Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity, and the rolling condition. First, the relations between these shape conditions are analyzed. Second, for the estimation of sets fulfilling a rolling condition, we obtain a result of ‘full consistency’ (i.e. consistency with respect to the Hausdorff metric for the target set and for its boundary). Third, the class of uniformly bounded compact sets whose reach is not smaller than a given constant r is shown to be a P-uniformity class (in Billingsley and Topsøe's (1967) sense) and, in particular, a Glivenko-Cantelli class. Fourth, under broad conditions, the r-convex hull of the sample is proved to be a fully consistent estimator of an r-convex support in the two-dimensional case. Moreover, its boundary length is shown to converge (almost surely) to that of the underlying support. Fifth, the above results are applied to obtain new consistency statements for level set estimators based on the excess mass methodology (see Polonik (1995)).

In Groeneboom (1988) a central limit theorem for the number of vertices Nn of the convex hull of a uniform sample from the interior of a convex polygon was derived. To be more precise, it was shown that {Nn - (2/3)rlogn} / {(10/27)rlogn}1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken. In the unpublished preprint Nagaev and Khamdamov (1991) a central limit result for the joint distribution of Nn and An is given, where An is the area of the convex hull, using a coupling of the sample process near the border of the polygon with a Poisson point process as in Groeneboom (1988), and representing the remaining area in the Poisson approximation as a union of a doubly infinite sequence of independent standard exponential random variables. We derive this representation from the representation in Groeneboom (1988) and also prove the central limit result of Nagaev and Khamdamov (1991), using this representation. The relation between the variances of the asymptotic normal distributions of the number of vertices and the area, established in Nagaev and Khamdamov (1991), corresponds to a relation between the actual sample variances of Nn and An in Buchta (2005). We show how these asymptotic results all follow from one simple guiding principle. This corrects at the same time the scaling constants in Cabo and Groeneboom (1994) and Nagaev (1995).

A stochastic ordering approach is applied with Stein's method for approximation by the equilibrium distribution of a birth-death process. The usual stochastic order and the more general s-convex orders are discussed. Attention is focused on Poisson and translated Poisson approximations of a sum of dependent Bernoulli random variables, for example, k-runs in independent and identically distributed Bernoulli trials. Other applications include approximation by polynomial birth-death distributions.

Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation, such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulae for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulae derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.

The ‘coupon collection problem’ refers to a class of occupancy problems in which j identical items are distributed, independently and at random, to n cells, with no restrictions on multiple occupancy. Identifying the cells as coupons, a coupon is ‘collected’ if the cell is occupied by one or more of the distributed items; thus, some coupons may never be collected, whereas others may be collected once or twice or more. We call the number of coupons collected exactly r times coupons of type r. The coupon collection model we consider is general, in that a random number of purchases occurs at each stage of collecting a large number of coupons; the sample sizes at each stage are independent and identically distributed according to a sampling distribution. The joint behavior of the various types is an intricate problem. In fact, there is a variety of joint central limit theorems (and other limit laws) that arise according to the interrelation between the mean, variance, and range of the sampling distribution, and of course the phase (how far we are in the collection processes). According to an appropriate combination of the mean of the sampling distribution and the number of available coupons, the phase is sublinear, linear, or superlinear. In the sublinear phase, the normalization that produces a Gaussian limit law for uncollected coupons can be used to obtain a multivariate central limit law for at most two other types — depending on the rates of growth of the mean and variance of the sampling distribution, we may have a joint central limit theorem between types 0 and 1, or between types 0, 1, and 2. In the linear phase we have a multivariate central limit theorem among the types 0, 1,…, k for any fixed k.

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {Xn, n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component Sn takes values on the real line , and is adjoined to the chain such that {Sn, n ≥ 1} is a Markov random walk. Assume that Sn = ∑k=1nξk, and that {ξn, n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of Sn / √n, we prove a self-normalized central limit theorem for Sn under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

In this paper we analyze different forms of fractional relaxation equations of order ν ∈ (0, 1), and we derive their solutions in both analytical and probabilistic forms. In particular, we show that these solutions can be expressed as random boundary crossing probabilities of various types of stochastic process, which are all related to the Brownian motion B. In the special case ν = ½, the fractional relaxation is shown to coincide with Pr{sup0≤s≤tB(s) < U} for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the gamma density, we obtain more and more complicated fractional equations.

Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

We extend Goldie's (1991) implicit renewal theorem to enable the analysis of recursions on weighted branching trees. We illustrate the developed method by deriving the power-tail asymptotics of the distributions of the solutions R to and similar recursions, where (Q, N, C1, C2,…) is a nonnegative random vector with N ∈ {0, 1, 2, 3,…} ∪ {∞}, and are independent and identically distributed copies of R, independent of (Q, N, C1, C2,…); here ‘∨’ denotes the maximum operator.

Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ − 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.