Showing papers in "Probability Theory and Related Fields in 1988"

Brownian motion on the Sierpinski gasket

Abstract: We construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of ℝ2, and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, p t(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on p t(x,y).

Stochastic calculus with anticipating integrands

Abstract: We study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended Ito formula. We also introduce an extension of Stratonovich's integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement.

On the limit of the largest eigenvalue of the large dimensional sample covariance matrix

Abstract: In this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity. The above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products (SP) matrix exist.

On the exceedance point process for a stationary sequence

Abstract: It is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.

Stochastic partial differential equations for some measure-valued diffusions

Abstract: We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R 1, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.

Uniqueness in law for pure jump Markov processes

Abstract: Let A be the operator defined on C2 functions by $$Af\left( x \right) = \smallint \left[ {f\left( {x + h} \right) - f\left( x \right) - f'\left( x \right)h 1_{([ - 1, 1])} \left( h \right)} \right]v\left( {x,dh} \right).$$ Sufficient conditions are given for existence and uniqueness for the martingale problem associated with A. In the case of stable-like processes, where v(x, dh) is equal to the Levy measure for the stable symmetric process of index α(x) for each x, the conditions reduce to α(x) continuous for existence and α(x) Dini continuous for uniqueness.

On bilinear forms in Gaussian random variables and Toeplitz matrices

Abstract: We improve a result of Szegő on the asymptotic behaviour of the trace of products of Toeplitz matrices.

KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees

Abstract: If R t is the position of the rightmost particle at time t in a one dimensional branching brownian motion, u(t, x)=P(R t >x) is a solution of KPP equation: $$\frac{{\partial u}}{{\partial t}} = \frac{1}{2}\frac{{\partial ^2 u}}{{\partial x^2 }} + f(u)$$ where f(u)=α(1-u-g(1-u)) g is the generating function of the reproduction law and α the inverse of the mean lifetime; if m=g′(1)>1 and g(0)=0, it is known that: $$\frac{{R_t }}{t}\xrightarrow{P}c_0 = \sqrt {2a(m - 1)} ,{\text{ when }}t \to + \infty .$$ For the general KPP equation, we show limit theorems for u(t, ct+ζ), c>c 0 , ξ ∈ ℝ, t → +∞. Large deviations for R t and probabilities of presence of particles for the branching process are deduced: (where Z t denotes the random point measure of particles living at time t) and a Yaglom type theorem is proved. The conditional distribution of the spatial tree, given {Z t (]ct, +∞[)>0}, is studied in the limit as t → +∞.

When are small subgraphs of a random graph normally distributed

Abstract: LetG be a graph and letX n count copies ofG in a random graphK(n,p). The random variable\(\left( {X_n - E\left( {X_n } \right)} \right)/\sqrt {Var\left( {X_n } \right)} \) is asymptotically normally distributed if and only ifnp m →∞ andn2(1-p)→∞, wherem=max {e(H)/|H|:H∪G}. In addition to, and in connection with this main result we investigate the formula for Var (X n ) and the Poisson convergence ofX n .

A boundary property of semimartingale reflecting Brownian motions

Abstract: We consider a class of reflecting Brownian motions on the non-negative orthant inR K . In the interior of the orthant, such a process behaves like Brownian motion with a constant covariance matrix and drift vector. At each of the (K-1)-dimensional faces that form the boundary of the orthant, the process reflects instantaneously in a direction that is constant over the face. We give a necessary condition for the process to have a certain semimartingale decomposition, and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces. This boundary property plays an essential role in the derivation (performed in a separate work) of an analytical characterization of the stationary distributions of such semimartingale reflecting Brownian motions.

Entropy Formula for Random Transformations

Abstract: We exhibit random strange attractors with random Sinai-Bowen-Ruelle measures for the composition of independent random diffeomorphisms.

Random non-linear wave equations: Smoothness of the solutions

Abstract: We show existence and uniqueness for the solution of a onedimensional wave equation with non-linear random forcing. Then we give sufficient conditions for the solution at a given time and a given point, to have a density and for this density to be smooth. The proof uses the extension of the Malliavin calculus to the two parameters Wiener functionals.

