We propose a generalized Cramér–Lundberg model of the risk theory of non-life insurance and study its ruin probability. Our model is an extension of that of Dubey (1977) to the case of multiple insureds, where the counting process is a mixed Poisson process and the continuously varying premium rate is determined by a Bayesian rule on the number of claims. We use two proofs to show that, for each fixed value of the safety loading, the ruin probability is the same as that of the classical Cramér–Lundberg model and does not depend on either the distribution of the mixing variable of the driving mixed Poisson process or the number of claim contracts.

For independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

Consider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

As an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.

For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, …, d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k < d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

Kipnis and Varadhan (1986) showed that, for an additive functional, Sn say, of a reversible Markov chain, the condition E[Sn2] / n → κ ∈ (0, ∞) implies the convergence of the conditional distribution of Sn / √E[Sn2], given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, E[Sn2] = nl(n), where l is a slowly varying function. It is shown by example that the conditional distributions of Sn / √E[Sn2] need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly nonstandard) normal distribution are developed.

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

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.

Saddlepoint approximations are derived for the conditional cumulative distribution function and density of where is the sample mean of n i.i.d. bivariate random variables and g(x, y) is a non-linear function. The relative error of order O(n–1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.

For independent random variables X and Y, if the conditional distribution of X given X + Y satisfies certain conditions, then the joint distribution of X and Y is a member of a certain one-parameter exponential family. Extensions for n independent random variables are given. A characterization for independent random variables involving order statistics is also given.