Showing papers in "Probability Theory and Related Fields in 1991"

The decomposition and measurement of the interdependency between second-order stationary processes

Abstract: For a given pair of multivariate stationary processes, the process of one-way effect is extracted from each of the processes. Each process is decomposed into two orthogonal processes, namely, into the process generated by the one-way effect of the other process and the process orthogonal to it. Based on the decomposition, three measures characterizing the interdependency of the pair of processes are introduced. They are the measure of association, the measure of one-way effect and the measure of reciprocity. Each of the measures is defined as overall as well as frequencywise measure. The paper shows that the measure of association is equal to the sum of the others. It discusses the relationships of those measures to the ones proposed by Gel'fand-Yaglom and by Geweke.

Stochastic differential equations in infinite dimensions : solutions via Dirichlet forms

Abstract: Using the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type $$dX_t = dW_t + \beta (X_t )dt$$ for possibly very singular drifts β. Here (X t ) t ≧0 takes values in some topological vector spaceE and (W t ) t ≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

Self-similar processes with independent increments

Abstract: A stochastic process {X t ∶t ≧0} onR d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X ct } and {aX t +b(t)} have common finite-dimensional distributions. If {X t } is widesense self-similar with independent increments, stochastically continuous, andX 0=const, then, for everyt, the distribution ofX t is of classL. Conversely, if μ is a distribution of classL, then, for everyH>0, there is a unique process {X } selfsimilar with exponentH with independent increments such thatX 1 has distribution μ. Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X } (called the process of classL with exponentH induced by μ) are compared with those of the Levy process {Y t } such thatY 1 has distribution μ. Results are generalized to operator-self-similar processes and distributions of classOL. A process {X t } onR d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA c and a functionb c (t) such that {X ct } and {A c X t +b c (t)} have common finite-dimensional distributions. It is proved that, if {X t } is wide-sense operator-self-similar and stochastically continuous, then theA c can be chosen asA c =c Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].

Decroissance exponentielle du noyau de la chaleur sur la diagonale (I)

Abstract: We give some conditions for the heat kernel to have an asymptotic expansion in small time such that all coefficients vanish, although the phenomenon seems difficult to understand by large deviations theory The fact that the leading term is not zero is strongly related to Bismut's condition These examples are related to the Varadhan estimates of the density of a dynamical system submitted to small random perturbations To understand that type of asymptotic, one must modify the definition of the distance by adding the Bismut condition (unnoticed, but hidden, in classical cases)

A probabilistic approach to one class of nonlinear differential equations

Abstract: We establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.

Lower bounds for bandwidth selection in density estimation

Abstract: This paper establishes asymptotic lower bounds which specify, in a variety of contexts, how well (in terms of relative rate of convergence) one may select the bandwidth of a kernel density estimator. These results provide important new insights concerning how the bandwidth selection problem should be considered. In particular it is shown that if the error criterion is Integrated Squared Error (ISE) then, even under very strong assumptions on the underlying density, relative error of the selected bandwidth cannot be reduced below ordern −1/10 (as the sample size grows). This very large error indicates that any technique which aims specifically to minimize ISE will be subject to serious practical difficulties arising from sampling fluctuations. Cross-validation exhibits this very slow convergence rate, and does suffer from unacceptably large sampling variation. On the other hand, if the error criterion is Mean Integrated Squared Error (MISE) then relative error of bandwidth selection can be reduced to ordern −1/2, when enough smoothness is assumed. Therefore bandwidth selection techniques which aim to minimize MISE can be much more stable, and less sensitive to small sampling fluctuations, than those which try to minimize ISE. We feel this indicates that performance in minimizing MISE, rather than ISE, should become the benchmark for measuring performance of bandwidth selection methods.

On convergence rates of suprema

Abstract: It is shown that the convergence rate of suprema of stationary Gaussian and related processes, such as processes defined by the empirical distribution function, is logarithmically slow, even if the rates are to be uniform over as few as three points. It is proved that the bootstrap approximation provides a substantial improvement.

