Showing papers in "Probability Theory and Related Fields in 1992"

Size-biased sampling of Poisson point processes and excursions

Abstract: Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

Smoothed cross-validation

Abstract: For bandwidth selection of a kernel density estimator, a generalization of the widely studied least squares cross-validation method is considered. The essential idea is to do a particular type of “presmoothing” of the data. This is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error. Analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance. The rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions. The method is more appealing than other selectors with this property, because its motivation is not heavily dependent on precise asymptotic analysis, and because its form is simple and intuitive. Theory is also given for choice of the amount of presmoothing, and this is used to derive a data-based method for this choice.

Asymptotics in the random assignment problem

Abstract: We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.

Transition densities for Brownian motion on the Sierpinski carpet

Abstract: Upper and lower bounds are obtained for the transition densitiesp(t, x, y) of Brownian motion on the Sierpinski carpet. These are of the same form as those which hold for the Sierpinski gasket. In addition, the joint continuity ofp(t, x, y) is proved, the existence of the spectral dimension is established, and the Einstein relation, connecting the spectral dimension, the Hausdorff dimension and the resistance exponent, is shown to hold.

Dirichlet forms on fractals: Poincaré constant and resistance

Abstract: We study Dirichlet forms associated with random walks on fractal-like finite grahs. We consider related Poincare constants and resistance, and study their asymptotic behaviour. We construct a Markov semi-group on fractals as a subsequence of random walks, and study its properties. Finally we construct self-similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries.

White noise driven quasilinear SPDEs with reflection

Abstract: We study reflected solutions of the heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by an additive space-time white noise. Roughly speaking, at any point (x, t) where the solutionu(x, t) is strictly positive it obeys the equation, and at a point (x, t) whereu(x, t) is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. An existence and uniqueness result is proved, which relies heavily on new results for a deterministic variational inequality.

Shock fluctuations in asymmetric simple exclusion

Abstract: The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities ϱ<λ to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (ϱ=0) we prove, in the scale t1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Abstract: The asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the Malliavin calculus. From this the second order efficiency of the maximum likelihood estimator is proved.

Ergodic theorems for the multitype contact process

Abstract: This paper studies a process involving competition of two types of particles (1 and 2) for the empty space (0). Each site of the latticeZ d is therefore in one of three possible states: 0, 1, or 2. Particles of each type die with rate 1, while an empty site becomes occupied by a particle of typei with rate λ i (proportion of neighbors of typei). The set of neighbors of a sitex is of the form {y:‖x−y‖≦J}, for a positive integerJ and a norm ‖·‖. Assuming there are only 0's and 1's present at the beginning, the process reduces to the contact process, with the critical rate of survival of 1's being λ c . The basic problem we address is the existence of equilibria in which both types of particles coexist. Without loss of generality, one can restrict to the case λ2 ≧ λ1 > λ c and in this case we show: (1) If λ2 > λ1, and the initial state is translation invariant and contains infinitely many 2's, then the 1's go away and the process approaches the invariant measure of the contact process with only 2's and 0's present, (2) If λ2 = λ1, andd≦2, then clustering occurs: starting from a translation invariant initial measure with no mass on all 0's, the process converges weakly to a convex combination of the two invariant measures obtained with only one type of particles present, and (3) If λ2 = λ1, andd≧3, then there is a one-parameter family of invariant measures including both types.

Asymptotics for Euclidean minimal spanning trees on random points

Abstract: Asymptotic results for the Euclidean minimal spanning tree onn random vertices inRd can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in allRd. In particular we prove a conjecture of Robert Bland: the sum of thed'th powers of the edge-lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofRd tends to a constant asn→∞.

Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

Abstract: We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤ d or ℤ d × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤ d or on a slab ℤ d × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.

Vertex-reinforced random walk

Abstract: This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix,R, which is real, symmetric and nonnegative. LetS i (n) keep track of the number of visits to statei up to timen, and form the fractional occupation vector,V(n), where $$v_i (n) = {{S_i (n)} \mathord{\left/ {\vphantom {{S_i (n)} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}} \right. \kern- ulldelimiterspace} {\left( {\sum\limits_{j = 1}^d {S_j (n)} } \right)}}$$ . It is shown thatV(n) converges to to a set of critical points for the quadratic formH with matrixR, and that under nondegeneracy conditions onR, there is a finite set of points such that with probability one,V(n)→p for somep in the set. There may be more than onep in this set for whichP(V(n)→p)>0. On the other handP(V(n)→p)=0 wheneverp fails in a strong enough sense to be maximum forH.

Comparison methods for a class of function valued stochastic partial differential equations

Abstract: A comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDE's) with Lipschitz coefficients driven by cylindrical and regular Hilbert space valued Brownian motions. Moreover, we obtain necessary and sufficient conditions for the positivity of the mild solutions of the SPDE's where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDE's, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDE's on the cone of positive functions with non-Lipschitz drift parts including the case of arbitrary polynomialsR(x) withR(O)≧O and leading negative coefficient.

Behavior of droplets for a class of Glauber dynamics at very low temperature

Abstract: We consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion.

Brownian motion on a homogeneous random fractal

Abstract: We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd s=2d f/dw. A branching process in a random environment can be used to deduce some of the sample path properties of the process.

Some asymptotics for multimodality tests based on kernel density estimates

Abstract: A test due to B.W. Silverman for modality of a probability density is based on counting modes of a kernel density estimator, and the idea of critical smoothing. An asymptotic formula is given for the expected number of modes. This, together with other methods, establishes the rate of convergence of the critically smoothed bandwidth. These ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values.

