Convergence Of Finite-dimensional Distributions Research Articles

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in (0,1)∪(1,4/3) in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Mosco Type Convergence of Bilinear Forms and Weak Convergence of n-Particle Systems

It is well known that Mosco (type) convergence is a tool in order to verify weak convergence of finite dimensional distributions of sequences of stochastic processes. In the present paper we are concerned with the concept of Mosco type convergence for non-symmetric stochastic processes and, in particular, n-particle systems in order to establish relative compactness.

Number Variance for Hierarchical Random Walks and Related Fluctuations

We study an infinite system of independent symmetric random walks on a hierarchical group, in particular, the <em>c</em>-random walks. Such walks are used, e.g., in mathematical physics and population biology. The number variance problem consists in investigating if the variance of the number of “particles” $N_n(L)$ lying in the ball of radius $L$<em></em> at a given step $n$<em></em> remains bounded, or even better, converges to a finite limit, as $L\to\infty$. We give a necessary and sufficient condition and discuss its relationship to transience/recurrence property of the walk. Next we consider normalized fluctuations of $N_n(L)$ around the mean as $n\to\infty$ and $L$<em></em> is increased in an appropriate way. We prove convergence of finite dimensional distributions to a Gaussian process whose properties are discussed. As the $c$<em></em>-random walks mimic symmetric stable processes on $\mathbb{R}$, we compare our results with those obtained by Hambly and Jones (2007, 2009), who studied the number variance problem for an infinite system of independent symmetric stable processes on $\mathbb{R}$. Since the hierarchical group is an ultrametric space, corresponding results for symmetric stable processes and hierarchical random walks may be analogous or quite different, as has been observed in other contexts. An example of a difference in the present context is that for the stable processes a fluctuation limit process is a Gaussian process which is not Markovian and has long range dependent stationary increments, but the counterpart for hierarchical random walks is Markovian, and in a special case it has independent increments.

On the closability and convergence of dirichlet forms

We construct a measure µ on ℝ2 for which the gradient quadratic form is closable, whereas partial quadratic forms are not closable. We obtain new sufficient conditions for the Mosco convergence of Dirichlet forms. This gives effective conditions for the weak convergence of finite-dimensional distributions of diffusion processes.

Convergence of Dirichlet forms with changing speed measures on ℝ d

We consider a sequence of vaguely convergent measures μn = σn dx → σ dx = μ on ℝd and a sequence of symmetric Dirichlet forms {ℰn}, ℰn(f, g) = ∑di,j=1 ∫ℝdani,j∂i f ∂j g dx, where every ℰ n is defined on L2(σn dx). We apply a functional analytic theory of Mosco convergence on changing L2-spaces recently developed by K. Kuwae and T. Shioya and obtain some new results about convergence of {ℰn} and weak convergence of finite dimensional distributions of associated stochastic processes.

Fluctuations for a ∇φ interface model with repulsion from a wall

We consider a ∇φ interface model on a one-dimensional lattice with repulsion from a hard wall. We suppose that the repulsion is of the form cφ−α−1, where c,α>0 and φ denotes the height of the interface from the wall. We prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a SPDE with singular drift. If c→0 the system becomes the Funaki-Olla ∇φ interface model with reflection at the wall, whose equilibrium fluctuations converge to the solution of a SPDE with reflection. We give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional Dirichlet Form, obtained in a previous paper. Our method is based on a study of integration by parts formulae w.r.t. the equilibrium measure of the interface model and allows to avoid the proof of the so called Boltzmann-Gibbs principle. We also obtain convergence of finite dimensional distributions of non-equilibrium fluctuations around the stationary hydrodynamic limit 0.

Convergence of finite dimensional distributions of heat kernel measures on loop groups

In this paper we consider heat kernel measure on loop groups associated to the H 1/2-metric. Unlike H s -case ( s>1/2), there is a difficulty that H 1/2 is not contained in the space of continuous loops. So we take limits. There are two limiting methods. One is to use delta functions and to let s go down to 1/2. The other is to fix s at 1/2 and to approximate the delta functions. For the second approach, a generalization of heat kernel measures is needed. Then, the first approach can be obtained as a special case of the second one. The limit in the sense of finite dimensional distribution is the fictitious infinite dimensional Haar measure.

A central limit theorem for nonuniform φ-mixing random fields with infinite variance

For set-indexed partial sums processes of stationary mixing random fields, convergence of finite dimensional distributions to a Brownian motion is proved, extending to infinite variance previous results of the author and a Central Limit Theorem of Nahapetian. Gibbs fields are considered.

Ergodicity of a polling network with an infinite number of stations

A Markov polling system with infinitely many stations is studied. The topic is the ergodicity of the infinite-dimensional process of queue lengths. For the infinite-dimensional process, the usual type of ergodicity cannot prevail in general and we introduce a modified concept of ergodicity, namely, weak ergodicity. It means the convergence of finite-dimensional distributions of the process. We give necessary and sufficient conditions for weak ergodicity. Also, the “usual” ergodicity of the system is studied, as well as convergence of functionals which are continuous in some norm.

Asymptotic normality of spherical means of nonlinear functionals of Gaussian random fields

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.

