Let M be a bounded domain of ∝^d with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

For a Borel set A and a homogeneous Poisson point process η in ∝^d of intensity λ>0, define the Poisson--Voronoi approximation A_η of A as a union of all Voronoi cells with nuclei from η lying in A. If A has a finite volume and perimeter, we find an exact asymptotic of E Vol(AΔ A_η) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(A_η) and Vol(AΔ A_η) together with their asymptotics for large λ are obtained as well.

For stationary fiber processes, the estimation of the directional distribution is an important task. We consider a stereological approach, assuming that the intersection points of the process with a finite number of test hyperplanes can be observed in a bounded window. The intensity of these intersection processes is proportional to the cosine transform of the directional distribution. We use the approximate inverse method to invert the cosine transform and analyze asymptotic properties of the estimator in growing windows for Poisson line processes. We show almost-sure convergence of the estimator and derive Berry--Esseen bounds, including formulae for the variance.

Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.

Cloud-computing shares a common pool of resources across customers at a scale that is orders of magnitude larger than traditional multiuser systems. Constituent physical compute servers are allocated multiple `virtual machines' (VMs) to serve simultaneously. Each VM user should ideally be unaffected by others' demand. Naturally, this environment produces new challenges for the service providers in meeting customer expectations while extracting an efficient utilization from server resources. We study a new cloud service metric that measures prolonged latency or delay suffered by customers. We model the workload process of a cloud server and analyze the process as the customer population grows. The capacity required to ensure that the average workload does not exceed a threshold over long segments is characterized. This can be used by cloud operators to provide service guarantees on avoiding long durations of latency. As part of the analysis, we provide a uniform large deviation principle for collections of random variables that is of independent interest.

We propose a simple stochastic volatility model which is analytically tractable, very easy to simulate, and which captures some relevant stylized facts of financial assets, including scaling properties. In particular, the model displays a crossover in the log-return distribution from power-law tails (small time) to a Gaussian behavior (large time), slow decay in the volatility autocorrelation, and multiscaling of moments. Despite its few parameters, the model is able to fit several key features of the time series of financial indexes, such as the Dow Jones Industrial Average, with remarkable accuracy.

We consider a model of fixed size N = 2^l in which there are l generations of daughter cells and a stem cell. In each generation i there are 2^i-1 daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generation i daughter cells become two cells of generation i+1. The last generation is removed from the population. A stem cell acquires first and second mutations at rates u₁ and u₂, and a daughter cell acquires first and second mutations at rates v₁ and v₂. We find the distribution for the time it takes to acquire two mutations as N goes to ∞ and the mutation rates go to 0. The mutation rates may tend to 0 at different speeds. We also find the distribution for the locations of the mutations. In particular, we determine whether or not the mutations occur on a stem cell and if not, at what generation in the daughter cells they occur. Several outcomes are possible, depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer.

We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L² of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Let {X_i} be a sequence of independent, identically distributed random variables with an intermediate regularly varying right tail F̄. Let (N, C₁, C₂,...) be a nonnegative random vector independent of the {X_i} with N∈ℕ∪ {∞}. We study the weighted random sum S_N =∑_{i=1}^N C_iX_i, and its maximum, M_N=sup_{{1≤k N+1}∑_i=1^k C_iX_i. This type of sum appears in the analysis of stochastic recursions, including weighted branching processes and autoregressive processes. In particular, we derive conditions under which P(M_N > x)∼ P(S_N > x)∼ E[∑_i=1^N F̄(x/C_i)] as x→∞. When E[X₁]>0 and the distribution of Z_N=∑ _i=1^NC_i is also intermediate regularly varying, we obtain the asymptotics P(M_N > x)∼ P(S_N > x)∼ E[∑_i=1^N F̄}(x/C_i)] +P(Z_N > x/E[X₁]). For completeness, when the distribution of Z_N is intermediate regularly varying and heavier than F̄, we also obtain conditions under which the asymptotic relations P(M_N > x) ∼ P(S_N > x)∼ P(Z_N > x / E[X₁] hold.

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.