Convex hulls of random walks: expected number of faces and face probabilities

Consider a sequence of partial sums Si=ξ1+…+ξi, 1≤i≤n, starting at S0=0, whose increments ξ1,…,ξn are random vectors in Rd, d≤n. We are interested in the properties of the convex hull Cn:=Conv(S0,S1,…,Sn). Assuming that the tuple (ξ1,…,ξn) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of Cn is given by the formula E[fk(Cn)]=2⋅k!n!∑l=0∞[n+1d−2l]{d−2lk+1}, for all 0≤k≤d−1, where [nm] and {nm} are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0≤i1<…<ik+1≤n, the points Si1,…,Sik+1 form a k-dimensional face of Conv(S0,S1,…,Sn). This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξk's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types An−1 and Bn. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract. Let A be an n by N real-valued matrix with n &lt; N; we count the number of k-faces fk(AQ) when Q is either the standard N-dimensional hypercube IN or else the positive orthant RN +. To state results simply, consider a proportional-growth asymptotic, where for fixed δ, ρ in (0, 1), we have a sequence of matrices An,Nn and of integers kn with n/Nn → δ, kn/n → ρ as n → ∞. If each matrix An,Nn has its columns in general position, then fk(AIN)/fk(I N) tends to zero or one depending on whether ρ&gt; min(0, 2 − δ−1) or ρ &lt; min(0, 2 − δ−1). Also, if each An,Nn is a random draw from a distribution which is invariant under right multiplication by signed permutations, then fk(ARN +)/fk(RN +) tends almost surely to zero or one depending on whether ρ&gt; min(0, 2 − δ−1) or ρ &lt; min(0, 2 − δ−1). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermine

Bayesian Nonparametric Inference for Queueing Systems

The present thesis deals with statistical inference for queueing models. Thereby, the considered approaches follow the Bayesian methodology. The work divides into three main parts which are related to each other by the underlying queueing model. The similarity lies in the assumption of continuous-time systems which are fed by a homogenous Poisson arrival stream of customers. Moreover, throughout the thesis no finite-dimensional parametric constraint is placed on the distribution of the service times. The latter leads to Bayesian nonparametric statistics. The distinction among the parts arise from the assumed number of servers as well as from the observational setups. The first two parts are about the single server queue while the last part deals with infinitely many servers. In the first part the arrival and service processes are taken as observations. Thereby, the main interest is in inference for the distributions of the waiting times of the customers and the occupation of the system, respectively. Besides the elaboration of the key properties of the statistical methods, their further examinations lead to new results for the theory of Bayesian statistics itself. The second part is also about the same system but a different observational setup is used. This means that the assumption of observations of the arrival and service process is dropped. Instead merely the customer's departure stream is assumed to be observable. Within this setup the main interest is in making inference for the service time distribution. This is done by studying the probabilistic structure of the system in more depth. The emerging theoretical considerations about the sufficient statistic of this inner structure make it possible to think about mixtures of such systems. These mixtures build the basis for the development of further Bayesian nonparametric inference for the service time distribution within this framework. The last part departs from the previous two in such that the queue with infinitely many servers is considered. This means that a separate server is assigned to each customer and no queue builds up. The interest is again in the service time distribution. The observations are indirect, meaning that merely the instants when customers arrive to and depart from the system, respectively, are recorded. The distribution of the emerging raw data has a known relationship to the service time distribution. This relationship was exploited in the past in order to make statistical inference for the service time distribution using the frequentist methodology. However, this raised several questions which have not been answered yet. This thesis provides answers to these questions which are necessary to deal with the issue from a Bayesian perspective. Since the raw data forms a general stationary process, the major problem consists in finding a suitable parametrization of the shift-ergodic measures. That is mainly due to the fact that this enables one to formalize mixtures of ergodic measures which in turn generate the observed data. Such a parametrization is developed and subsequently used for making Bayesian inference for stationary data