Exact and asymptotic results for the uniform random labelled tree on $n$ vertices have been studied extensively by combinatorialists. Here we treat asymptotics from a modern stochastic process viewpoint. There are three limit processes. One is an infinite discrete tree. The other two are most naturally represented as continuous two-dimensional fractal tree-like subsets of the infinite-dimensional space $l_1$. One is compact; the other is unbounded and self-similar. The proofs are based upon a simple algorithm for generating the finite random tree and upon weak convergence arguments. Distributional properties of these limit processes will be discussed in a sequel.

We consider single equations and systems of reaction-diffusion equations depending on small parameter. These equations are generalizations of the Kolmogorov-Petrovskii-Piskunov equation. Using the large deviation principle, we describe the asymptotic behavior of solutions.

We consider the jump-type Markov processes which are small random perturbations of dynamical systems and their empirical processes. We prove large deviations theorems for empirical measures which are marginal measures of empirical processes at the exit time of Markov processes from a bounded domain in a $d$-dimensional Euclidean space $\mathscr{R}^d$.

We define a renormalized intersection local time to describe the amount of self-intersection of the Brownian motion on a two-dimensional Riemannian manifold $M$. The second order asymptotics of the area of the Wiener sausage of radius $\varepsilon$ on $M$ are described in terms of the renormalized intersection local time.

We develop explicit stochastic integral representations for the renormalized triple intersection local time of planar Brownian motion. Our representations involve a new type of double stochastic integral, the bilateral stochastic integral, which is developed in detail.

Let $\mathbf{X}(t), t \geq 0$, be a vector Gaussian process in $R^m$ whose components are i.i.d. copies of a real Gaussian process $X(t)$ with stationary increments. Under specified conditions on the spectral distribution function used in the representation of the incremental variance function, it is shown that the self-intersection local time of multiplicity $r$ of the vector process is jointly continuous. The dimension of the self-intersection set is estimated from above and below. The main tool is the concept of local nondeterminism.

The Brownian density process is a distribution-valued process that arises either via a limiting operation on an infinite collection of Brownian motions or as the solution of a stochastic partial differential equation. It has a (self-) intersection local time, that is formally defined through an operation involving delta functions, much akin to the better studied intersection local time of measure-valued ("super") processes. Our main aim is to show that this formal definition not only makes sense mathematically, but can also be understood, at least in two and three dimensions, via the intersection local times of simple Brownian motions. To show how useful this way of looking at the Brownian density intersection local time can be, we also derive a Tanaka-like evolution equation for it in the two-dimensional case.

It is shown that all Levy processes on the line whose paths are of bounded variation have a closed range over any finite time interval that is nowhere dense except for those processes having positive (negative) drift with Levy measure finite on $(0, \infty)$ [finite on $(-\infty, 0)$].

The one-dimensional nearest neighbor asymmetric simple exclusion process has been used as a microscopic approximation for the Burgers equation. This equation has travelling wave solutions. In this paper we show that those solutions have a microscopic structure. More precisely, we consider the simple exclusion process with rate $p$ (respectively, $q = 1 - p)$ for jumps to the right (left), $\frac{1}{2} < p \leq 1$, and we prove the following results: There exists a measure $\mu$ on the space of configurations approaching asymptotically the product measure with densities $\rho$ and $\lambda$ to the left and right of the origin, respectively, $\rho < \lambda$, and there exists a random position $X(t) \in \mathbb{Z}$, such that, at time $t$, the system "as seen from $X(t)$," remains distributed according to $\mu$, for all $t \geq 0$. The hydrodynamical limit for the simple exclusion process with initial measure $\mu$ converges to the travelling wave solution of the inviscid Burgers equation. The random position $X(t)/t$ converges strongly to the speed $\nu = (1 - \lambda - \rho)(p - q)$ of the travelling wave. Finally, in the weakly asymmetric hydrodynamical limit, the stationary density profile converges to the travelling wave solution of the Burgers equation.

Hydrodynamic behavior of one-dimensional homogeneous exclusion processes with speed change on periodic lattices $\mathbb{Z}/N\mathbb{Z}, N = 1,2,3,\ldots$, is studied. For every reversible exclusion process with nearest neighbor jumps and local interactions of gradient type it is shown that under diffusion-type scaling in space and time the empirical density fields of the processes converge to a weak solution of a nonlinear diffusion equation as $N$ goes to infinity. Two classes of examples of exclusion processes as stated are given.

