Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998     

ON THE UNIFORM CONSISTENCY OF THEKAPLAN-MEIER ESTIMATOR UNDER HEAVY CENSORING

Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)＜1)＞to=sup(t2G(t)＜1) The sufficient condition is discussed.     

A class of risk processes with reserve-dependent premium rate: sample path large deviations and importance sampling

Let (X(t)) be a risk process with reserve-dependent premium rate, delayed claims and initial capital u. Consider a class of risk processes {(X ^ε (t)): ε > 0} derived from (X(t)) via scaling in a slow Markov walk sense, and let Ψ_ε(u) be the corresponding ruin probability. In this paper we prove sample path large deviations for (X ε (t)) as ε → 0. As a consequence, we give exact asymptotics for log Ψ_ε(u) and we determine a most likely path leading to ruin. Finally, using importance sampling, we find an asymptotically efficient law for the simulation of Ψ_ε(u). AMS Subject Classifications 60F10, 91B30 This work has been partially supported by Murst Project “Metodi Stocastici in Finanza Matematica”     

Robust Approximation in a Filtering Problem with Real State Space and Counting Observations

Let (X _t ,Y _t ) be a pure jump Markov process, where X _t takes values in \bf R and Y _t is a counting process. We compare the filter of this system and a filter of a suitably modified system. We compute an explicit bound for the distance in the so-called bounded Lipschitz metric between the two filters. Finally we show how to use this bound to construct a discrete space approximation of the filter. Accepted 7 December 1999     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

PRV property and the <Emphasis Type="Italic">ϕ</Emphasis>-asymptotic behavior of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007.     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

Any two irreducible Markov chains are finitarily orbit equivalent

Two invertible dynamical systems (X, gA, μ, T) and (Y, ℬ, ν, S), where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X ₀ of X of full measure to a subset Y ₀ of Y of full measure such that ϕ|x ₀ is continuous in the relative topology on X ₀, ϕ ⁻¹|Y ₀ is continuous in the relative topology on Y ₀ and ϕ(Orb _T (x)) = Orb _Sϕ (x) for μ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains.     

MULTIPLIERS OF MATRIX SPACES

Abstract

Suppose X and Y are FK spaces in which ? the span of the coordinate vectors (e_n) is dense. Let L(X,Y) denote the space of all matrices of the form E_i(T(e_j)) as T ranges over all continuous linear operators from X into Y; here e_i represents the i^th coordinate vector and E_i represents the i^th coordinate functional. Let M(L(X, Y)) denote the space of all matrices B such that (B(i,j)A(i,j)) is in L(X,Y) whenever A is in L(X,Y). In this paper we shall show how the summability properties of X and Y determine the extent of M(L(X,Y)) and conversely how the extent of M(L(X,Y)) determines the summability properties of both X and Y.     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L ² quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

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Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕ_ɛ(u=ɛ⁻¹ ϕ(u/ɛ) be an approximate identity. Setting X _ɛ(t)=X*ϕ_ɛ(t) and letting N _ɛ(T) be the number of zeros of X _ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N _ɛ(T) converges in L ² as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Topological Representation of Sheaf Cohomology of Sites

For a site & (with enough points), we construct a topological space X(&) and a full embedding ^* of the category of sheaves on & into those on X _(&) (i.e., a morphism of toposes :Sh (X_(&)) Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H^*(&,A) = H^*(X_(&), ^*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X _(Y) and a functorial isomorphism between the étale cohomology H^*(Y _ét,A) and the ordinary sheaf cohomology H^*(X(_(Y),),^*A), for any sheaf A for the étale topology on Y.     

Probabilistic Representations of Solutions of the Forward Equations

In this paper we prove a stochastic representation for solutions of the evolution equation

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.     

A simple formula for an analogue of conditional wiener integrals and its applications II

Let C[0, T] denote the space of real-valued continuous functions on the interval [0, T] with an analogue w _ϕ of Wiener measure and for a partition 0 = t ₀ < t ₁ < ... < t _n < t _n+1 = T of [0, T], let X _n : C[0, T] → ℝ ⁿ⁺¹ and X ⁿ⁺¹: C[0, T] → ℝ ⁿ⁺² be given by X _n (x) = (x(t ₀), x(t ₁), ..., x(t _n )) and X _n+1(x) = (x(t ₀), x(t ₁), ..., x(t _n+1)), respectively. In this paper, using a simple formula for the conditional w _ϕ-integral of functions on C[0, T] with the conditioning function X _n+1, we derive a simple formula for the conditional w _ϕ-integral of the functions with the conditioning function X _n . As applications of the formula with the function X _n , we evaluate the conditional w _ϕ-integral of the functions of the form F _m (x) = ∫₀ ^T (x(t)) ^m for x ∈ C[0, T] and for any positive integer m. Moreover, with the conditioning X _n , we evaluate the conditional w _ϕ-integral of the functions in a Banach algebra which is an analogue of the Cameron and Storvick’s Banach algebra . Finally, we derive the conditional analytic Feynman w _ϕ-integrals of the functions in .      

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.