Showing papers in "Annals of Probability in 1992"

Strict Stationarity of Generalized Autoregressive Processes

Abstract: In this paper we consider the multivariate equation $X_{n+1} = A_{n+1}X_n + B_{n+1}$ with i.i.d. coefficients which have only a logarithmic moment. We give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future. As an application we characterize the multivariate ARMA equations with general noise which have such a solution.

On the Behavior of Some Cellular Automata Related to Bootstrap Percolation

Abstract: convergence to total occupancy, and irc, the threshold for this convergence to occur exponentially fast. We locate these critical points for all the bootstrap percolation models, showing that they are both 0 when 1 d. For certain rules in which the orientation is important, we show that 0 < Pc = ir- < 1, by relating these systems to oriented site percolation. Finally, these oriented models are used to obtain an estimate for a critical exponent of these models. 1. Introduction. The fields of interacting particle systems, mathematical statistical mechanics and percolation have benefited very much from their interrelations. Here we study a family of models which has arisen in the interface among these areas. Cellular automata, such as those studied in this article, may be considered as interacting particle systems [see Liggett (1985) for a survey of this field]. The relations between the models that we consider and percolation will become clear in many of the proofs given, but can already be guessed from the fact that some of these systems are known as "bootstrap percolation." Finally, relations with statistical mechanics, while not so explicit in this article, were clearly present, for instance, in the article by Chalupa, Leath and Reich (1979), where bootstrap percolation was introduced in connection to disordered magnetic systems. Also in Aizenman and Lebowitz (1988) the motivation for studying these systems came from the (nonequilibrium statistical mechanics) problem of metastability. Our main concern in this article will be with the critical behavior of our models, that is, how their behavior changes qualitatively as some parameters cross certain values (critical points). As usual, one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

Uniform Convergence of Martingales in the Branching Random Walk

Abstract: In a discrete-time supercritical branching random walk, let $Z^{(n)}$ be the point process formed by the $n$th generation. Let $m(\lambda)$ be the Laplace transform of the intensity measure of $Z^{(1)}$. Then $W^{(n)}(\lambda) = \int e^{-\lambda x}Z^{(n)}(dx)/m(\lambda)^n$, which is the Laplace transform of $Z^{(n)}$ normalized by its expected value, forms a martingale for any $\lambda$ with $|m(\lambda)|$ finite but nonzero. The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit $W(\lambda)$ is actually an analytic function of $\lambda$. The uniform convergence results are used to obtain extensions of known results on the growth of $Z^{(n)}(nc + D)$ with $n$, for bounded intervals $D$ and fixed $c$. This forms the second part of the paper, where local large deviation results for $Z^{(n)}$ which are uniform in $c$ are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.

Random Walks, Capacity and Percolation on Trees

Abstract: A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a.s. In particular, we resolve a conjecture of Griffeath on the necessity of the Nash-Williams criterion. Fifth, for this last purpose, we establish several new basic results on branching processes in varying environments.

Internal Diffusion Limited Aggregation

Abstract: We study the asymptotic shape of the occupied region for an interacting lattice system proposed recently by Diaconis and Fulton. In this model particles are repeatedly dropped at the origin of the $d$-dimensional integers. Each successive particle then performs an independent simple random walk until it "sticks" at the first site not previously occupied. Our main theorem asserts that as the cluster of stuck particles grows, its shape approaches a Euclidean ball. The proof of this result involves Green's function asymptotics, duality and large deviation bounds. We also quantify the time scale of the model, establish close connections with a continuous-time variant and pose some challenging problems concerning more detailed aspects of the dynamics.

