Recursive self-similarity for a random object is the property of being decomposable into independent rescaled copies of the original object. Certain random combinatorial objects--trees and triangulations--possess approximate versions of recursive self-similarity, and then their continuous limits possess exact recursive self-similarity. In particular, since the limit continuum random tree can be identified with Brownian excursion, we get a nonobvious recursive self-similarity property for Brownian excursion.

Let $X_t$ be an $\alpha$-self-similar Markov process on $(0, \infty)$ killed when hitting 0. $\alpha$-self-similar extensions of $X(t)$ to $\lbrack 0, \infty)$ are studied via Ito execusion theory (entrance laws). We give a condition that guarantees the existence of an extension, which either leaves 0 continuously (a.s.) or (a.s.) jumps from 0 to $(0, \infty)$ according to the "jumping in" measure $\eta(dx) = dx/x^{\beta+1}$. Two applications are given: the diffusion case and the "reflecting barrier process" of S. Watanabe.

For each $a \in (0, \frac{1}{2})$, there exists a random measure $\beta_a$ which is supported on the set of points where two-dimensional Brownian motion spends $a$ units of local time. The measure $\beta_a$ is carried by a set which has Hausdorff dimension equal to $2 - a$. A Palm measure interpretation of $\beta_a$ is given.

Both standard and functional laws of the iterated logarithm are obtained for the local time of a symmetric Levy process, at a fixed point in its state space, as time goes to infinity. Similar results are also obtained for the difference of the local times at two points in the state space. These results are sharp if the exponent of the characteristic function that defines the Levy process is regularly varying at zero with index $1 < \beta \leq 2$. The results are given in terms of the $\alpha$-potential density at zero, considered as a function of $\alpha$. Without additional effort our methods give essentially the same results for the number of visits of a symmetric random walk to a point in its state space and for the difference of the number of visits to two points in the state space. A limit theorem for the sequence of times that a random walk returns to its initial point is obtained as an application of the functional laws.

Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = F(X^\epsilon(\tau), \tau/\epsilon, \omega) \text{subject to} X^\epsilon(0) = x_0$, where $\{F(x, t, \omega), t \geq 0\}$ are stochastic processes indexed by $x$ in $\mathfrak{R}^d$, and the dependence on $x$ is sufficiently regular to ensure that the equation has a unique solution $X^\epsilon(\tau, \omega)$ over the interval $0 \leq \tau \leq 1$ for each $\epsilon > 0$. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: $\dot{x}^0(\tau) = \overline{F}(x^0(\tau)) \text{subject to} x^0(0) = x_0,$ such that $\lim_{\epsilon\rightarrow 0} \sup_{0\leq\tau\leq 1}E|X^\epsilon(\tau) - x^0(\tau)| = 0$. In this article we show that as $\epsilon \rightarrow 0$ the random function $(X^\epsilon(\cdot) - x^0(\cdot))/\sqrt{2\epsilon\log\log\epsilon^{-1}}$ almost surely converges to and clusters throughout a compact set $K$ of $C\lbrack 0, 1\rbrack$.

We prove that a sufficient condition for a semimartingale reflecting Brownian motion in an orthant (SRBM) to be positive recurrent is that all solutions of an associated deterministic Skorokhod problem are attracted to the origin. To prove this result, we construct a Lyapunov function for the SRBM.

Let $P(h) = (p_{ij}(h), i, j = 1,2,\ldots, n), h \geq 0, n \geq 1$, be a transition matrix function defining an irreducible recurrent continuous parameter Markov process. Let $(S_i, i = 1,2,\ldots, n)$ be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs $A_{k\ell}$. Let $f_t$ be a rotation of length $t$ of the circle, and denote Lebesgue measure by $\leftthreetimes$. We generalize and prove for the transition matrix function $P(h)$ a theorem of Cohen $(n = 2)$ and Alpern $(n \geq 2)$ asserting that every recurrent stochastic $n \times n$ matrix $P$ is given by \begin{equation*}\tag{*}p_{ij} = \big(\leftthreetimes(S_i\cap f^{-1}_t(S_j)\big)/\leftthreetimes(S_i),\end{equation*} for some choice of rotation $f_t$ and partition $\{S_i\}$. We prove the existence of a continuous map $\Phi$ from the space of $n \times n$ irreducible stochastic matrices into $n$-partitions of [0, 1), such that every domain matrix $P$ is represented by $(\ast)$ with $\{S_i\} = \Phi(P)$ and $t = 1/n$!. Furthermore, the representing process $(f_t, \{S_i\})$ has not only the same transition probabilities but also the same probabilistic cycle distribution as the Markov process based on $P$.

