Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive or negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positively or negatively dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.

The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k–1 and Mk can be interpreted as the number of comparisons of ‘Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi), (Zi) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.

Optimization problems in cancer radiation therapy are considered, with the efficiency functional defined as the difference between expected survival probabilities for normal and neoplastic tissues. Precise upper bounds of the efficiency functional over natural classes of cellular response functions are found. The ‘Lipschitz' upper bound gives rise to a new family of probability metrics. In the framework of the ‘m hit-one target' model of irradiated cell survival the problem of optimal fractionation of the given total dose into n fractions is treated. For m = 1, n arbitrary, and n = 1, 2, m arbitrary, complete solution is obtained. In other cases an approximation procedure is constructed. Stability of extremal values and upper bounds of the efficiency functional with respect to perturbation of radiosensitivity distributions for normal and tumor tissues is demonstrated.