We consider a system of particles in $\mathbb{R}^d$ performing symmetric stable motion with exponent $\alpha, 0 < \alpha \leq 2$, and branching at the end of an exponential lifetime with offspring generating function $F(s) = s + \frac{1}{2}(1 - s)^{1+\beta}, 0 < \beta \leq 1$. (This includes binary branching Brownian motion for $\alpha = 2, \beta = 1$.) It is shown that, for an initial Poisson population with uniform intensity, the system goes to extinction if $d \leq \alpha/\beta$ and is "persistent" (i.e., preserves intensity in the large time limit) if $d > \alpha/\beta$. To this purpose a continuous-time version of Kallenberg's backward technique for computing Palm distributions of branching particle systems is developed, which permits us to adapt methods used by Dawson and Fleischmann in the study of discrete-space and discrete-time systems.

A critical branching Brownian motion in $R^d$ is studied where the initial state is either a single particle or a homogeneous field with finite or infinite density. Conditioned on survival in a bounded subset $B$ of $R^d$ at a large time $t$, some normalized limits of the number of particles in a bounded subset $A$ are obtained. When the initial state is a single particle, the normalization factor is a power of $t$ in low dimensions, a power of $\log t$ in the critical dimension and a constant in high dimensions. Extensions to the other initial states and/or more general critical offspring distributions are discussed. Both factors affect the critical dimension. The results are motivated by probabilistic consideration and are proved with the aid of analytic technique of differential equations.

Consider moving average processes of the form $X_t = \sum^\infty_{j=0} c_jZ_{t-j}$, where $\{Z_j\}$ are iid and nonnegative random variables and $c_j > 0$ are constants satisfying summability conditions at least sufficient to make the random series above converge. We suppose that the distribution of $Z_j$ is regularly varying near 0 and discuss lower tail behavior of finite and infinite linear combinations. The behavior is quite different in the two cases. For finite linear combinations, the lower tail is again regularly varying but for infinite moving averages, the lower tail is $\Gamma$-varying, i.e., it is in the domain of attraction of a type I extreme value distribution in the sense of minima. Convergence of point processes based on the moving averages is shown to hold in both the finite and infinite order cases and suitable conclusions are drawn from such convergences. A useful analytic tool is asymptotic normality of the Esscher transform of the common distribution of the $Z$'s. The extreme value results of this paper are in terms of minima of the moving average processes but results can be adapted to study maxima of moving averages of random variables in the domain of attraction of the type III extreme value distribution for maxima.

Let $X_1, X_2,\ldots$ be integrable, i.i.d. r.v.'s with common distribution function $F$ and let $\{v_n\}_{n \geq 1}$ be the sequence of optimal rewards or values in the associated optimal stopping problem, i.e., $v_n = \sup\{E(X_T): T \text{is a stopping time for} \{X_m\}_{m\geq 1} \text{and} T \leq n\}$ for $n \geq 1$. For distribution functions $F$ in the domain of attraction of one of the three classical extreme-value laws $G_I, G^\alpha_{II}$ or $G^\alpha_{III}$, it is shown that $\lim_n n(1 - F(v_n)) = 1, 1 - \alpha^{-1}$, or $1 + \alpha^{-1}$ if $F \in \mathscr{D}(G_1), F \in \mathscr{D}(G^\alpha_{II})$ and $\alpha > 1$, or $F \in \mathscr{D}(G^\alpha_{III})$ and $\alpha > 0$, respectively. From this result, the growth rate of $\{v_n\}_{n\geq 1}$ is obtained and compared to the growth rate of the expected maximum sequence. Also, the limit distribution of the optimal reward r.v.'s $\{X_{T^\ast_n}\}_{n\geq 1}$ is derived, where $\{T^\ast_n\}_{n\geq 1}$ are the optimal stopping times defined by $T^\ast_n \equiv 1$ if $n = 1$ and, for $n = 2,3,\ldots$, by $T^\ast_n = \min\{1 \leq k < n: X_k > v_{n-k}\}$ if this set is not equal to $\varnothing$ and equal to $n$ otherwise. This tail-distribution growth rate is shown to be sufficient for any threshold sequence to be asymptotically optimal.

For the secretary (or best-choice) problem with an unknown number $N$ of objects, minimax-optimal stop rules and (worst-case) distributions are derived, under the assumption that $N$ is a random variable with unknown distribution, but known upper bound $n$. Asymptotically, the probability of selecting the best object in this situation is of order of $(\log n)^{-1}$. For example, even if the only information available is that there are somewhere between 1 and 100 objects, there is still a strategy which will select the best item about one time in five.