The Contact Process on Trees

Abstract: The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter $\lambda$ is varied. For small values of $\lambda$ a single infection eventually dies out. For larger $\lambda$ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of $\lambda$, and the proof of this is much easier than it is for the contact process on $d$-dimensional integer lattices.) For still larger $\lambda$ the infection converges in distribution to a nontrivial invariant measure. For any $n$-ary tree, with $n$ large, the first of these transitions occurs when $\lambda \approx 1/n$ and the second occurs when $1/2\sqrt{n} < \lambda < e/\sqrt{n}$. Nonhomogeneous trees whose vertices have degrees varying between 1 and $n$ behave essentially as homogeneous $n$-ary trees, provided that vertices of degree $n$ are not too rare. In particular, letting $n$ go to $\infty$, Galton-Watson trees whose vertices have degree $n$ with probability that does not decrease exponentially with $n$ may have both phase transitions occur together at $\lambda = 0$. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.

Compound Poisson Approximation for Nonnegative Random Variables Via Stein's Method

Abstract: The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$( u)$, where $ u$ is a finite positive measure on $(0, \infty)$. A number of results related to these distributions are established. These in turn are used in a number of examples to give bounds for the error in the compound Poisson approximation to the distribution of a sum of random variables.

Large Deviations for a Reaction-Diffusion Equation with Non-Gaussian Perturbations

Abstract: In this paper we establish a large deviations principle for the non-Gaussian stochastic reaction-diffusion equation (SRDE) $\partial_t u^\varepsilon = \mathscr{L} u^\varepsilon + f(x, u^\varepsilon) + \varepsilon\sigma(x, u^\varepsilon)\ddot{W}_{tx}$ as a random perturbation of the deterministic RDE $\partial_t u^0 = \mathscr{L} u^0 + f(x, u^0)$. Here the space variable takes values on the unit circle $S^1$ and $\mathscr{L}$ is a strongly-elliptic second-order operator with constant coefficients. The functions $f$ and $\sigma$ are sufficiently regular so that there is a unique solution to the above SRDE for any continuous initial condition. We also assume that there are positive constants $m$ and $M$ such that $m \leq \sigma(x, y) \leq M$ for all $x$ in $S^1$ and all $y$ in $\mathbb{R}$. The perturbation $\ddot{W}_{tx}$ is the formal derivative of a Brownian sheet. It is known that if the initial condition is continuous, then the solution will also be continuous, and moreover, if the initial condition is assumed to be Holder continuous of exponent $\kappa$ for some $0 < \kappa < \frac{1}{2}$, then the solution will be Holder continuous of exponent $\kappa/2$ as a function of $(t, x).$ In this paper we establish the large deviations principle for $ u^\varepsilon$ in the Holder norm of exponent $\kappa/2$ when the initial condition is Holder continuous of exponent $\kappa$ for any $0 < \kappa < \frac{1}{2}$, and when the initial condition is assumed only to be continuous, we establish the large deviations principle for $ u^\varepsilon$ in the supremum norm. Moreover, we prove that these large deviations principles are uniform with respect to the initial condition.

Rough Large Deviation Estimates for Simulated Annealing: Application to Exponential Schedules

Abstract: Simulated annealing algorithms are time inhomogeneous controlled Markov chains used to search for the minima of energy functions defined on finite state spaces. The control parameters, the so-called cooling schedule, control the probability that the energy should increase during one step of the algorithm. Most of the studies on simulated annealing have dealt with limit theorems, such as characterizing convergence conditions on the cooling schedule, or giving an equivalent of the law of the process for one fixed cooling schedule. In this paper we derive finite time estimates. These estimates are uniform in the cooling schedule and in the energy function. With new technical tools, we gain a new insight into the algorithm. We give a sharp upper bound for the probability that the energy is close to its minimum value. Hence we characterize the optimal convergence rate. This involves a new constant, the "difficulty" of the energy landscape. We calculate two cooling schedules for which our bound is almost reached. In one case it is reached up to a multiplicative constant for one energy function. In the other case it is reached in the sense of logarithmic equivalence uniformly in the energy function. These two schedules are both triangular: There is one different schedule for each finite simulation time. For each fixed finite time the second schedule has the currently used but previously mathematically unjustified exponential form. Finally, the title is "Rough large deviation estimates" because we have computed sharper ones (i.e., with sharp multiplicative constants) in two other papers.

The Cycle Structure of Random Permutations

Abstract: The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.

