Showing papers in "Probability Theory and Related Fields in 1987"

A central limit theorem for generalized quadratic forms

Abstract: Random variables of the form $$W(n) = \mathop \sum \limits_{1 \leqq i \leqq n} \mathop \sum \limits_{{\text{ }}1 \leqq j \leqq n} w_{ijn} (X_i ,X_j )$$ are considered with X i independent (not necessarily identically distributed), and w ijn (·, ·) Borel functions, such that w ijn (X i , X j ) is square integrable and has vanishing conditional expectations: $$E(w_{ijn} (X_i ,X_j )|X_i = E(w_{ijn} (X_i ,X_j )|X_j ) = 0,{\text{ a}}{\text{.s}}{\text{.}}$$ A central limit theorem is proved under the condition that the normed fourth moment tends to 3. Under some restrictions the condition is also necessary. Finally conditions on the individual tails of w ijn (X i , X j ) and an eigenvalue condition are given that ensure asymptotic normality of W(n).

Stochastic differential equations for multi-dimensional domain with reflecting boundary

Abstract: In this paper we prove that there exists a unique solution of the Skorohod equation for a domain inR d with a reflecting boundary condition. We remove the admissibility condition of the domain which is assumed in the work [4] of Lions and Sznitman. We first consider a deterministic case and then discuss a stochastic case.

Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation

Abstract: Let h o, ĥ o and ĥ c be the windows which minimise mean integrated square error, integrated square error and the least-squares cross-validatory criterion, respectively, for kernel density estimates. It is argued that ĥ o, not h o, should be the benchmark for comparing different data-driven approaches to the determination of window size. Asymptotic properties of h o-ĥ o and ĥ c -ĥ o, and of differences between integrated square errors evaluated at these windows, are derived. It is shown that in comparison to the benchmark ĥ o, the observable window ĥ c performs as well as the so-called “optimal” but unattainable window h o, to both first and second order.

Best-possible bounds for the distribution of a sum — a problem of Kolmogorov

Abstract: Recently, in answer to a question of Kolmogorov, G.D. Makarov obtained best-possible bounds for the distribution function of the sumX+Y of two random variables,X andY, whose individual distribution functions,FX andFY, are fixed. We show that these bounds follow directly from an inequality which has been known for some time. The techniques we employ, which are based on copulas and their properties, yield an insightful proof of the fact that these bounds are best-possible, settle the question of equality, and are computationally manageable. Furthermore, they extend to binary operations other than addition and to higher dimensions.

Statistically self-similar fractals

Abstract: LetX be a complete separable bounded metric space and μ a Borel probability measure on the space Con(X) N of allN-tuples of contractions ofX with the topology of pointwise convergence. Then there exists a unique μ-self-similar probability measureP μ on the spaceK(X) of all non-empty compact subsets ofX. Here a measureP onK(X) is called μ-self-similar if, for every Borel setB ⊂K(X), $$P(B) = \int {P^N } \left( {(K_O ,...,K_{N - 1} )\left| {\bigcup\limits_{i = 0}^{N - 1} {S_i (K_i ) \in B} } \right.} \right)d\mu (S_O ,...,S_{N - 1} ).$$ If, for μ-a.e. (S 0, ..., SN-1), eachS i has an inverse which satisfies a Lipschitz condition then there is an α≧0 such that, forP μ-a.e.K∈K(X), the Hausdorff dimensionH-dim(K) is equal to α. IfX⊂ℝ d is compact and has non-empty interior and if μ-a.e. (S 0, ..., SN-1) consists of similarities which satisfy a certain disjointness condition w.r.t.X then α is determined by the equation $$\int {\sum\limits_{i = 0}^{N - 1} {Lip(S_i )^\alpha } } d\mu (S_O ,...,S_{N - 1} ) = 1,$$ where Lip(S i) denotes the (smallest) Lipschitz constant forS i. Under fairly general assumptions the α-dimensional Hausdorff measure ofP μ-a.e.K∈K(X) equals O. If μ andX are chosen in a rather special way thenP μ-a.e.K∈K(X) is the graph of a homeomorphism of [0, 1] (or a curve or the graph of a continuous function).