Asymptotic behavior of Brownian polymers

Abstract: We consider a system that models the shape of a growing polymer. Our basic problem concerns the asymptotic behavior ofX t , the location of the end of the polymer at timet. We obtain bounds onX t in the (physically uninteresting) case thatd=1 and the interaction functionf(x)≥0. If, in addition,f(x) behaves for largex likeCx −β with β<1 we obtain a strong law that gives the exact growth rate.

The compact support property for solutions to the heat equation with noise

Abstract: We consider all solutions of a martingale problem associated with the stochastic pde $$u_t = \tfrac{1}{2}u_{xx} + u^\gamma \dot W$$ and show thatu(t,·) has compact support for allt≧0 ifu(0,·) does and if γ<1. This extends a result of T. Shiga who derived this compact support property for γ≦1/2 and complements a result of C. Mueller who proved this property fails if γ≧1.

Bootstrap, wild bootstrap, and asymptotic normality

Abstract: We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend onn. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap.

The Wigner semi-circle law and eigenvalues of matrix-valued diffusions

Abstract: A system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent Ornstein-Uhlenbeck processes is derived. This corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter-particle repulsion. The associated empirical distribution of particles is shown to converge weakly (as the number of particles tends to infinity) to a limiting measure-valued process which may be characterized as the weak solution of a deterministic ODE. The Wigner semi-circle density is found to be one of the equilibrium points of this limiting equation.

Branching random walk in random environment: phase transitions for local and global growth rates

Abstract: Let (η n ) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionF x and each of its members independently jumps to sitex±1 with probability (1±h)/2,h∈[0, 1]. The sequence {F x } is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη 0 is shift invariant and ergodic. Two quantities are considered:

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Abstract: In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) ∂ t v e =ℒv e +f(x,v e )+eσ(x,v e ) $$\ddot W_{tx} $$ . Here ℒ is a strongly-elliptic second-order operator with constant coefficients, ℒh:=DH xx-αh, and the space variablex takes values on the unit circleS 1. The functionsf and σ are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0

Lifetimes of conditioned diffusions

Abstract: We investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non-divergence form can be found. The critical value of the parameter is found for each of the following classes of domains:L p -domains (p=n−1), uniformly regular twistedL p -domains (p=n−1), and twisted Holder domains (α=1/3). A related parabolic boundary Harnack principle is proved.

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Abstract: The Skorohod oblique reflection problem for (D, Γ, w) (D a general domain in ℝ d , Γ(x),x∈∂D, a convex cone of directions of reflection,w a function inD(ℝ+,ℝ d )) is considered. It is first proved, under a condition on (D, Γ), corresponding to Γ(x) not being simultaneously too large and too much skewed with respect to ∂D, that given a sequence {wn} of functions converging in the Skorohod topology tow, any sequence {(xn, ϕn)} of solutions to the Skorohod problem for (D, Γ, wn) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, Γ, w). Next it is shown that if (D, Γ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, Γ, w) exist for everyw∈D(ℝ+,ℝ d ) with small enough jump size. The requirement is met in the case when ∂D is piecewiseC b 1 , Γ is generated by continuous vector fields on the faces ofD and Γ(x) makes and angle (in a suitable sense) of less than π/2 with the cone of inward normals atD, for everyx∈∂D. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.

Some exact equivalents for the Brownian motion in Hölder norm

Abstract: Some exact equivalents of small probabilities are given for the Wiener measure on spaces of Holder paths. It turns out that most of them are easier to derive than their counterparts in the uniform norm because of a classical result of Z. Ciesielski which makes the Brownian motion on these spaces easy to handle. In particular we study the equivalents of the probability of eB in a fixed ball, ofB in a small ball and we give applications to the speed of clustering in Strassen law.

Large deviations for Markov processes with discontinuous statistics, II: random walks

Abstract: Letμ 1 andμ 2 be Borel probability measures on ℝ d with finite moment generating functions. The main theorem in this paper proves the large deviation principle for a random walk whose transition mechanism is governed byμ 1 when the walk is in the left halfspace Λ1 = {x∈ℝ d :x 1≦0} and whose transition mechanism is governed byμ 2 when the walk is in the right halfspace Λ2 = {x∈ℝ d :x 1>0}. When the measuresμ 1 andμ 2 are equal, the main theorem reduces to Cramer's Theorem.

On reflected Dirichlet spaces

Abstract: Reflecting diffusion processes on smooth domains in Euclidean space are well understood. Silverstein in [22] and [23] developed two variant procedures for constructing the reflected processes for a general class of symmetric Hunt processes from a Dirichlet space point of view. A direct approach is given in this paper and these two variant procedures are shown to yield the same result. Only the techniques of martingales and ordinary Markov processes are used.

Local asymptotic mixed normality for semimartingale experiments

Abstract: We give conditions for local asymptotic mixed normality of experiments when the observed process is a semimartingale and the observation time increases to infinity. As a consequence we obtain asymptotic efficiency of various estimators. Several special models for counting process,s, diffusion processes and diffusions with jumps are studied.

Skorohod stochastic differential equations of diffusion type

Abstract: Leta, b beC2(R1)-functions with bounded derivatives of first and second order. We study stochastic differential equations $$dX_t = a(X_t )dW_t + b(X_t )dt,0 \leqq t \leqq 1,$$ whose initial valueX0 is a Frechet differentiable random variable which may depend on the whole path of the driving Brownian motion (Wt). This anticipation requires to pass from the Ito-integral to the Skorohod-integral. We show that the equation has a unique local solution {Xt(ω), 0≦t≦t0(ω)}, for sufficiently smallt0(ω)>0, and we provide conditions for the existence of a global solution {Xt(ω), 0≦t≦1}.

Local times on curves and uniform invariance principles

Abstract: Sufficient conditions are given for a family of local times |L t µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and μ. Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ℝ2 and ℝ2.