Dietmar Pfeifer

CHARACTERIZATIONS OF EXPONENTIAL DISTRIBUTIONS BY INDEPENDENT NON-STATIONARY RECORD INCREMENTS

A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.

MODELING AND GENERATING DEPENDENT RISK PROCESSES FOR IRM AND DFA

Modern Integrated Risk Management (IRM) and Dynamic Financial Analysis (DFA) rely in great part on an appropriate modeling of the stochastic behavior of the various risky assets and processes that influence the performance of the company under consideration. A major challenge here is a more substantial and realistic description and modeling of the various complex dependence structures between such risks showing up on all scales. In this presentation, we propose some approaches towards modeling and generating (simulating) dependent risk processes in the framework of collective risk theory, in particular w.r.t. dependent claim number processes of Poisson type (homogeneous and non-homogeneous), and compound Poisson processes.

On a relationship between Uspensky's theorem and poisson approximations

In this paper we show that Uspenskys expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.

Extremal processes, secretary problems and the 1/ e law

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a timetransformed extremal process it is shown that the asymptotic winning probability is again lle as in the classical situation. Extensions of the problem to more than one choice are also considered.

A new semigroup technique in poisson approximation

We present a unified and self-contained approach to Poisson approximation problems for independent Bernoulli summands with respect to several metrics by a general semigroup technique, expanding and completing earlier work on this subject by the first two authors [4], [5], [6].

Solvency II: stability problems with the SCR aggregation formula

One of the central issues in the Solvency II process will be an appropriate calculation of the Solvency Capital Requirement (SCR). This is the economic capital that an insurance company must hold in order to guarantee a one-year ruin probability of at most 0.5%. In the so-called standard formula, the overall SCR is calculated from individual SCRs in a particular way that imitates the calculation of the standard deviation for a sum of normally distributed risks (SCR aggregation formula). However, in order to cope with skewness in the individual risk distributions, this formula must be calibrated accordingly in order to maintain the prescribed level of confidence. In this paper, we want to show that the methods proposed and discussed thus far still show stability problems within the general setup.

Poisson approximations of multinomial distributions and point processes

We present an extension to the multinomial case of former estimations for univariate Poisson binomial approximation problems and generalize a result obtained by [1], 805-810). As an application, we evaluate the total variation distance between superpositions of independent Bernoulli point processes and a suitable Poisson process. The main tool will be a multiparameter semigroup approach.

Some remarks on Nevzorov's record model

We give a simple proof for the independence of record indices in Nevzorovs record model which is based on ranks. An application of these results to a probabilistic analysis of a particular searching algorithm with non-equiprobable orderings is also discussed.

Approximation-theoretic aspects of probabilistic representations for operator semigroups

Abstract In the present paper explicit sharp estimations are provided for the rate of convergence of basically all known representation formulae for operator semigroups which in an earlier paper have been shown to arise from a single general probabilistic representation theorem based on a special version of the weak law of large numbers. As a main tool, sharp estimations for moments and moment-generating functions of suitable random variables are used. Some of the results are applied to exponential operators as well as to a class of Poisson approximation theorems in probability theory.

THE STRUCTURE OF ELEMENTARY PURE BIRTH PROCESSES

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.