Uniqueness for diffusions with piecewise constant coefficients

Abstract: LetL be a second-order partial differential operator inRd. LetRd be the finite union of disjoint polyhedra. Suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron, and that the drift coefficient is bounded and measurable. We show that the martingale problem associated withL is well-posed.

On the gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions

Abstract: On etudie le taux d'approximation de distribution de sommes de vecteurs aleatoires independants par des distributions gaussiennes correspondantes

Rates of growth and sample moduli for weighted empirical processes indexed by sets

Abstract: Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-Cervonenkis class C of sets. These inequalities are particularly useful under the assumption P(∪{C∈C:P(C)

Reflected Brownian motion with skew symmetric data in a polyhedral domain

Abstract: This paper is concerned with the characterization and invariant measures of certain reflected Brownian motions (RBM's) in polyhedral domains. The kind of RBM studied here behaves like d-dimensional Brownian motion with constant drift μ in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit. Under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in Harrison-Williams [9], it is shown that such an RBM can be characterized in terms of a family of submartingales and that it reaches non-smooth parts of the boundary with probability zero. In [9], a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained in [9] is shown to be the density of an invariant measure for the RBM. Furthermore, if the density is integrable over the polyhedral state space, then it yields the unique stationary distribution for the RBM. In the proofs of these results, a key role is played by a dual process for the RBM and by results in [9] for reflected Brownian motions on smooth approximating domains.

Chains with infinite connections: Uniqueness and Markov representation

Abstract: If for a process $$(\xi _n )_{n = - \infty }^\infty$$ the conditional distribution of ξ n given the past does not depend on n for e.g. n≧0, then the process may be called a chain with infinite connections. Under a well-known continuity condition on this conditional distribution the process is shown to be distributed as an instantaneous function of a countable state Markov chain. Also under a certain weaker continuity condition uniqueness of the distributions of the stationary chains is obtained.

Majoration en temps petit de la densité d'une diffusion dégénérée

Abstract: Nous majorons en temps petit le logarithme de la densite d'une diffusion degeneree a l'aide d'une distance semi-riemanienne. Nous obtenons ainsi par une methode purement probabiliste, basee sur le calcul de Malliavin [K-S] et la theorie des grandes deviations [A], des resultats generalisant en partie ceux obtenus par Varadhan dans le cas non degenere [V].

Exit times from cones in ℝ n of Brownian motion

Abstract: For a fairly general class of cones inn dimensions (n≧3) we determine the corresponding distributions of Brownian first exit times. Asymptotic results may then be read off.

A two-sided stochastic integral and its calculus

Abstract: Let X be a forward diffusion and Y a backward diffusion, both defined on [0,1], X t and Y t being respectively adapted to the past of a Wiener process W (·), and to its future increments. We construct a “two-sided” stochastic integral of the form. $$\mathop \smallint \limits_0^t \Phi (u,X_u ,Y^u )dW(u)$$ which generalizes the backward and forward Ito integrals simultaneously. Our construction is quite intuitive, and leads to a generalized stochastic calculus. It is also shown that for each fixed t, our integral coincides with that defined by Skorohod in [18].

An infinite dimensional stochastic differential equation with state spaceC(ℝ)

Abstract: We consider a time evolution of unbounded continuous spins on the real line. The evolution is described by an infinite dimensional stochastic differential equation with local interaction. Introducing a condition which controls the growth of paths at infinity, we can construct a diffusion process taking values inC(ℝ). In view of quantum field theory, this is a time dependent model ofP(φ)1 field in Parisi and Wu's scheme.

