We propose a trajectorial construction of a class of measure-valued Markov processes, called superprocesses or measure-valued branching processes, which have been studied extensively in the last few years. These processes were originally defined as weak limits of systems of branching particles. The basic idea of our construction is to use the branching structure of excursions of a linear Brownian motion to model the branching mechanism of the superprocess. Without any additional effort, our approach leads to the so-called historical process, which contains more information than the superprocess in the sense that it keeps track of the individual paths followed by the particles. We emphasize the relationship between the properties of the historical process and the corresponding results of excursion theory. We also give a description of the support of the superprocess at a fixed time, using a simple tree model. Finally, we use our construction to recover certain pathwise properties recently obtained by Perkins.

Particles placed in $N$ cells on the unit interval give birth or die according to linear rates. Adjacent cells are coupled by diffusion with a rate proportional to $N^2$. Cell numbers are divided by a density parameter to represent concentrations, and the resulting space-time Markov process is compared to a corresponding deterministic model, the solution to a partial differential equation. The models are viewed as Hilbert space valued processes and compared by means of a law of large numbers and central limit theorem. New and nearly optimal results are obtained by exploiting the Ornstein-Uhlenbeck type structure of the stochastic model.

Diffusion approximations for stochastic congested networks, both open and closed, are described in terms of the networks' bottlenecks. The approximations arise as limits of functional central limit theorems. The limits are driven by reflected Brownian motions on the nonnegative orthant (for open networks) and on the simplex (for closed ones). The results provide, in particular, invariance principles for Jackson's open queueing networks, Gordon and Newell's closed networks and some of Spitzer's finite particle systems with zero-range interaction.

For independent percolation models, it is shown that if the diagrammatic "triangle condition" is satisfied, then the critical exponents $\delta$ and $\hat\beta$ exist and take their mean-field values, generalizing the criterion introduced in 1984 by Aizenman and Newman for the mean-field value of $\gamma$ in nonoriented percolation. The results apply to a broad class of nonoriented, as well as oriented, weakly homogeneous models, in which the range of the connecting bonds need not be bounded. For the nonoriented case, the condition reduces to the finiteness at the critical point of $\nabla \equiv \sum_{x,y}\tau(0,x)\tau(x,y)\tau(y,0)$ [with $\tau(u,\nu)$ the probability that the site $u$ is connected to $\nu$], which was recently established by Hara and Slade for models with sufficiently spread out connections in $d > 6$ dimensions. Our analysis proceeds through the derivation of complementary differential inequalities for the percolation order parameter $M(\beta,h)$--whose value at $h = 0+$ yields the percolation density, with $\beta$ parametrizing the bond, or site, occupation probabilities and with $h, h \geq 0$, a "ghost field." The conclusion is that under the triangle condition, in the vicinity of the critical point $(\beta_c,0), M(\beta,0+) \approx (\beta - \beta_c)^{\hat\beta}_+$ and $M(\beta_c,h) \approx h^{1/\delta}$, with $\hat \beta = 1$ and $\delta = 2$.

We prove conservation of local equilibrium, away from the shock, for some attractive asymmetric particle systems on $\mathbb{Z}^d$. The method applies to a class of particle processes which includes zero-range and simple exclusion processes. The main point in the proof is to exploit attractiveness. The hydrodynamic equation obtained is a first-order nonlinear partial differential equation which presents shock waves.

Consider the nearest neighbor graph for the integer lattice $\mathbf{Z}^d$ in $d$ dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for $\mathbf{Z}^d$. This is shown to be a tree if and only if $d \leq 4$. In this case, the tree has only one topological end, that is, there are no doubly infinite paths. When $d \geq 5$ the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

Based on two notions of almost subadditivity which were introduced by Derriennic and Schurger, two a.s. limit theorems are proved which both generalize Kingman's subadditive ergodic theorem. These results, being valid under weak moment conditions, are obtained by short proofs. One of these proofs is completely elementary and does not even make use of Birkhoff's ergodic theorem which, instead, is obtained as a by-product. Finally, an improvement of Liggett's a.s. limit theorem is given.

