Let $(T, d)$ be a compact metric space of diameter $D$, and $\|\cdot \|_\Phi$ be an Orlicz norm. When is it true that all (separable) processes $(X_t)_{t \in T}$ that satisfy the increment condition $\|X_t - X_s\|_\Phi \leq d(t, s)$ for all $s, t$ in $T$ are sample bounded? We give optimal necessary conditions and optimal sufficient conditions in terms of the existence of a probability measure $m$ on $T$ that satisfies an integral condition $\int^D_0 f(\varepsilon, m(B(x, \varepsilon))) d\varepsilon \leq K$ for each $x$ in $T$, where $f$ is a function suitably related to $\Phi$. When $T$ is a compact group and $d$ is translation invariant, we are able to compute the necessary and sufficient condition in several cases.

A class of diffusions on the line is considered whose drifts are generalized functions (Schwartzian distributions). A probability measure is put on this space of drifts, giving a diffusion in a random environment. An invariance principle is then proven for the rescaled diffusion, generalizing a result of De Masi, Ferrari, Goldstein and the author.

A stationary $l^2$-valued Ornstein-Uhlenbeck process is considered which is given formally by $dX_t = -AX_t dt + \sqrt 2a dB_t$, where $A$ is a positive self-adjoint operator on $l^2, B_t$ is a cylindrical Brownian motion on $l^2$ and $a$ is a positive diagonal operator on $l^2$. A simple criterion is given for the almost-sure continuity of $X_t$ in $l^2$ which is shown to be quite sharp. Furthermore, in certain special cases, we obtain simple necessary and sufficient conditions for the almost-sure continuity of $X_t$ in $l^2$.

We complete the results of M. Marcus and G. Pisier by showing that a strongly stationary 1-stable process $(X_t)_{t \in G}$ defined on a locally compact group has a version with sample continuous paths if (and only if) the entropy integral $\int^\infty_0 \log^+ \log N(K, d_X, \varepsilon) d\varepsilon$ is finite, where $K$ is a given neighborhood of the unit and $d_X$ is the distance induced by the process.

Let $\{\xi(t)\}_{t \geq 0}$ be a stationary stochastic process, with one-dimensional distribution function $G$. We develop a method to determine an asymptotic expression for $\Pr\{\sup_{0 \leq t \leq h} \xi(t) > u\}$, when $u \uparrow \sup\{v: G(v) < 1\}$, applicable when $G$ belongs to a domain of attraction of extremes, and we show that if $G$ belongs to such a domain, then so does the distribution function of $\sup_{0 \leq t \leq h} \xi(t)$. Applications are given to hitting probabilities for small sets for $\mathbb{R}^m$-valued Gaussian processes and to extrema of Rayleigh processes. Further, we prove the Gumbel, Frechet and Weibull laws, for maxima over increasing intervals, when $G$ is type I-, type II- and type III-attracted, respectively, and we establish the asymptotic Poisson character of $\varepsilon$-upcrossings and local $\varepsilon$-maxima.

Mason and van Zwet gave an approximation of the uniform empirical process by a Brownian bridge. Their result is a refinement of a result of Komlos, Major and Tusnady in the case when the process is considered in a small interval. In this note we show that in such cases a much better Poissonian approximation is possible which seems to be better applicable in certain cases. We also prove a multidimensional version of this result, where a sequence of uniform empirical processes is simultaneously approximated by partial sums of independent Poisson processes.

Let $X$ be a Harris recurrent Markov process (in discrete or continuous time). We give a functional law of the iterated logarithm for the additive functionals of $X$ which are (close to) square integrable martingales with respect to the invariant measure of $X$. The proof is based on the Skorokhod embedding technique and the construction of an atom for a Harris chain. In contrast with the positive recurrent case, "the suitable normalizations" are random in the null recurrent case. Moreover it is shown from two examples how to use the law of the iterated logarithm to get the rate of almost sure convergence of an estimator.

Functional laws of the iterated logarithm are established for a modified version of the classical product-limit estimator of a distribution function when the data are subject to random censorship or truncation. These functional laws are shown to hold for the entire interval $I$ over which the distribution function can be consistently estimated, under basically the same assumptions that have been used in the literature to establish the weak convergence of the normalized estimator in $D(I)$. Making use of stochastic integral representations and empirical process theory, strong approximations involving i.i.d. continuous-parameter martingales are developed for the product-limit estimator, and these strong approximations are then applied to derive the functional laws of the iterated logarithm.

This paper studies the average cost for controlled systems via the martingale problem for their generator. The stationary distributions for these processes on the product of the state and control spaces are characterized in terms of integration of the generator and existence of a stationary solution is established. This gives existence of a stationary control.

This paper studies the average cost for controlled systems given as solutions of the martingale problem for their generator. The control problem is reformulated as a linear programming problem and conditions are given for the existence of an optimal solution. It is further shown that the optimal control can be taken to depend only on the history of the system and that this cost remains optimal for systems with different initial distributions.

We give a short new proof of the existence of optimal solutions to a continuous time formulation of the two-armed bandit problem, using a new topological embedding of the set of randomized optional increasing paths. We do not make any hypothesis on the two-parameter filtration, other than completeness and right-continuity.