Connectivity Properties of Mandelbrot's Percolation Process

Abstract: In 1974, Mandelbrot introduced a process in [0, 1]2 which he called “canonical curdling” and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or “percolation” properties of these sets, proving all of the claims he made in Sect. 23 of the “Fractal Geometry of Nature” and a new one that he did not anticipate: There is a probability pc∈(0,1) so that if p

A convex analytic approach to Markov decision processes

Abstract: This paper develops a new framework for the study of Markov decision processes in which the control problem is viewed as an optimization problem on the set of canonically induced measures on the trajectory space of the joint state and control process. This set is shown to be compact convex. One then associates with each of the usual cost criteria (infinite horizon discounted cost, finite horizon, control up to an exit time) a naturally defined occupation measure such that the cost is an integral of some function with respect to this measure. These measures are shown to form a compact convex set whose extreme points are characterized. Classical results about existence of optimal strategies are recovered from this and several applications to multicriteria and constrained optimization problems are briefly indicated.

Large deviations for Gibbs random fields

Abstract: A large deviation principle for Gibbs random fields on Zd is proven and a corresponding large deviations proof of the Gibbs variational formula is given. A generalization of the Lanford theory of large deviations is also obtained.

Generalized Poisson Functionals

Abstract: With the aim of treating nonlinear systems with inputs being discrete and outputs being generalized functions, generalized Poisson functional are defined and analysed, where the -transforms and the renormalizational play essential roles. For Poisson functionals, the differential operators with respect to a Poisson white noise $$\dot P$$ (t), their adjoint operators and the multiplication operators by $$\dot P$$ (t) are defined. Since these operators involve the time parameter explicitly, they can be used to obtain information concerning the Poisson functional at each point in time. As an example, a new method for measuring the Wiener kernels of such functionals is outlined.

Limit theorems for convex hulls

Abstract: It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk.

The strong p-variation of martingales and orthogonal series

Abstract: Let 1≦p<∞ and letx=(x n)n≧0 be a sequence of scalars. The strongp-variation ofx, denoted byW p (x), is defined as $$W_p (x) = \sup \left\{ {\left( {|x_0 |^p + \sum\limits_{k = 1}^\infty {|x_{n_k } - x_{n_{k - 1} } |^p } } \right)^{1/p} } \right\}$$ where the supremum runs over all increasing sequences of integers 0=n 0 ≦n 1 ≦n 2 ≦... Let 1≦p<2 and letM=(M n ) n≧0 be a martingale inL p . Our main results are as follows: If $$\Sigma \mathbb{E}|M_n - M_{n - 1} |^p< \infty $$ , thenW p (M) is finite a.s. and we have $$\mathbb{E}W_p (M)^p \leqq C(\mathbb{E}|M_0 |^p + \sum\limits_{n \geqq 1} {\mathbb{E}|M_n - M_{n - 1} |^p )} $$ for some constantC depending only onp. On the other hand, let (ϕ n be an arbitrary orthonormal system of functions inL 2, considerx=(x n ) n≧0 inl 2 and letS n =Σ 0 x i ϕ i andS=(S n ) n≧0. We prove that ifΣ|x n | p <∞ (1≦p<2) thenW p (S(t))<∞ for a.e.t and ∥W p (S)∥2≦C(Σ|x n | p )1/p for some constantC. Each of these results is an extension of a result proved by Bretagnolle for sums of independent mean zero r.v.'s. The casep>2 in also discussed. Our proofs use the real interpolation method of Lions-Peetre. They admit extensions in the Banach space valued case, provided suitable assumptions are imposed on the Banach space.

Shocks in the asymmetric exclusion process

Abstract: In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density λ at -∞ and ⌕ at +∞. Earlier results described the limiting behavior in all cases except for 0<λ<1/2, λ+⌕=1. Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in 1975 [7]. The convergence proof is based on coupling and symmetry considerations.

A diffusion limit for a class of randomly-growing binary trees

Abstract: Binary trees are grown by adding one node at a time, an available node at height i being added with probability proportional to c -i, c>1. We establish both a “strong law of large numbers” and a “central limit theorem” for the vector X(t)=(X i(t)), where X i(t) is the proportion of nodes of height i that are available at time t. We show, in fact, that there is a deterministic process x i(t) such that $$\sum {|X_i (t) - x_i (t)|} {\text{ converges to 0 a}}{\text{.s}}{\text{.,}}$$ and such that if c2 $$\tfrac{1}{2}$$ , $$Z_i^n (t) = 2^{n/2} \{ X_{n + 1} (tc^n ) - x_{n + 1} (tc^n )\} ,$$ and Z n(t)=(Z i n (t)), then Z n(t) converges weakly to a Gaussian diffusion Z(t). The results are applied to establish asymptotic normality in the unbiased coin-tossing case for an entropy estimation procedure due to J. Ziv, and to obtain results on the growth of the maximum height of the tree.