Random Walk in a Random Environment and First-Passage Percolation on Trees

Abstract: We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches per vertex. This generalizes and unifies previous work of the authors. It also shows that the point of phase transition for edge-reinforced random walk is likewise determined by the branching number of the tree. Finally, we show that the branching number determines the rate of first-passage percolation on trees, also known as the first-birth problem. Our techniques depend on quasi-Bernoulli percolation and large deviation results.

Universal schemes for prediction, gambling and portfolio selection'

Abstract: We discuss universal schemes for portfolio selection. When such a scheme is used for investment in a stationary ergodic market with unknown distribution, the compounded capital will grow with the same limiting rate as could be achieved if the infinite past and hence of the distribution of the market were known to begin with. By specializing the market to a Kelly horse race, we obtain a universal scheme for gambling on a stationary ergodic process with values in a finite set. We point out the connection between universal gambling schemes and universal modeling schemes that are used in noiseless data compression. We also discuss a universal prediction scheme to learn, from past experience, the conditional distribution given the infinite past of the next outcome of a stationary ergodic process with values in a Polish space. This generalizes Ornstein's scheme for finite-valued processes. Although universal prediction schemes can be used to obtain universal gambling and portfolio schemes, they are not necessary.

Weak Convergence of Sums of Moving Averages in the $\alpha$-Stable Domain of Attraction

Abstract: Skorohod has shown that the convergence of sums of i.i.d. random variables to an $\alpha$-stable Levy motion, with $0 < \alpha < 2$, holds in the weak-$J_1$ sense. $J_1$ is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-$J_1$ convergence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-$M_1$ convergence. $M_1$ is weaker than $J_1$, but it is strong enough for the $\sup$ and $\inf$ functionals to be continuous.

Sample Path Properties of the Local Times of Strongly Symmetric Markov Processes Via Gaussian Processes

Abstract: Necessary and sufficient conditions are obtained for the almost sure joint continuity of the local time of a strongly symmetric standard Markov process $X$. Necessary and sufficient conditions are also obtained for the almost sure global boundedness and unboundedness of the local time and for the almost sure continuity, boundedness and unboundedness of the local time in the neighborhood of a point in the state space. The conditions are given in terms of the 1-potential density of $X$. The proofs rely on an isomorphism theorem of Dynkin which relates the local times of Markov processes related to $X$ to a mean zero Gaussian process with covariance equal to the 1-potential density of $X$. By showing the equivalence of sample path properties of Gaussian processes with the related local times, known necessary and sufficient conditions for various sample path properties of Gaussian processes are carried over to the local times. The results are used to obtain examples of local times with interesting sample path behavior.

A Generalization of Holder's Inequality and Some Probability Inequalities

Abstract: The main result of this article is a generalization of the generalized Holder inequality for functions or random variables defined on lowerdimensional subspaces of n-dimensional product spaces. It will be seen that various other inequalities are included in this approach. For example, it allows the calculation of upper bounds for the product measure of n-dimensional sets with the help of product measures of lower-dimensional marginal sets. Furthermore, it yields an interesting inequality for various cumulative distribution functions depending on a parameter n e N. 1. Introduction. We first recall the generalized Holder inequality in terms of a measure-theoretic approach. Let (fQ, X, /,t) be a measure space and let LP(fb A, But) be the set of p-integrable (1 1 with E i lp-1= 1 and let f;ELp~(fVf ,), jM=1,...,m.

Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes

Abstract: Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(t), 0 \leq t \leq 1\}$ be the empirical and quantile processes generated by the first $n$ observations from an i.i.d. sequence of uniformly distributed random variables on (0,1). Let $0 < a_n < 1$ be a sequence of constants such that $a_n \rightarrow 0$ as $n \rightarrow \infty$. We investigate the strong limiting behavior as $n \rightarrow \infty$ of the increment functions $\{\alpha_n(t + a_ns) - \alpha_n(t), 0 \leq s \leq 1\}$ and $\{\beta_n(t + a_ns) - \beta_n(t), 0 \leq s \leq 1\},$ where $0 \leq t \leq 1 - a_n$. Under suitable regularity assumptions imposed upon $a_n$, we prove functional laws of the iterated logarithm for these increment functions and discuss statistical applications in the field of nonparametric estimation.

