In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

Рассматривается отрицательная биномиальная аппроксимация сумм независимых Z +-значных случайных величин. С помощью метода Стейна устанавливаются границы ошибок. Свертка отрицательного биномиального и пуассоновского распределений используется в качестве трехпараметрической аппроксимации. Ключевые слова и фразы: отрицательное биномиальное распределение, отрицательное биномиальное возмущение, метод Стейна, расстояние по вариации.

In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.2010 Mathematics Subject Classification. 60G55; 60G51.

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