We give several examples for Poisson approximation of quantities of interest in the analysis of algorithms: the distribution of node depth in a binary search tree, the distribution of the number of losers in an election algorithm and the discounted profile of a binary search tree. A simple and well-known upper bound for the total variation distance between the distribution of a sum of independe...

The Stein-Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to ap...

2 Poisson approximation 6 2.1 A coupling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Stein’s method for Poisson approximation . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Independent summands . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Dependent summands: the local approach . . . . . . . . . . . . . . . . . 10 2.2.3 Size biasing and coup...

In order to study the relationship between random Boolean sets and some explanatory variables, this paper introduces a Propagation model. This model can be applied when corresponding Poisson process of the Boolean model is related to explanatory variables and the random grains are not affected by these variables. An approximation for the likelihood is used to find pseudo-maximum likelihood esti...

We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

Poisson approximation using Stein’s method has been extensively studied in the literature. The main focus has been on bounding the total variation distance. This paper is a first attempt on moderate deviations in Poisson approximation for right-tail probabilities of sums of dependent indicators. We obtain results under certain general conditions for local dependence as well as for size-bias cou...

A well-known approximation of the aggregate claims distribution in the individual risk theory model with mutually independent individual risks is the compound Poisson approximation. In this paper, we relax the assumption of independency and show that the same compound Poisson approximation will still perform well under certain circumstances.

The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Was...

We propose an original approximation method, which is based on the Stein’s method and the zero bias transformation, to calculate CDO tranches in the general factor framework. We establish first-order correction terms for the Gaussian and the Poisson approximations respectively and we estimate the approximation errors. The application to the CDOs pricing consists of combining the two approximati...

The saddlepoint approximation to the probabilities of a general time-homogenous birth process, as derived by Daniels 6 , is revisited. Of interest is the accuracy of the approximation for extended Poisson process models constructed from state-dependent birth processes. Numerical calculations are used to examine the accuracy of the probability approximation for a range of state-dependent models,...