Percolation in half-spaces: equality of critical densities and continuity of the percolation probability

Abstract: Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets ℕ of ℤ d ,d≧2, yielding: Corollaries of these results include uniqueness of the infinite cluster for such ℕ's and sufficiency of the following for proving continuity of the full-space phase transition: showing that percolation in the full-space at densityp implies percolation in the half-space at thesame density.

Quantum random walk on the dual of SU (n)

Abstract: We study a quantum random walk onA(SU(n)), the von Neumann algebra of SU(n), obtained by tensoring the basic representation of SU(n). Two classical Markov chains are derived from this quantum random walk, by restriction to commutative subalgebras ofA(SU(n)), and the main result of the paper states that these two Markov chains are related by means of Doob'sh-processes.

The asymptotic distributions of generalized U-statistics with applications to random graphs

Abstract: We consider the random variable $$S_{n,v} (f) = \sum\limits_{i_1< ...< i_v \leqq n} {f(X_{i_1 } ,...,X_{i_v } ,Y_{i_1 i_2 } ,...,Y_{i_{v - 1} i_v } ),}$$

Path processes and historical superprocesses

Abstract: A superprocessX over a Markov process ξ can be obtained by a passage to the limit from a branching particle system for which ξ describes the motion of individual particles.The historical process\(\hat \xi \) for ξ is the process whose state at timet is the path of ξ over time interval [0,t]. The superprocess\(\hat X\) over\(\hat \xi \)the historical superprocess over ξ—reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process\(\hat \xi \) is that\(\hat \xi _s \) is a function of\(\hat \xi _t \) for alls

Diffusion equation techniques in stochastic monotonicity and positive correlations

Abstract: A diffusion equation approach is investigated for the study of stochastic monotonicity, positive correlations and the preservation of Lipschitz functions. Necessary and sufficient conditions are given for diffusion semigroups to be stochastically monotonic and to preserve the class of positively correlated measures. Applications are given which discuss the shape of the ground state for Schrodinger operators-Δ+V with FKG potentialsV.

Limit distributions for minimal displacement of branching random walks

Abstract: We study the minimal displacement (X n ) of branching random walk with non-negative steps. It is shown that (X n −EX n ) is tight under a mild moment condition on the displacements. For supercritical B.R.W. (X n ) converges almost surely. For critical B.R.W. we determine the possible limit points of (X n −EX n ), and we prove a generalization of Kolmogorov's theorem on the extinction probability of a critical branching process. Finally we generalize Bramson's results on the almost sure convergence ofX n log 2/log logn.

Long time existence for the heat equation with a noise term

Abstract: We consider the equationu t =u xx +u γ W forx on a finite interval, with Dirichlet boundary conditions. W is spacetime white noise. The initial condition is continuous and nonnegative. We show existence and uniqueness for all time, provided 1 ≧γ<3/2.

Some dimension-free features of vector-valued martingales

Abstract: Given any local maringaleM inR d orl 2, there exists a local martingaleN inR 2, such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR 2 which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|2 and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.

A note on superprocesses

Abstract: Subject to a mild restriction onA, generator of the one-particle motion, we show theA-Fleming-Viot superprocess can be obtained from theA-Dawson-Watanabe superprocess by conditioning the latter to have constant total mass.

Linear skorohod stochastic differential equations

Abstract: Let σ ∈ C 4 b(R1). We provide assumptions on the random variable G and the process b = (b t (x)) possibly anticipating the driving Wiener process (W t ) under which the anticipative stochastic differential equation with Skorohod integral 21-1

Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales

Abstract: In this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors. In the first part we prove conditional versions of the Strassen-Dudley theorem. We apply these in the second part of the paper to obtain strong invariance principles for vector-valued martingales which, when properly normalized, converge in law to a mixture of Gaussian distributions.

Anticipative Girsanov transformations

Abstract: The transformations of measures induced by\(\left( {W + \int\limits_0^ \bullet {K_s ds} } \right)\) with (Ks) possibly anticipating the Wiener process (Ws) is discussed and a Girsanovtype theorem under rather weak assumptions on (Ks) is derived.