We consider a sequence of $\mathbb{R}^d$-valued semimartingales $\{X_n\}$ satisfying $X_n(t) = X_n(0) + \int^t_0\sigma_n(X_n(s-))dZ_n(s) + \int^t_0F(X_n(s-))dA_n(s),$ where $\{Z_n\}$ is a "well-behaved" sequence of $\mathbb{R}^e$-valued semimartingales, $\sigma_n$ is a continuous $d \times e$ matrix-valued function, $F$ is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points $\Gamma$, and $A_n$ is a nondecreasing process which asymptotically puts infinite mass on every interval. Many Markov processes with lower dimensional diffusion approximations can be written in this form. Intuitively, if $X_n(0)$ is close to $\Gamma$, the drift term $F dA_n$ forces $X_n$ to stay close to $\Gamma$, and any limiting process must actually stay on $\Gamma$. If $X_n(0)$ is only in the domain of attraction of $\Gamma$ under the flow of $F$, then the drift term immediately carries $X_n$ close to $\Gamma$ and forces $X_n$ to stay close to $\Gamma$. We make these ideas rigorous, give conditions under which $\{X_n\}$ is relatively compact in the Skorohod topology and give a stochastic integral equation for the limiting process(es).

Two algorithms for calculating rate functions for a family of measures $\{\mu_\varepsilon\}$ on a $B$-space $X$ are considered. The first one is a relaxed version of the Fenchel transform type theorem for convex rate functions. The second gives conditions under which $\{\mu_\varepsilon\}$ can be replaced by a more convenient family $\{\mu^x_\varepsilon\}$ near admissible points $x \in X$ such that rate functions for both families coincide near $x$. As an example, we apply both techniques to investigate large deviation properties of some reaction-diffusion equations with quick random noise.

Let $\{\mathbf{X}_j, 1 \leq j \leq n\}$ be a sequence of iid random vectors in $\mathbb{R}^d$ and $S_n = \{\mathbf{X}_j/b_n, 1 \leq j \leq n\}$. When do there exist scaling constants $b_n \rightarrow \infty$ such that $S_n$ converges to some compact set $S$ in $\mathbb{R}^d$ almost surely (in probability)? We show that a limit set $S$ is star-shaped (i.e., $\mathbf{x} \in S$ implies $t\mathbf{x} \in S$, for $0 \leq t \leq 1$) so that after a polar coordinate transformation the limit set is the hypograph of an upper semicontinuous function. We specify necessary and sufficient conditions for convergence to a particular limit set. Some examples are also given.

We consider an estimation problem in which the signal is modelled by a continuous Gaussian random field and is observed through smooth and bounded nonlinear sensors. A nonhomogeneous Markov process is defined in order to sample the conditional distribution of the signal given the observations. At any finite time the process takes values in a finite-dimensional space, although the dimension goes to infinity in time. We prove that the empirical averages of any bounded functional continuous w.p.1 converge in the mean square to the conditional expectation of the functional.

Let $f$ be a real martingale and $s(f)$ its conditional square function. Then the following inequalities are sharp: $\|f\|_p \leq \sqrt{\frac{2}{p}}\|s(f)\|_p,\quad 0 < p \leq 2,$ $\sqrt{\frac{2}{p}}\|s(f)\|_p \leq \|f\|_p,\quad p \geq 2.$ The second inequality is still sharp if $f$ is replaced by the maximal function $f^\ast$. Let $S(f)$ denote the square function of $f$. Then the following inequalities are also sharp: $\|S(f)\|_p \leq \sqrt{\frac{2}{p}}\|s(f)\|_p,\quad 0 < p \leq 2,$ $\sqrt{\frac{2}{p}}\|s(f)\|_p \leq \|S(f)\|_p,\quad p \geq 2.$ These inequalities hold for Hilbert-space-valued martingales and are strict inequalities in all of the nontrivial cases.

The problem of approximating boundary hitting times for diffusion processes, and in particular Brownian motion, is considered. Using a combination of probabilistic and function-analytic techniques, approximations for conditioned diffusion distributions are obtained. These lead to approximations for the distribution of the hitting time itself. The approximations are split into three cases depending on whether the boundary is upper case, approximation square root or lower case, and one-sided boundaries are also considered separately.

Let $\bar{X}_n$ denote the mean of an i.i.d. sequence of random vectors $X_1,X_2,X_3,\ldots$ taking values in $\mathbf{R}^d$. If $\lambda$ denotes the convex conjugate of the logarithm of the moment generating function for $X_1$, then $\lim\sup\frac{1}{n}\log P(\bar{X}_n \in C) \leq -\inf\{\lambda(\nu): \nu \in C\}$ when $C \subset \mathbf{R}^d$ is closed and the moment generating function for $X_1$ is finite in a neighborhood of the origin. An example is given in which this upper bound fails for a certain closed set in $\mathbf{R}^3$ and the moment generating function for $X_1$ is not finite in a neighborhood of the origin. An example is also given in which this upper bound is valid for all closed sets but the moment generating function for $X_1$ is not finite in a neighborhood of the origin.