This paper presents almost sure uniform large deviation principles for the empirical distributions and empirical processes of Markov chains with random transitions. The results are derived under assumptions that generalize assumptions earlier used for time-homogeneous chains. The rate functions for the skew chain are expressed in terms of the Donsker-Varadhan functional and relative entropy. The sample chain rates are different, but they have natural upper and lower bounds in terms of familiar rate functions.

Recently, one of the authors (van den Berg) has obtained a uniqueness condition for Gibbs measures, in terms of disagreement percolation involving two independent realizations. In the present paper we study the dependence of Markov fields on boundary conditions by taking a more suitable coupling. This coupling leads to a new uniqueness condition, which improves the one mentioned above. We also compare it with the Dobrushin uniqueness condition. In the case of the Ising model, our coupling shares certain properties with the Fortuin-Kasteleyn representation: It gives an explicit expression of the boundary effect on a certain vertex in terms of percolation-like probabilities.

The threshold voter models considered in this paper are special cases of the nonlinear voter models which were introduced recently by Cox and Durrett. They are spin systems on $Z^d$ with transition rates \begin{equation*}c(x, \eta) = \begin{cases}1, \text{if there is a} y \text{with} \|x - y\| \leq N \text{and} \eta(x) \neq \eta(y), \\ 0, \text{otherwise}.\end{cases}\end{equation*} This system is known to cluster if $N = d = 1$, and to coexist if $N \geq 4$ in one dimension and if $N$ is reasonably large in other dimensions. Cox and Durrett conjectured that it coexists in all cases except $N = d = 1$. In this paper, we prove this conjecture. The proof is based on comparisons with threshold contact processes. The hard part of the proof consists of showing that the second nearest neighbor threshold contact process in one dimension with parameter 1 survives. The proof of this result is modeled after the proof by Holley and Liggett that the critical value of the basic contact process in one dimension is at most 2. By comparison with that proof, however, the fact that the interaction is not of nearest neighbor type presents substantial additional difficulties. In fact, part of the proof is computer aided.

Consider a special independent bond percolation model on $Z^2$, in which all bonds with vertices in the $X$ axis are open with probability $\delta$ and closed with probability $1 - \delta$, and all other bonds are open with probability $p$ and closed with probability $1 - p$. In this paper we show that no percolation occurs at $p_c$ for any $\delta < 1$. The method allows us also to show no percolation at $p_c$ in a more general inhomogeneous case.

We compute the diffusion coefficient of the current of particles through a fixed point in the one-dimensional nearest neighbor asymmetric simple exclusion process in equilibrium. We find $D = |p - q|\rho(1 - \rho)|1 - 2\rho|$, where $p$ is the rate at which the particles jump to the right, $q$ is the jump rate to the left and $\rho$ is the density of particles. Notice that $D$ vanishes if $p = q$ or $\rho = 1/2$. Laws of large numbers and central limit theorems are also proven. Analogous results are obtained for the current of particles through a position travelling at a deterministic velocity $r$. As a corollary we get that the equilibrium density fluctuations at time $t$ are a translation of the fluctuations at time 0. We also show that the current fluctuations at time $t$ are given, in the scale $t^{1/2}$, by the initial density of particles in an interval of length $|(p - q)(1 - 2\rho)|t$. The process is isomorphic to a growth interface process. Our result means that the equilibrium growth fluctuations depend on the general inclination of the surface. In particular, they vanish for interfaces roughly perpendicular to the observed growth direction.