Self-avoiding random walk: A Brownian motion model with local time drift

Abstract: A natural model for a ‘self-avoiding’ Brownian motion inR d, when specialised and simplified tod=1, becomes the stochastic differential equation $$X_t = B_t - \int\limits_0^t g (X_s ,L(s,X_s ))ds$$ , where {L(t, x):t≧0,x∈R} is the local time process ofX. ThoughX is not Markovian, an analogue of the Ray-Knight theorem holds for {L(∞,x):x∈R}, which allows one to prove in many cases of interest that $$\mathop {\lim }\limits_{t \to \infty } X_t /t$$ exists almost surely, and to identify the limit.

The rate of escape for anisotropic random walks in a tree

Abstract: Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z n on G or T which applies to the distance of Z n from a fixed point, as well as to the distribution of the last R letters in Z n . For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z n with a finitely supported step distribution is derived and discussed.

Convergence rates in the strong law for bounded mixing sequences

Abstract: Speed of convergence is studied for a Marcinkiewicz-Zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate. Though α-mixing is also considered, the most interesting result concerns absolutely regular sequences. The results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible. Another application sharpens a result for averaging a function along a random walk.

Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant

Abstract: The paper studies the pathwise asymptotic behaviour of stochastic algorithms of the following general form $$\theta _{n + 1} = \theta _n + \gamma _{n + 1} f(\theta _n ,Y_{n + 1} ),$$ the hypotheses allowing discontinuities on the adaptation termf. The process (Yn)n≧0 is a Markov chain “controlled by (θn)”. For each θ fixed the Markov chain (Ynθ)n≧0 is essentially of positive recurrent type.

Elliptically Contoured Distributions

Abstract: Given two covariance matricesR andS for a given elliptically contoured distribution, we show how simple inequalities between the matrix elements imply thatE R(f)≦E S(f), e.g., whenx=(xi1,i2,...,in) is a multiindex vector and $$f(x) = \mathop {\min }\limits_{i_1 } \mathop {\max }\limits_{i_2 } \mathop {\min }\limits_{i_3 } \max ...x_{i_{1,...,} i_n } ,$$ orf(x) is the indicator function of sets such as $$\mathop \cap \limits_{i_1 } \mathop \cup \limits_{i_2 } \mathop \cap \limits_{i_3 } \cup ...[x_{i_{1,...,} i_n } \mathop< \limits_ = \lambda _{i_{1,...,} i_n } ]$$ of which the well known Slepian's inequality (n=1) is a special case.

Uniform measure results for the image of subsets under Brownian motion

Abstract: The paper obtains bounds on the Hausdorff and packing measures of the imageX(E) of a Borel setE by a transient strictly stable processX t which as hold for allE and for every measure function $$h_{\beta ,\gamma } (s) = s^\beta \left| {\log s} \right|^{\gamma ^ \star }$$ In some cases examples are constructed to show that the bounds are sharp

Splitting intervals II: Limit laws for lengths

Abstract: In the processes under consideration, a particle of size L splits with exponential rate L α, 0<α<∞, and when it splits, it splits into two particles of size LV and L(1-V) where V is independent of the past with d.f. F on (0, 1). Let Z tbe the number of particles at time t and L tthe size of a randomly chosen particle. If α=0, it is well known how the system evolves: e -tZtconverges a.s. to an exponential r.v. and −L t≈t + Ct 1/2 X where X is a standard normal t.v. Our results for α>0 are in sharp contrast. In “Splitting Intervals” we showed that t -1/α Z tconverges a.s. to a constant K>0, and in this paper we show $$log L_t = \frac{1}{\alpha }log t + 0(1).$$ . We show that the empirical d.f. of the rescaled lengths, $$Z_t^{ - 1} \sum I \{ t^{^{^{1/\alpha } } } L_i \underline \leqslant \cdot \} ,$$ , converges a.s. to a non-degenerate limit depending on F that we explicitly describe. Our results with α=2/3 are relevant to polymer degradation.

A useful estimate in the multidimensional invariance principle

Abstract: An estimate of the convergence speed in the multidimensional invariance principle is obtained. Using this estimate, we can prove strong invariance principles for partial sums of independent not necessarily identically distributed multidimensional random vectors.