Let $(X_i,U_i)$ be pairs of i.i.d. bounded real-valued random variables ($X_i$ and $U_i$ are generally mutually dependent). Assume $E\lbrack X_i\rbrack < 0$ and $\Pr\{X_i > 0\} > 0$. For the (rare) partial sum segments where $\sum^l_{i=k}X_i \rightarrow \infty$, strong limit laws are derived for the sums $\sum^l_{i=k}U_i$. In particular a strong law for the length $(l - k + 1)$ and the empirical distribution of $U_i$ in the event of large segmental sums of $\sum X_i$ are obtained. Applications are given in characterizing the composition of high scoring segments in letter sequences and for evaluating statistical hypotheses of sudden change points in engineering systems.

Let $A_1,A_2,\ldots,A_n$ be generated governed by an $r$-state irreducible Markov chain and suppose $(X_i,U_i)$ are real valued independently distributed given the sequence $A_1,A_2,\ldots,A_n$, where the joint distribution of $(X_i,U_i)$ depends only on the values of $A_{i-1}$ and $A_i$ and is of bounded support. Where $A_0$ is started with its stationary distribution, $E\lbrack X_1\rbrack < 0$ and the existence of a finite cycle $C = \{A_0 = i_0,\ldots,A_k = i_k = i_0\}$ such that $\Pr\{\sum^m_{i=1}X_i > 0, m = 1,\ldots,k; C\} > 0$ is assumed. For the partial sum realizations where $\sum^l_{i=k}X_i \rightarrow \infty$, strong laws are derived for the sums $\sum^l_{i=k}U_i$. Examples with $r = 2, X \in \{-1, 1\}$ and the cases of Brownian motion and Poisson process with negative drift are worked out.

Let $\{N(t), t \geq 0\}$ be the (generalized) renewal process associated with an i.i.d. sequence $X_1,X_2,\ldots$ of random variables having finite moment generating function on some left-sided neighborhood of the origin. Some strong limiting results are proved for the maximal increments $\sup_{0\leq t\leq T-K} (N(t + K) - N(t))$, where $K = K_T$ is a function of $T$ such that $K_T \uparrow \infty$, but $K_T/\log T \downarrow 0$ as $T \rightarrow \infty$. These provide analogs to a recent extension due to Mason (1989) of the Erdos-Renyi strong law of large numbers for partial sums.

We show that if $\mathbf{X} = (X_1,\ldots,X_d)$ is a vector in $\mathbb{R}^d$ and all linear combinations $\sum^d_{i=1}C_iX_i$ are 1-stable random variables, then $\mathbf{X}$ is itself 1-stable. More generally, a probability measure $\mu$ on a vector space whose univariate marginals are 1-stable is itself 1-stable. This settles an outstanding problem of Dudley and Kanter.

We show that if $f$ is a probability density on $R^n$ wrt Lebesgue measure (or any absolutely continuous measure) and $0 \leq f \leq 1$, then there is another density $g$ with only the values 0 and 1 and with the same $(n - 1)$-dimensional marginals in any finite number of directions. This sharpens, unifies and extends the results of Lorentz and of Kellerer. Given a pair of independent random variables $0 \leq X,Y \leq 1$, we further study functions $0 \leq \phi \leq 1$ such that $Z = \phi(X,Y)$ satisfies $E(Z\mid X) = X$ and $E(Z\mid Y) = Y$. If there is a solution then there also is a nondecreasing solution $\phi(x,y)$. These results are applied to tomography and baseball.

This paper establishes the best constant $c_q$ appearing in inequalities of the form $\mathbb{E}S_\infty \leq c_q\sup_{t\geq 0}\|M_t\|_q,$ where $M$ is an arbitrary nonnegative submartingale and $S_t = \sup_{s\leq t}M_s.$ The method of proof is via the Lagrangian for a version of the problem $\sup_\tau\mathbb{E}\{\lambda S_t - \lambda^qM^q_t\},$ where $M \equiv |B|, B$ a Brownian motion. More general inequalities of the form $\mathbb{E}S_\infty \leq C_\Phi\sup_{t\geq 0}\|M_t\|_\Phi$ and $\mathbb{E}S_\infty \leq C_\Phi\sup_{t\geq 0}\||M_t\||_\Phi$ (where $\|\cdot\|_\Phi$ and $\||\cdot\||_\Phi$ are, respectively, the Luxemburg norm and its dual, the Orlicz norm, associated with a Young function $\Phi$) are established under suitable conditions on $\Phi$. A simple proof of the John-Nirenberg inequality for martingales is given as an application.