Variable window width kernel estimates of probability densities

Abstract: Kernel density estimators which allow different amounts of smoothing at different locations are studied. Modifications of estimators proposed by Breiman, Meisel and Purcell (1977) and Abramson (1982a), which have variable window widths, are seen to have very fast rates of convergence. These rates have traditionally been obtained using a less natural higher order kernel, which has the disadvantage of allowing an estimator which takes on negative values.

Large deviations and stochastic flows of diffeomorphisms

Abstract: Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal Δ in M2 of the associated two point motion. Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M2-Δ if the top Lyapunov exponent is positive.

Remark on exit times from cones in \(\mathbb{R}^n \)of Brownian motion

Abstract: Our purpose is to show how the asymptotics in Corollary 1.3 of [2] can be obtained under much weaker hypotheses. It turns out the problem essentially reduces to showing that if R(s) is a Bessel process, u>0 and α>0, then $$P\left( {\mathop f\limits_0^1 R(s)^{ - 2} ds \leqq u} \right) = O(t^{ - \alpha } )$$ as t→∞. We provide a simple proof of this fact.

Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized $L$-statistics

Abstract: We study a nonclassical form of empirical df Hnwhich is of U-statistic structure and extend to Hnthe classical exponential probability inequalities and Glivenko-Cantelli convergence properties known for the usual empirical df. An important class of statistics is given byT(Hn), where T(·) is a generalized form of L-functional. For such statisticswe prove almost sure convergence using an approach which separates the functional-analytic and stochastic components of the problem and handles the latter component by application of Glivenko-Cantelli type properties.Classical results for U-statistics and L-statistics are obtained as special cases without addition of unnecessary restrictions.Many important new types of statistics of current interest are covered as well by our result.

High density limit theorems for nonlinear chemical reactions with diffusion

Abstract: The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX v,N (law of large numbers (LLN)).X v,N is constructed on a gridS N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: υ → ∞, asN → ∞, and for the CLT:\(\frac{N}{\upsilon } \to 0\), asN → ∞. The limitY =Y X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY X in dependence ofX is also investigated.

The strong law of large numbers for k-means and best possible nets of Banach valued random variables

Abstract: Let B be a uniformly convex Banach space, X a B-valued random variable and k a given positive integer number. A random sample of X is substituted by the set of k elements which minimizes a criterion. We found conditions to assure that this set converges a.s., as the sample size increases, to the set of k-elements which minimizes the same criterion for X.

Self-similar random measures

Abstract: A set is called self-similar if it is decomposable into parts which are similar to the whole. This notion was generalized to random sets. In the present paper an alternative, axiomatic approach is given which makes precise the following idea (using Palm distribution theory): A random set is statistically self-similar if it is statistically scale invariant with respect to any center chosen at random from that set. For these sets Hausdorff dimension coincides with an intrinsic self-similarity index.

A simple proof of the stability criterion of Gray and Griffeath

Abstract: Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having “all 0's” and “all 1's” as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the “all 1's” equilibrium under small perturbations. In this paper we use a renormalized site construction to give a much simpler proof. Our new approach can be used in many situations as a substitute for the contour method.

Weighted sums of I.I.D. Random variables attracted to integrals of stable processes

Abstract: Under the assumption of the convergence of partial sum processesZ n (t) of i.i.d. random variables to an α-stable Levy processZ(σ)(t), 0<α≦2, the convergence of weighted sums ∫f n (u)dZ n (u) to ∫f(u)dZ (α)(u) is studied. The general convergence result is then applied to examine the domain of attraction of the fractional stable process.

Pseudo-free energies and large deviations for non gibbsian FKG measures

Abstract: A large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on {0, 1{ Z d is proven. In this way a pseudo-free energy and pressure is defined. For ergodic systems the large deviations property holds with the usual scaling. The case of non ergodic systems is also discussed. A similar result holds for occupation times. The perturbation by an external field is treated.