Let $x(t) = \{x_i(t), i \in \mathbb{Z}^d\}$ be the solution of the system of stochastic differential equations $dx_i(t) = \bigg(\sum_{j\in\mathbb{Z}^d}a(i,j)x_j(t) - x_i(t)\bigg) dt + \sqrt{2g (x_i(t))} dw_i(t), \quad i \in \mathbb{Z}^d.$ Here $g: \lbrack 0, 1\rbrack \rightarrow \mathbb{R}^+$ satisfies $g > 0$ on (0, 1), $g(0) = g(1) = 0, g$ is Lipschitz, $a(i,j)$ is an irreducible random walk kernel on $\mathbb{Z}^d$ and $\{w_i(t), i \in \mathbb{Z}^d\}$ is a family of standard, independent Brownian motions on $\mathbb{R}; x(t)$ is a Markov process on $X = \lbrack 0, 1\rbrack^{\mathbb{Z}^d}$. This class of processes was studied by Notohara and Shiga; the special case $g(v) = v(1 - v)$ has been studied extensively by Shiga. We show that the long term behavior of $x(t)$ depends only on $\hat{a}(i,j) = (a(i,j) + a(j, i))/2$ and is universal for the entire class of $g$ considered. If $\hat{a}(i,j)$ is transient, then there exists a family $\{\nu_\theta, \theta \in \lbrack 0, 1\rbrack\}$ of extremal, translation invariant equilibria. Each $\nu_\theta$ is mixing and has density $\theta = \int x_0 d\nu_\theta$. If $\hat{a}(i,j)$, is recurrent, then the set of extremal translation invariant equilibria consists of the point masses $\{\delta_0, \delta_1\}$. The process starting in a translation invariant, shift ergodic measure $\mu$ on $X$ with $\int x_0 d\mu = \theta$ converges weakly as $t \rightarrow \infty$ to $\nu_\theta$ if $\hat{a}(i,j)$ is transient, and to $(1 - \theta)\delta_0 + \theta\delta_1$ if $\hat{a}(i,j)$ is recurrent. (Our results in the recurrent case remove a mild assumption on $g$ imposed by Notohara and Shiga.) For the case $\hat{a}(i,j)$ transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case $\hat{a}(i,j)$, recurrent we use a duality comparison argument.

In this paper we consider a multilevel branching diffusion particle system and its diffusion approximation, which can be characterized as an $M(M(R^d))$-valued process. The long term behavior of the limiting process is studied. The main results are that if $d \leq 4$, then the two level $M(M(R^d))$-valued process suffers local extinction, and if $d = 4$, then the process has a self-similarity property.

We consider an ordinary, symmetric, continuous-time random walk on the two-dimensional lattice $\mathbb{Z}^2$. The distribution of the walk is transformed by a density which discounts exponentially the number of points visited up to time $T$. This introduces a self-attracting interaction of the paths. We study the asymptotic behavior for $T \rightarrow \infty$. It turns out that the displacement is asymptotically of order $T^{1/4}$. The main technique for proving the result is a refined analysis of large deviation probabilities. A partial discussion is given also for higher dimensions.

Consider two independent sequences $(X_i)_{i\leq n}$ and $(X'_i)_{i\leq n}$ that are independent and uniformly distributed over $\lbrack 0, 1\rbrack^d, d \geq 3$. Under mild regularity conditions, we describe the convex functions $\varphi$ such that, with large probability, there exists a one-to-one map $\pi$ from $\{1,\ldots,n\}$ to $\{1,\ldots,n\}$ for which $\sum_{i\leq n}\frac{1}{n}\varphi\big(\frac{X_i - X'_{\pi(i)}}{n^{-1/d}K_\varphi}\big) \leq 1,$ where $K_\varphi$ depends on $\varphi$ only.

The $k$th-order joint distribution for an ergodic finite-alphabet process can be estimated from a sample path of length $n$ by sliding a window of length $k$ along the sample path and counting frequencies of $k$-blocks. In this paper the problem of consistent estimation when $k = k(n)$ grows as a function of $n$ is addressed. It is shown that the variational distance between the true $k(n)$-block distribution and the empirical $k(n)$-block distribution goes to 0 almost surely for the class of weak Bernoulli processes, provided $k(n) \leq (\log n)/(H + \epsilon)$, where $H$ is the entropy of the process. The weak Bernoulli class includes the i.i.d. processes, the aperiodic Markov chains and functions thereof and the aperiodic renewal processes. A similar result is also shown to hold for functions of irreducible Markov chains. This work sharpens prior results obtained for more general classes of processes by Ornstein and Weiss and by Ornstein and Shields, which used the $\bar{d}$-distance rather than the variational distance.