For a continuous martingale $M$, let $\langle M, M\rangle$ denote the increasing process. Let $I_0, I_1,\ldots$ denote the iterated stochastic integrals of $M$. We prove the inequalities of Burkholder-Davis-Gundy type, $A_{p,n}\|\langle M, M \rangle^{1/2}_t\|^n_{np} \leq \|I_n(t)\|_p \leq B_{p,n}\|\langle M, M \rangle^{1/2}_t\|^n_{np}$, where $\ln A_{p,n} \sim \ln B_{p,n} \sim -(n/2)\ln n$ as $n \rightarrow \infty$. Our proof requires the sharp constant $b_p$ in Burkholder-Davis-Gundy inequalities $\|M\|_p \leq b_p\|\langle M, M\rangle^{1/2}\|_p$. In the Appendix we prove $\sup_{p\geq 1}(b_p/\sqrt p) = 2$. We apply our inequality to the study of the $L^p$ convergence of the Neuman series $\sum I_n(t)$ for exponential martingales.

We prove that if $(\theta_k)$ is a sequence of i.i.d. real random variables then, for $1 < q < p$, the linear combinations of $(\theta_k)$ have comparable $p$th and $q$th moments if and only if the joint distribution of $(\theta_k)$ is $(p, q)$-hypercontractive. We elaborate hypercontraction methods in a new proof of the inequality $\bigg(E\bigg\|\sum_i X_i\bigg\|^p\bigg)^{1/p} \leq C_p\bigg(E\big\|\sum_i X_i\bigg\| + \big(E\sup_i\|X_i\|^p\big)^{1/p}\bigg),$ where $(X_i)$ is a sequence of independent zero-mean random variables with values in a normed space, and $C_p \approx p/\ln p$.

We study some generalizations of the LIL when self-normalizations are used. Two particular results proved are: (1) an extension of the Kolmogorov-Erdos test for partial sums of symmetric i.i.d. random variables having finite second moments; this result eliminates distinctions required when nonrandom normalizers are used and $E(X^2I(|X| > t))$ is not $O((L_2t)^{-1})$, and (2) an extension of a universal bounded LIL of Marcinkiewicz to nonsymmetric random variables. An interesting corollary of this work is a short new proof of the classical LIL avoiding truncation methods.

It is shown that a perturbed Bernoulli probability transition kernel yields an explicit example of an orthogonality preserving kernel which is not completely orthogonal. In statistical language, such a kernel defines models $P_\theta, \theta \in \lbrack 0, 1 \rbrack$, in which there is no estimate that estimates $\theta$ perfectly for all $\theta$, but there is, for any given prior distribution on $\theta$ and hypothesis $H_0 \subset \lbrack 0, 1 \rbrack$, a perfect test for $H_0$ against its complement $\lbrack 0, 1\rbrack\backslash H_0$. It is also demonstrated with an analysis and an application of sets and maps with the Baire property that there are continuum many nonisomorphic atomless orthogonality preserving transition kernels which are not completely orthogonal. Our methods may be regarded as refinements of those used by Blackwell.

Herein boundary crossing behavior of conditional random walks is studied. Asymptotic distributions of the exit time and the excess over the boundary are derived. In the course of derivation, two results of independent interest are also obtained: Lemma 4.1 shows that a conditional random walk behaves like an unconditional one locally in a very strong sense. Theorem B.1 describes a class of distributions over which the renewal theorem holds uniformly. Applications are given for modified repeated significance tests and change-point problems.

Starting at $x$ in a Polish space $X$, a player selects the distribution $\sigma_0$ of the next state $x_1$ from the collection $\Gamma (x)$ of those distributions available and then selects the distribution $\sigma_1(x_1)$ for $x_2$ from $\Gamma(x_1)$ and so on. Suppose the player wins if every $x_i$ in the stochastic process $x_1, x_2,\ldots$ lies in a given Borel subset $A$ of $X$, that is, if the process stays in $A$ forever. If $\{(x, \gamma): \gamma \in \Gamma (x)\}$ is a Borel subset of $X \times \mathbb{P}(X)$, where $\mathbb{P}(X)$ is the natural Polish space of probability measures on $X$, and if $0 \leq p \leq 1$, then a player can stay in $A$ forever with probability at least $p$ if and only if the player can stay in $A$ up to time $t$ with probability at least $p$ for every Borel stop rule $t$. A similar result holds when the object of the game is to visit $A$ infinitely often.

In 1971, Simons and Johnson showed that the classical theorem of binomial to Poisson convergence is actually stronger than in the usual sense. Their result was proved valid also for the distributions of sums of independent, but not necessarily identically distributed, Bernoulli random variables by Chen in 1974. Here we show that their result is indeed true for a much larger class of random variables. The limiting distribution is generalized to a compound Poisson distribution.