Phase-Type Representations in Random Walk and Queueing Problems

Abstract: The distributions of random walk quantities like ascending ladder heights and the maximum are shown to be phase-type provided that the generic random walk increment $X$ has difference structure $X = U - T$ with $U$ phase-type, or the one-sided assumption of $X_+$ being phase-type is imposed. As a corollary, it follows that the stationary waiting time in a GI/PH/1 queue with phase-type service times is again phase-type. The phase-type representations are characterized in terms of the intensity matrix $\mathbf{Q}$ of a certain Markov jump process associated with the random walk. From an algorithmic point of view, the fundamental step is the iterative solution of a fix-point problem $\mathbf{Q} = \psi(\mathbf{Q})$, and using a coupling argument it is shown that the iteration typically converges geometrically fast. Also, a variant of the classical approach based upon Rouche's theorem and root-finding in the complex plane is derived, and the relation between the approaches is shown to be that $\mathbf{Q}$ has the Rouche roots as its set of eigenvalues.

Ergodic Theory of Stochastic Petri Networks

Abstract: Stochastic Petri nets are a general formalism for describing the dynamics of discrete event systems. The present paper focuses on a subclass of stochastic Petri nets called stochastic event graphs. Under the assumption that the variables used for the timing of an event graph form stationary and ergodic sequences of random variables, we make use of an associated stochastic recursive equation in order to construct its stationary and ergodic regime. In particular, we determine the conditions under which the existence of this regime is guaranteed.

Decoupling and Khintchine's Inequalities for $U$-Statistics

Abstract: In this paper we introduce a fairly general decoupling inequality for $U$-statistics. Let $\{X_i\}$ be a sequence of independent random variables in a measurable space $(S, \mathscr{J})$, and let $\{\tilde{X}_i\}$ be an independent copy of $\{X_i\}$. Let $\Phi(x)$ be any convex increasing function for $x \geq 0$. Let $\Pi_{ij}$ be families of functions of two variables taking $(S \times S)$ into a Banach space $(D, \|\cdot\|)$. If the $f_{ij} \in \Pi_{ij}$ are Bochner integrable and $\max_{1\leq i eq j\leq n} E\Phi\big(\sup_{f_{ij}\in\Pi_{ij}}\|f_{ij}(X_i, X_j)\|\big) < \infty,$ then, under measurability conditions, $E\Phi\big(\sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i eq j\leq n} f_{ij}(X_i, X_j)\big\|\big) \leq E\Phi\big(8 \sup_{\mathbf{f}\in\mathbf{\Pi}}\big\|\sum_{1\leq i eq j\leq n} f_{ij}(X_i, \tilde{X}_j)\big\|\big),$ where $\mathbf{f} = (f_{ij}, 1 \leq i eq j \leq n)$ and $\mathbf{\Pi} = (\Pi_{ij}, 1 \leq i eq j \leq n)$. In the case where $\mathbf{\Pi}$ is a family of functions of two variables satisfying $f_{ij} = f_{ji}$ and $f_{ij}(X_i, X_j) = f_{ij}(X_j, X_i)$, the reverse inequality holds (with a different constant). As a corollary, we extend Khintchine's inequality for quadratic forms to the case of degenerate $U$-statistics. A new maximal inequality for degenerate $U$-statistics is also obtained. The multivariate extension is provided.