We consider a system of several interconnected queues (with a single class of customers) and model the state $(X_t)$ of the system as a jump Markov process. The problem of interest is to estimate the large deviations behavior of the rescaled system $X^\varepsilon_t = \varepsilon X_{t/\varepsilon}$, corresponding to large time and large excursions of the original (unscaled) system. The techniques employed are those of the theory of viscosity solutions to Hamilton-Jacobi equations. From the point of view of large deviation theory, the interesting new problem here is the treatment of the process when one or more of the queues are nearly empty, since an abrupt change in the jump measure occurs. From the point of view of viscosity solutions, the discontinuity of the jump measure leads to nonlinear boundary conditions on domains with corners for the associated partial differential equations. Much of the paper is devoted to proving uniqueness of viscosity solutions for these equations, and these sections are of independent interest. While our use of test functions in proving the uniqueness is an adaptation of the usual technique, the construction of the test functions themselves via the Legendre transform is new. We obtain a representation for the solution of the equation in terms of a nonstandard optimal control problem, which suggests the correct integrand in the large deviation "rate" functional. Since it is the treatment of the effects due to the "boundaries" that is novel, we devote the majority of the paper to the detailed development of a simple two-dimensional system that exhibits all the essential new features. However, the arguments may be applied to queueing systems that are considerably more general, and we attempt to indicate this generality as well.

In this paper we use a central limit theorem for entropy due to Ibragimov to obtain limit theorems for linear normalizations of the log minimum distance when observations are sampled from measures belonging to a family of fractal distributions. It is shown that in almost all cases the limit distribution is Gaussian with parameters determined in part by the Hausdorff dimension associated with the underlying measure. Exceptions to this rule include absolutely continuous measures which obey the classical extreme value limit laws.

We consider critical branching Bessel processes initially at $r \gg 1$ and stopped at $r = 1$. Let $N$ be the number of descendants hitting $r = 1$. We give the norming constant $k(r)$ and prove convergence, as $r \rightarrow \infty$, of $N/\lbrack k(r) \rbrack$ conditioned on $\{N > 0\}$. The distribution of conditioned limit laws is also investigated. A feature of this study is an interplay between probabilistic insights and analytic techniques for Emden-Fowler's equation.

We study populations consisting of individuals whose birth and death rates depend on both their age and the age-structure of the whole population. In particular, as the population size tends to infinity, we derive a law of large numbers and a central limit theorem for the empirical processes.

Random tree-type partitions for finite sets are used as a model of a chemical polymerization process when ring formation is forbidden. Technically, our series of three papers studies the asymptotic behavior (in the thermodynamic limit as $n \rightarrow \infty$) of a particular probability distribution on the set of all forests of trees on a set of $n$ elements (monomers). The study rigorously establishes the existence of three stages of polymerization dependent upon the ratio of association and dissociation rates of monomers. The subcritical stage has been analyzed in the other two papers of this series. The present paper, second in the series, concentrates on the analysis of the near-critical and supercritical stages. In the supercritical stage we discover that the molecular weight of the largest connected component (gel) has the Holtsmark distribution. Our study combines elements of a classical Flory-Stockmayer polymerization theory with the spirit of more recent developments in the Erdos-Renyi theory of random graphs. Although this paper has a chemical motivation, conceptually similar mathematical models have been found useful in other disciplines, such as computer science and biology, etc.

It is proved that for a wide class of point processes indexed by the positive quadrant of the plane, and for a class of compact sets in this quadrant, the germ $\sigma$-field is equal to the $\sigma$-field generated by the values of the process on the set. Therefore, there exists a large family of point processes in the plane (and among them the spatial Poisson process) which satisfy the sharp Markov property in the sense of P. Levy. The strong Markov property with respect to stopping lines is also studied. Some examples are obtained by taking transformations of the probability measure.

There are many ways of describing positive dependence, for example the strong FKG inequalities and association. It is known that for Bernoulli random variables the strong FKG inequalities are equivalent to all the conditional distributions being associated, which is in turn equivalent to all the conditional distributions having positively correlated marginals. These and similar definitions are extended to point processes on $\mathbb{R}^d$. Examples are given to show that, unlike the analogous Bernoulli random variable case, these conditions are no longer equivalent, although some are implied by others.

We consider quantities such as the probability that a two-dimensional random walk crosses the ordinate $y$ for the first time to the left of the abscissa $x$, and describe the asymptotic behaviour as $x$ and $y$ tend to $\infty$. The result is applied to the risk reserve process of insurance mathematics as well as to one-dimensional random walks.

Suppose that $U$ is a $U$-statistic of degree 2 based on a simple random sample of size $n$ selected without replacement from a finite population of $N$ elements. A bound for the difference between the distribution function of a standardized version of $U$ and its single-term Edgeworth expansion is given. We apply these results to obtain an Edgeworth expansion for the variance estimator in a finite population. Some simulation results are reported in this case.

General results on series representations, involving arrival times in a Poisson process, are established for infinitely divisible Banach space valued random vectors without Gaussian components. Applying these results, various generalizations of LePage's representation are obtained in a unified way. Certain conditionally Gaussian infinitely divisible random vectors are characterized and some problems related to a Gaussian randomization method are investigated.