Structure des cocycles markoviens sur l'espace de Fock

Abstract: We show that strongly continuous unitary Markov cocycles on Fock space are solutions of a quantum stochastic Schrodinger equation and give their explicit form through a decomposition of Fock space on the eigenspaces of the number operator. We also give necessary and sufficient conditions for a generalized Hamiltonian to be the generator of such a cocycle. This generalizes the work of Hudson and Parthasarathy in the norm-continuous case.

Mouvement brownien, cônes et processus stables

Abstract: LetW=(W t, t≧0) denote a two-dimensional Brownian motion starting at 0 and, for 0<α<π, letC α be a wedge in ℝ2 with vertex 0 and angle 2α. We consider the set of timest's such that the path ofW, up to timet, stays inside the translated wedgeW t-Cα. It follows from recent results of Burdzy and Shimura that this set, which we denote byH α, contains nonzero times if, and only if, α>π/4. Here we construct a measure, a local time, supported onH α. For π/4<α≦α/2, the Brownian motionW, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2-π/2α. This results extends Spitzer's construction of the Cauchy process, which is recovered by taking α=π/2. A formula which describes the behaviour ofW before a timet∈H α is established and applied to the proof of a conjecture of Burdzy. We also obtain a two-dimensional version of the famous theorem of Levy concerning the maximum process of linear Brownian motion. Precisely, for 0<α<π/2, letS t denote the vertex of the smallest wedge of the typez-C α which contains the path ofW up to timet. The processS t-Wt is shown to be a reflected Brownian motion in the wedgeC α, with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion inR d, d≧3. LetC Ω be the cone associated with an open subset Ω of the sphereS d-1, and letH Ω be defined asH α above. Sufficient conditions are given forH Ω to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on Ω.

Integration dans la fibre associée a une diffusion dégénérée

Abstract: Nous montrons que la densite d'une diffusion hypoelliptique se calcule en temps petit par integration dans la fibre lorsqu'on est en dehors du cut-locus: a cette fin, nous utilisons la methode mise au point par Bismut dans son livre «Large deviations and Malliavin calculus», en la simplifiant grâce a une utilisation adequate du calcul de Malliavin. De plus, sans utiliser le calcul de Malliavin, nous reobtenons par cette methode la minoration de Varadhan de la densite en temps petit de la diffusion au moyen de la distance semigeodesique associee.

Computable examples of the maximal Lyapunov exponent

Abstract: Some new examples are given of sequences of matrix valued random variables for which it is possible to compute the maximal Lyapunov exponent. The examples are constructed by using a sequence of stopping times to group the original sequence into commuting blocks. If the original sequence is the outcome of independent Bernoulli trials with success probability p, then the maximal Lyapunov exponent may be expressed in terms of power series in p, with explicit formulae for the coefficients. The convexity of the maximal Lyapunov exponent as a function of p is discussed, as is an application to branching processes in a random environment.

Diffusion with interactions and collisions between coloured particles and the propagation of chaos

Abstract: One considers a system ofN particles on the real line which are of two different types (colours). Their dynamics is given by a stochastic differential equation with constant diffusion part; the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant; further a reflection condition is imposed so that particles of different type are not allowed to cross each other. The article studies the Vlasov-McKean limit of the system asN→∞: propagation of chaos and an evolution equation for the limiting empirical measures is established, from where in particular an equation for the separating front between the two types follos.

Central limit theorems for stochastic processes under random entropy conditions

Abstract: Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array (φ nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a “random entropy” condition, analogous to one used by Gine and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Ĉervonenkis property and each φ nj (f)fdνnj is of the form νnjc for some reasonable random finite signed measure v nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-Ĉervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.

Laplace approximations for sums of independent random vectors. II: Degenerate maxima and manifolds of maxima

Abstract: We consider expressions of the form Z n =E(exp nF(S n /n)) where S n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z n =sup x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.