a Hepatitis C model

The deterministic stability of a model of Hepatitis C which includes a term defining the effect of immune system is studied on both local and global scales. Random effect is added to the model to investigate the random behavior of the model. The numerical characteristics such as the expectation, variance and confidence interval are calculated for random effects with two different distributions from the results of numerical simulations. In addition, the compliance of the random behavior of the model and the deterministic stability results is examined.

Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

ABSTRACT A semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζn) → ∞ when the random variable ζn has a generalized beta distribution with parameters (s, S, α, β);  α, β > 1, 0  ⩽ s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.

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

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.