Large Deviations for Exchangeable Random Vectors

Abstract: Say that a family $\{P_\theta^n: \theta \in \Theta\}$ of sequences of probability measures is exponentially continuous if whenever $\theta_n \rightarrow \theta$, the sequence $\{P_{\theta_n}^n\}$ satisfies a large deviation principle with rate function $\lambda_\theta$. If $\Theta$ is compact and $\{P_\theta^n\}$ is exponentially continuous, then the mixture $P^n(A) =: \int_\Theta P_\theta^n(A)d\mu(\theta)$ satisfies a large deviation principle with rate function $\lambda(x) =: \inf\{\lambda_\theta(x): \theta \in S(\mu)\}$, where $S(\mu)$ is the support of the mixing measure $\mu$. If $X_1,X_2,\ldots$ is a sequence of i.i.d. random vectors, $\{\bar{X}_n\}$ the corresponding sequence of sample means and $P_\theta^n =: P_\theta\circ\bar{X}^{-1}_n$, then $\{P_\theta^n\}$ is exponentially continuous if the classical rate function $\lambda_\theta( u)$ is jointly lower semicontinuous and a uniform integrability condition introduced by de Acosta is satisfied. These results are applied in Section 4 to derive a large deviation theory for exchangeable random variables; the resulting rate functions are typically nonconvex. If the parameter space $\Theta$ is not compact, then examples can be constructed where a full large deviation principle is not satisfied because the upper bound fails for a noncompact set.

Stability in Distribution for a Class of Singular Diffusions

Abstract: A verifiable criterion is derived for the stability in distribution of singular diffusions, that is, for the weak convergence of the transition probability $p(t; x, dy)$, as $t \rightarrow \infty$, to a unique invariant probability. For this we establish the following: (i) tightness of $\{p(t; x, dy): t \geq 0\}$; and (ii) asymptotic flatness of the stochastic flow. When specialized to highly nonradial nonsingular diffusions the results here are often applicable where Has'minskii's well-known criterion fails. When applied to traps, a sufficient condition for stochastic stability of nonlinear diffusions is derived which supplements Has'minskii's result for linear diffusions. We also answer a question raised by L. Stettner (originally posed to him by H. J. Kushner): Is the diffusion stable in distribution if the drift is $Bx$ where $B$ is a stable matrix, and $\sigma(\cdot)$ is Lipschitzian, $\sigma(\underline{0}) eq 0$? If not, what additional conditions must be imposed?

An Extension of Pitman's Theorem for Spectrally Positive Levy Processes

Abstract: If $X$ is a spectrally positive Levy process, $\bar{X}^c$ the continuous part of its maximum process, and $J$ the sum of the jumps of $X$ across its previous maximum, then $X - 2\bar{X}^c - J$ has the same law as $X$ conditioned to stay negative. This extends a result due to Pitman, who links the real Brownian motion and the three-dimensional Bessel process. Several other relations between the Brownian motion and the Bessel process are extended in this setting.

Self-Avoiding Walk on a Hierarchical Lattice in Four Dimensions

Abstract: We define a Levy process on a $d$-dimensional hierarchical lattice. By construction the Green's function for this process decays as $|x|^{2-d}$. For $d = 4$, we prove that the introduction of a sufficiently weak self-avoidance interaction does not change this decay provided the mass $\equiv$ "killing" rate is chosen in a special way, so that the process is critical.

Frechet Differentiability, $p$-Variation and Uniform Donsker Classes

Abstract: Classically, a stastistical functional is defined on a space of distribution F on the real line with the supremum norn. The values may be real or themselves functions such as the quantile function \(F^{-1}\). Nonlinear functionals are studied via their derivatives. This paper and a related one [Dudley (1991a)] will show that the differentiability properties originally proved by Reeds (1976), also treated in Fernholz (1983), can be improved substantially.

Asymptotic Series and Exit Time Probabilities

Abstract: This paper is concerned with accurate asymptotic estimates for exit time probabilities associated with nearly deterministic Markov diffusions. The exit time probabilities are expressed as asymptotic series of WKB type in a small parameter, which measures the strength of the random Brownian motion inputs. This series is valid in certain regions in which the minimum action function $u(x,s)$ is a smooth function of state $x$ and time $s$. The function $u$ is a solution to the corresponding Hamilton-Jacobi PDE of first order.