An isoperimetric property of exponential distributions with respect to the supremum distance in $\mathbb{R}^n$ is proved and applied to stochastic processes linearly generated by i.i.d. positive random values.

This paper contains sharp norm, maximal, escape and exponential inequalities for stochastic integrals in which the integrator is either a nonnegative submartingale or a nonnegative supermartingale. Analogous inequalities hold for Ito processes and for smooth functions on Euclidean domains.

Let $\{X, X_n; n \geq 1\}$ be a sequence of i.i.d. random variables with values in a separable Banach spacc $B$ and set, for each $n, S_n = X_1 + \cdots + X_n$. We give necessary and sufficient conditions in order that $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=1}^n\|S_i\|^p < \infty \mathrm{a.s.},$ $\lim\sup_{n\rightarrow\infty} n^{-1-(p/2)}(2L_2n)^{-(p/2)}\sum_{i=0}^n\|S_n - S_i\|^p < \infty \mathrm{a.s.},$ where $p \geq 1$. Furthermore, the exact values of the above $\lim \sup$ are obtained. Some results are the extensions of Strassen's work to the vector settings and some are new even on the real line. The proofs depend on the construction of an independent sequence with values in $l_p(B)$ and appear as an illustration of the power of the limit law in Banach space.

Let $X_1, X_2,\ldots,X_n$ be independent mean-zero random variables and let $a_{ij}, 1 \leq i, j \leq n$, be an array of constants with $a_{ii} \equiv 0$. We present a method of obtaining the order of magnitude of $E\Phi(\sum_{1\leq i,j\leq n}a_{ij}X_iX_j)$ for any such $\{X_i\}$ and $\{a_{ij}\}$ and any nonnegative symmetric (convex) function $\Phi$ with $\Phi(0) = 0$ such that, for some integer $k \geq 0, \Phi(x^{2-k})$ is convex and simultaneously $\Phi(x^{2^{-k-1}})$ is concave on $\lbrack 0, \infty)$. The approximation is based on decoupling inequalities valid for all such mean-zero $\{X_i\}$ and reals $\{a_{ij}\}$ and a certain further "independentization" procedure.

In this article we derive a law of the logarithm for the maximal deviation between two kernel-type quantile density estimators and the true underlying quantile density function in the randomly right-censored case. Extensions to higher derivatives are included. The results are applied to get optimal bandwidths with respect to almost sure uniform convergence.

Let $\mathbf{X} = \{X_k\}$ be an i.i.d. real random sequence, let $\epsilon = \{\epsilon_k\}$ be a Rademacher sequence independent of $\mathbf{X}$ and let $\mathbf{a} = \{a_k\}$ be a deterministic real sequence. The aim of this paper is to prove that the mutual absolute continuity of probability measures induced by $\{X_k\}$ and $\{X_k + a_k\epsilon_k\}$ implies $\mathbf{a} \in \ell_4$. This is a generalization of a result of Shepp.

Suppose the random variables $(X^N, Y^N)$ on the probability space $(\Omega^N, \mathscr{F}^N, P^N)$ converge in distribution to the pair $(X, Y)$ on $(\Omega, \mathscr{F}, P)$, as $N \rightarrow \infty$. This paper seeks conditions which imply convergence in distribution of the conditional expectations $E^{P^N}\{F(X^N)\mid Y^N\}$ to $E^P\{F(X)\mid Y\}$, for all bounded continuous functions $F$. An absolutely continuous change of probability measure is made from $P^N$ to a measure $Q^N$ under which $X^N$ and $Y^N$ are independent. The Radon-Nikodym derivative $dP^N/dQ^N$ is denoted by $L^N$. Similarly, an absolutely continuous change of measure from $P$ to $Q$ is made, with Radon-Nikodym derivative $dP/dQ = L$. If the $Q^N$-distribution of $(X^N, Y^N, L^N)$ converges weakly to the $Q$-distribution of $(X, Y, L)$, convergence in distribution of $E^{P^N}\{F(X^N)\mid Y^N\}$ (under the original distributions) to $E^P\{F(X)\mid Y\}$ follows. Conditions of a uniform equicontinuity nature on the $L^N$ are presented which imply the required convergence. Finally, an example is given, where convergence of the conditional expectations can be shown quite easily.