Asymptotic behaviour of densities of stable semigroups of measures

Abstract: We prove that densities of the measures in a strictly stable semigroup (ht) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour: $$\mathop {\lim |}\limits_{|x| \to \infty } x|^{Q + \alpha } \cdot h_1 (x) = k(\bar x),$$ where α is the characteristic exponent,\(\bar x = |x|^{ - 1} x\), andk is the density of the Levy measure associated to the semigroup. Moreover, if\(k(\bar x) = 0\) a more precise description is given.

Gaussian fluctuations of connectivities in the subcritical regime of percolation

Abstract: We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp pc, the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.

Rates of clustering for some Gaussian self-similar processes

Abstract: The analogue of Strassen's functional law of the iterated logarithm in known for many Gaussian processes which have suitable scaling properties, and here we establish rates at which this convergence takes place. We provide a new proof of the best upper bound for the convergence toK by suitably normalized Brownian motion, and then continue with this method to get similar bounds for the Brownian sheet and other self-similar Gaussian processes. The previous method, which produced these results for Brownian motion in ℝ1, was highly dependent on many special properties unavailable when dealing with other Gaussian processes.

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Abstract: This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.

A lim inf result in Strassen's law of the iterated logarithm

Abstract: For a functionf(.) from Strassen's class, we investigate the lim inf behaviour of its distance from the normalized trajectories of a Wiener process. The lim inf rate is expressed in terms of a certain functional off(.). In addition, we give a result on the lim inf behaviour of the distance of the normed trajectories from Strassen's class as a whole.

Convergence in distribution and Skorokhod Convergence for the general theory of processes

Abstract: This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X n ,ℱ n ),n∈ℕ, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y n ,ℊ n ),n∈ℕ, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X n ,ℊ n ) has the same stochastic properties as (X n ,ℱ n ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y n ,ℊ n ),n∈ℕ, and their limit (Y, ℊ) may be taken as all having the same fixed filtrationℊ n =ℊ. In the continuous time case, theY n ,ℊ n may require different filtrationsℊ n , which converge toℊ. To handle this, convergence of filtrations is defined and its theory developed.

Critical branching in a highly fluctuating random medium

Abstract: A model of one-dimensional critical branching (superprocess) is constructed in a random medium fluctuating both in time and space. The medium describes a moving system of point catalysts, and branching occurs only in the presence of these catalysts. Although the medium has an infinite overall density, the clumping features of the branching model can be exhibited by rescaling time, space, and mass by an exactly calculated scaling power which is stronger than in the constant medium case. The main technique used is the asymptotic analysis of a generalized diffusion-reaction equation in the space-time random medium, which (given the medium) prescribes the evolution of the Laplace transition functional of the Markov branching process.

Random recursive construction of self-similar fractal measures. The noncompact case

Abstract: The self-similarity of sets (measures) is often defined in a constructive way. In the present paper it will be shown that the random recursive construction model of Falconer, Graf and Mauldin/Williams for (statistically) self-similar sets may be generalized to the noncompact case. We define a sequence of random finite measures, which converges almost surely to a self-similar random limit measure. Under certain conditions on the generating Lipschitz maps we determine the carrying dimension of the limit measure.

Consistent polygonal fields

Abstract: We extend our results [3] on the construction of polygonal Markov fields on the plane taking finite number of values and having a given symmetric Markov process η as the marginal process on each line, to the case when η is reversible.

The Lifschitz singularity for the density of states on the Sierpinski gasket

Abstract: We prove the existence of the density of states for the Laplacian on the infinite Sierpinski gasket. Then the Lifschitz-type singularity of the density of states is established. We also investigate the long-time asymptotics of the Brownian trajectory on the Sierpinski gasket, getting bounds similar to those in the ℝd-case.

Itô's lemma without non-anticipatory conditions

Abstract: A stochastic integral (with respect to Brownian motion) which extends Ito's integral to anticipatory integrands is constructed and investigated. This stochastic integral is different from the Skorokhod integral. The Ito lemma is proved for this integral.