Large Deviations for a Class of Anticipating Stochastic Differential Equations

Abstract: Consider the family of perturbed stochastic differential equations on $\mathbb{R}^d$, $X^\varepsilon_t = X^\varepsilon_0 + \sqrt{\varepsilon} \int^t_0\sigma(X^\varepsilon_s)\circ dW_s + \int^t_0 b(X^\varepsilon_s) ds,$ $\varepsilon > 0$, defined on the canonical space associated with the standard $k$-dimensional Wiener process $W$. We assume that $\{X^\varepsilon_0, \varepsilon > 0\}$ is a family of random vectors not necessarily adapted and that the stochastic integral is a generalized Stratonovich integral. In this paper we prove large deviations estimates for the laws of $\{X^\varepsilon_., \varepsilon > 0\}$, under some hypotheses on the family of initial conditions $\{X^\varepsilon_0, \varepsilon > 0\}$.

Isoperimetric Inequalities and Transient Random Walks on Graphs

Abstract: recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of Z2 is transient. More precisely, if f(k) denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices, then the graph is transient if the infinite sum E f(k)-2 converges. This can be applied to parabolicity versus hyperbolicity of surfaces. 1. Introduction. Let G be a connected graph which is locally finite, that is, all vertices have finite degree. We consider a random walk starting at a vertex v, say, such that at any vertex u, the walk proceeds to a neighbour with probability 1/d(u), where d(u) is the degree (i.e., the number of neighbours) of u. The graph G is recurrent if we revisit v with probability 1. Otherwise G is transient. It is well known that the three-dimensional grid Z3 is transient while Z2 is recurrent. More generally, Nash-Williams [13] (see also [4, 10]) proved that any graph with smaller growth rate than Z2 is recurrent. Lyons [10] showed that certain subgraphs of grids are transient provided they grow just a little faster than Z2. Other results, in terms of isoperimetric inequalities, supporting the statement that Z2 is, in a sense, an "extreme" recurrent graph, can be derived from work of Fernandez [5], Grigor'yan [7] and Varopoulos [15]. (Varopoulos [16] used results of Gromov [8] to characterize completely the recurrent Cayley graphs.) We shall carry these results further. If V is a vertex set in G, then aV will denote the boundary of V, that is, the set of vertices of V having neighbours outside of V. Let f be a nondecreasing positive real function defined on the natural numbers. We say that G satisfies an f-isoperimetric inequality if there exists a constant c > 0 such that, for each finite vertex set V of G,

Superdiffusions and Parabolic Nonlinear Differential Equations

Abstract: We establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations. Analytic results due to Brezis and Friedman, Baras and Pierre and others are used to investigate the graphs of superdiffusions. A survey of the literature and general comments are presented in Section 4.

Automorphism Invariant Measures on Trees

Abstract: Consider a collection of real-valued random variables indexed by the integers. It is well known that such a process can be stationary, that is, translation invariant, and ergodic and yet have very strong associations: The one-sided tail field may determine the sample; the measure may fail to be mixing in any sense; the weak law of large numbers may fail on some infinite subset of the integers. The main result of this paper is that this cannot happen if the integers are replaced by an infinite homogeneous tree and the translations are replaced by all graph automorphisms. In fact, any automorphism-invariant process indexed by the tree is a mixture of extremal processes whose one-sided tail fields are trivial, from which the mixing properties follow.

Semi-Min-Stable Processes

Abstract: We define a semi-min-stable (SMS) process $Y(t)$ in $\lbrack 0,\infty)$ to be one which is stable under the simultaneous operations of taking the minima of $n$ independent copies of $Y(t)$ (pointwise over time $t$) and rescaling space and time. We show that the only possible rescaling of time is by a fixed power of $n$ and that SMS processes are essentially the only possible weak limits for large $m$ of a process obtained by taking the minimum, pointwise over $t$, of $m$ independent copies of a given process and then rescaling space and time. We describe the representation of a SMS process as the minimum of a Poisson process on a function space. We obtain a partial characterization of sample continuous SMS processes, similar to that of de Haan in the case of max-stable processes.