Approximating multiple itô integrals with "band limited" processes

Let denote a "wide-band width" vector valued process. The paper is concerned with limits of , and for the ra-multiple integral case. For the most important case, a weak convergence result is obtained, and the "correction'" terms exhibited. The method is such that the conditions used can readily be weakened. An application to a likelihood functional and hypothesis testing problem is given. There, the weak convergence result (rather than mere convergence of finite dimensional distributions) is essential if the limit approximation is to make sense as an approximation to the likelihood functional. The correction terms depend only on the limit (as a-0) of the correlation function of the(renormalized)

Approximating Multiple Itô Integrals with “Wide-Band Width” Processes, and Application to Filtering Theory

Let nε(·) denote a "wide-band widt h" vector valued process. We are concerned with limits of ∫0t∫0τnε'(τ)L(τ,s)nε(s)dτds, and for the m-multiple integral case. A weak convergence result is obtained, and the “correction” terms exhibited. An application to a likelihood functional and hypothesis testing problem is given. There, the weak convergence result (rather than mere convergence of finite dimensional distributions) is essential if the limit approximation is to make sense as an approximation to the likelihood functional. The correction terms depend only on the limit (as ε → 0) of the correlation function of the (renormalized) nε(·).

Weak convergence and relative compactness of martingale processes with applications to some nonparametric statistics

For forward and reverse martingale processes, weak convergence to appropriate stochastic (but, not necessarily, Wiener) processes is studied. In particular, it is shown that martingale processes are tight under a uniformly integrability condition, and also, convergence of finite dimensional distributions satisfying certain mild conditions implies the compactness of such processes. The theory is illustrated with the aid of a class of U-statistics and von Mises' differentiable statistical functions which need not be stationary of order zero. Weak convergence of the classical Cramér-von Mises goodness-of-fit statistic is also considered. The case of martingales with random indices is studied at the end.

Martingale Central Limit Theorems

The classical Lindeberg-Feller CLT for sums of independent random variables (rv's) provides more than the convergence in distribution of the sum to a normal law. The independence of summands also guarantees the weak convergence of all finite dimensional distributions of an a.e. sample continuous stochastic process (suitably defined in terms of the partial sums) to those of a Gaussian process with independent increments, namely, the Wiener process. Moreover, the distributions of said process converge weakly to Wiener measure on $C\lbrack 0, 1\rbrack$, the latter result being known as an invariance principle, or functional CLT, an idea originating with Erdos and Kac [10] and Donsker [5], then developed by Billingsley, Prohorov, Skorohod and others. The present work contains an invariance principle for a certain class of martingales, under a martingale version of the classical Lindeberg condition. In the case of sums of independent rv's, our results reduce to the conventional invariance principle (see, for example, Parthasarathy [16]) in the setting of the classical Lindeberg-Feller CLT. Theorem 1 contains a type of martingale characteristic function convergence which is strictly analogous to the classical CLT, while Theorem 2 provides weak convergence of finite dimensional distributions to those of a Wiener process, followed by (Theorem 3) the weak convergence of corresponding induced measures on $C\lbrack 0, 1\rbrack$ to Wiener measure, thus entailing an invariance principle for martingales. Notation and results are listed in Section 2. Section 3 defines the Lindeberg condition for martingales and gives it several equivalent forms. Sections 4 and 5 contain the proofs of Theorems 1 and 2, respectively, while Theorem 3 is proved in Section 6 by use of a martingale inequality derived from an upcrossing inequality of Doob [7]. Section 7 contains brief remarks. Among the large literature on CLT's for sums of dependent rv's, mention of a martingale CLT is first made by Levy [12], [13], followed by Doob [6] page 383. Billingsley [2] and Ibragimov [11] gave a version for stationary ergodic martingales, and Csorgo [4] considered related problems. The author also knows of Dvoretsky [8]. Invariance principles for various dependent rv's were found by Billingsley [1], and in [3] for stationary ergodic martingales, the latter result being given by Rosen [17] for bounded summands. The present Theorem 3 relaxes the stationarity and ergodicity requirements of Billingsley's Theorem (23.1) of [3]. Since preparing the original version of this paper, the author's attention has been drawn to Dvoretsky [9], which announces a result strongly resembling the present Theorem 2, and to the CLT [14] and invariance principle [15] for reversed martingales, due to Lyones. The methods of Section 5 are owed to Billingsley [2] (who in turn acknowledges a debt to Levy) and to ideas given by Dvoretsky [8]. Finally, I wish to thank Dr. C. C. Heyde, for making a vital remark, and Mr. David Scott, for helpful criticism.

Convergence in Distribution of Stochastic Integrals

Abstract : A type of convergence of stochastic processes, convergence in linear law, is introduced. It entails convergence of finite dimensional distributions and a condition on the covariance kernels of the processes. A method is presented under which a sequence of finite collections of random variables, with suitable covariance structure, may be embedded in a sequence of continuous time processes satisfying the covariance conditions for linear law convergence. The paper illustrates that certain functionals of stochastic processes can be shown to converge in distribution to the corresponding functional of the limiting process, without an analysis of sample paths. Applications to limiting distributions of linear combinations of order statistics and linear combinations of successive differences of order statistics are presented. Limiting distributions are also obtained for certain random variables arising in nuclear chemistry and reliability.