The recent increasing interplay between actuarial and financial mathematics has led to a surge in risk theoretic modeling. Especially actuarial ruin models under fairly general conditions on the underlying risk process have become a focus of attention. Motivated by applications such as the modeling of operational risk losses in financial risk management, we investigate the stability of classical asymptotic ruin estimates when claims are heavy, and this under variability of the claim intensity process. Various examples are discussed.

We study the asymptotics for the maximum on a random time interval of a random walk with a long-tailed distribution of its increments and negative drift. We extend to a general stopping time a result by Asmussen, simplify its proof and give some converses.

A single queueing station that serves K input streams is considered. Each stream is an independent renewal process, with customers having random lead times. Customers are served by processor sharing across streams. Within each stream, two disciplines are considered--earliest deadline first and first-in, first-out. The set of current lead times of the K streams is modeled as a K-dimensional vector of random counting measures on $\mathbb{R}$, and the limit of this vector of measure-valued processes is obtained under heavy traffic conditions.

Under fairly general assumptions on the arrival and service time processes, we prove fluid and heavy traffic limit theorems for the unfinished work, queue length, sojourn time and waiting time processes associated with a single station multiclass generalized processor sharing model. The fluid limit of the unfinished work process is characterized by the Skorokhod map associated with a Skorokhod problem formulation of the generalized processor sharing model, while the heavy traffic diffusion limit is characterized using the corresponding extended Skorokhod map. An interesting feature of the diffusion limits is that they may fail to be semimartingales.

We are interested in statistical solutions of McKean-Vlasov-Fokker-Planck equations. An example of motivation is the Navier-Stokes equation for the vorticity of a two-dimensional incompressible fluid flow. We propose an original and efficient numerical method to compute moments of such solutions. It is a stochastic particle method with random weights. These weights are defined through nonparametric estimators of a regression function and convey the uncertainty on the initial condition of the considered equation. We prove an existence and uniqueness result for a class of nonlinear stochastic differential equations (SDEs), and we study the relationship between these nonlinear SDEs and statistical solutions of the corresponding McKean-Vlasov equations. This result forms the foundation of our stochastic particle method where we estimate the convergence rate in terms of the numerical parameters: the number of simulated particles and the time discretization step.

One of the goals of this paper is to show that the infinite-alleles model with overdominant selection "looks like" the neutral infinite-alleles model when the selection intensity and mutation rate get large together. This rather surprising behavior was noticed by Gillespie (1999) in simulations. To make rigorous and refine Gillespie's observations, we analyze the limiting behavior of the likelihood ratio of the stationary distributions for the model under selection and neutrality, as the mutation rate and selection intensity go to $\infty$ together in a specified manner. In particular, we show that the likelihood ratio tends to 1 as the mutation rate goes to $\infty$, provided the selection intensity is a multiple of the mutation rate raised to a power less than $3/2$. (Gillespie's simulations correspond to the power 1.) This implies that we cannot distinguish between the two models in this setting. Conversely, if the selection intensity grows like a multiple of the mutation rate raised to a power greater than $3/2$, selection can be detected; that is, the likelihood ratio tends to 0 under neutrality and $\infty$ under selection. We also determine the nontrivial limit distributions in the case of the critical exponent $3/2$. We further analyze the limiting behavior when the exponent is less than $3/2$ by determining the rate at which the likelihood ratio converges to 1 and by developing results for the distributions of finite samples.

A problem of bounding the generalization error of a classifier %\break $f\in \conv(\mathcal{H})$, where $\mathcal{H}$ is a "base" class of functions (classifiers), is considered. This problem frequently occurs in computer learning, where efficient algorithms that combine simple classifiers into a complex one (such as boosting and bagging) have attracted a lot of attention. Using Talagrand's concentration inequalities for empirical processes, we obtain new sharper bounds on the generalization error of combined classifiers that take into account both the empirical distribution of "classification margins" and an "approximate dimension" of the classifiers, and study the performance of these bounds in several experiments with learning algorithms.

We investigate random distances in a random binary search tree. Two types of random distance are considered: the depth of a node randomly selected from the tree, and distance between randomly selected pairs of nodes. By a combination of classical methods and modern contraction techniques we arrive at a Gaussian limit law for normed random distances between pairs. The exact forms of the mean and variance of this latter distance are first derived by classical methods to determine the scaling properties, then used for norming, and the normed random variable is then shown by the contraction method to have a normal limit arising as the fixed-point solution of a distributional equation. We identify the rate of convergence in the limit law to be of the order $\Theta(1/\sqrt{\ln n})$ in the Zolotarev metric $\zeta_3$. In the analysis we need the rate of convergence in the central limit law for the depth of a node, as well. This limit law was derived before by various techniques. We establish the rate $\Theta(1/\sqrt{\ln n})$ in $\zeta_3$.

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers functionals of marked point processes, including statistics of Boolean models.

Consider the partial sums $\{S_t\}$ of a real-valued functional $F(\Phi(t))$ of a Markov chain $\{\Phi(t)\}$ with values in a general state space. Assuming only that the Markov chain is geometrically ergodic and that the functional $F$ is bounded, the following conclusions are obtained:

Spectral theory. Well-behaved solutions $\cf$ can be constructed for the "multiplicative Poisson equation" $(e^{\alpha F}P)\cf=\lambda\cf$, where P is the transition kernel of the Markov chain and $\alpha\in\Co$ is a constant. The function $\cf$ is an eigenfunction, with corresponding eigenvalue $\lambda$, for the kernel $(e^{\alpha F}P)=e^{\alpha F(x)}P(x,dy)$.

A "multiplicative" mean ergodic theorem. For all complex $\alpha$ in a neighborhood of the origin, the normalized mean of $\exp(\alpha S_t)$ (and not the logarithm of the mean) converges to $\cf$ exponentially, where $\cf$ is a solution of the multiplicative Poisson equation.

Edgeworth expansions. Rates are obtained for the convergence of the distribution function of the normalized partial sums $S_t$ to the standard Gaussian distribution. The first term in this expansion is of order $(1/\sqrt{t})$ and it depends on the initial condition of the Markov chain through the solution $\haF$ of the associated Poisson equation (and not the solution $\cf$ of the multiplicative Poisson equation).

Large deviations. The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line.

Exact large deviations asymptotics. Rates of convergence are obtained for the large deviations estimates above. The polynomial preexponent is of order $(1/\sqrt{t})$ and its coefficient depends on the initial condition of the Markov chain through the solution $\cf$ of the multiplicative Poisson equation.

Extensions of these results to continuous-time Markov processes are also given.

This paper concerns studies on continuous-time controlled Markov chains, that is, continuous-time Markov decision processes with a denumerable state space, with respect to the discounted cost criterion. The cost and transition rates are allowed to be unbounded and the action set is a Borel space. We first study control problems in the class of deterministic stationary policies and give very weak conditions under which the existence of $\varepsilon$-optimal ($\varepsilon\geq 0)$ policies is proved using the construction of a minimum Q-process. Then we further consider control problems in the class of randomized Markov policies for (1) regular and (2) nonregular Q-processes. To study case (1), first we present a new necessary and sufficient condition for a nonhomogeneous Q-process to be regular. This regularity condition, together with the extended generatorof a nonhomogeneous Markov process, is used to prove the existence of $\varepsilon$-optimal stationary policies. Our results for case (1) are illustrated by a Schlögl model with a controlled diffusion. For case (2), we obtain a similar result using Kolmogorov's forward equation for the minimum Q-process and we also present an example in which our assumptions are satisfied, but those used in the previous literature fail to hold.