Showing papers on "Central limit theorem published in 2000"

Random Walks on Infinite Graphs and Groups

Abstract: Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification of the recurrent quasi transitive graphs 6. More on recurrence Part II. The Spectral Radius: 7. Superharmonic functions and r-recurrence 8. The spectral radius 9. Computing the Green function 10. Spectral radius and strong isoperimetric inequality 11. A lower bound for simple random walk 12. Spectral radius and amenability Part III. The Asymptotic Behaviour of Transition Probabilities: 13. The local central limit theorem on the grid 14. Growth, isoperimetric inequalities, and the asymptotic type of random walk 15. The asymptotic type of random walk on amenable groups 16. Simple random walk on the Sierpinski graphs 17. Local limit theorems on free products 18. Intermezzo 19. Free groups and homogenous trees Part IV. An Introduction to Topological Boundary Theory: 20. Probabilistic approach to the Dirichlet problem, and a class of compactifications 21. Ends of graphs and the Dirichlet problem 22. Hyperbolic groups and graphs 23. The Dirichlet problem for circle packing graphs 24. The construction of the Martin boundary 25. Generalized lattices, Abelian and nilpotent groups, and graphs with polynomial growth 27. The Martin boundary of hyperbolic graphs 28. Cartesian products.

Limit Theorems of Probability Theory

Abstract: I. Classical-Type Limit Theorems for Sums of Independent Random Variables.- II. The Accuracy of Gaussian Approximation in Banach Spaces.- III. Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution.- IV. Refinements of the Central Limit Theorem for Homogeneous Markov Chains.- V. Limit Theorems on Large Deviations.- Name Index.

Determinantal random point fields

Abstract: This paper contains an exposition of both recent and rather old results on determinantal random point fields. We begin with some general theorems including proofs of necessary and sufficient conditions for the existence of a determinantal random point field with Hermitian kernel and of a criterion for weak convergence of its distribution. In the second section we proceed with examples of determinantal random fields in quantum mechanics, statistical mechanics, random matrix theory, probability theory, representation theory, and ergodic theory. In connection with the theory of renewal processes, we characterize all Hermitian determinantal random point fields on and with independent identically distributed spacings. In the third section we study translation-invariant determinantal random point fields and prove the mixing property for arbitrary multiplicity and the absolute continuity of the spectra. In the last section we discuss proofs of the central limit theorem for the number of particles in a growing box and of the functional central limit theorem for the empirical distribution function of spacings.

On the functional central limit theorem for stationary processes

Abstract: We give a sufficient condition for a stationary sequence of square-integrable and real-valued random variables to satisfy a Donsker-type invariance principle. This condition is similar to the L 1 -criterion of Gordin for the usual central limit theorem and provides invariance principles for α-mixing or β-mixing sequences as well as stationary Markov chains. In the latter case, we present an example of a non irreducible and non α-mixing chain to which our result applies.

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Abstract: Let {X i, 1≤i≤n} be a negatively associated sequence, and let {X* i , 1≤i≤n} be a sequence of independent random variables such that X* i and X i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(∑ n i=1 X i)≤Ef(∑ n i=1 X* i ) for any convex function f on R 1 and that Ef(max1≤k≤n ∑ n i=k X i)≤Ef(max1≤k≤n ∑ k i=1 X* i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.

Linear functionals of eigenvalues of random matrices

Abstract: Let Mn be a random n×n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n → ∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.

Stationary arch models: dependence structure and central limit theorem

Abstract: This paper studies a broad class of nonnegative ARCH(∞) models Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure A moving average representation in martingale differences is established, and the central limit theorem is proved

Stochastic Processes: From Physics to Finance

Abstract: A First Glimpse of Stochastic Processes.- A Brief Survey of the Mathematics of Probability Theory.- Diffusion Processes.- Beyond the Central Limit Theorem: Levy Distributions.- Modeling the Financial Market.- Stable Distributions Revisited.- Hyperspherical Polar Coordinates.- The Weierstrass Random Walk Revisited.- The Exponentially Truncated Levy Flight.- Put-Call Parity.- Geometric Brownian Motion.

Value at Risk Calculations, Extreme Events, and Tail Estimation

Abstract: Value at risk has become a standard approach for estimating and expressing a firm9s exposure to market risk Unlike the traditional risk measure, standard deviation, VaR focuses only on the tail of the distribution of outcomes - the extreme events This makes a lot of sense in theory, but a major problem arises in practice, because empirical returns distributions tend to have tails that look quite different from those of the normal and lognormal distributions that we typically assume in finance Extreme value theory offers an elegant solution Like the familiar central limit theorem, which proves that under general conditions the distribution of the average of n identically distributed random variables will converge to the normal as n grows large, extreme value theory provides similar convergence results for the extreme tails of distributions In this article, Neftci gives a clear and intuitive introduction to extreme value theory and demonstrates how to use it In an example applying it to forecasting the tails of the distributions of interest rate and exchange rate changes, extreme value theory is shown to be much more accurate than the standard value at risk calculated from the normal distribution

Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices

Abstract: Conditions are derived for the consistency of kernel estimators of the covariance matrix of a sum of vectors of dependent heterogeneous random variables, which match those of the currently best-known conditions for the central limit theorem, as required for a unified theory of asymptotic inference. These include finite moments of order no more than 2 + for > 0, trending variances, and variables which are near-epoch dependent on a mixing process, but not necessarily mixing. The results are also proved for the case of sample-dependent bandwidths.

The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities

Abstract: Author(s): Soshnikov, Alexander | Abstract: We discuss CLT for the global and local linear statistics of random matrices from classical compact groups. The main part of our proofs are certain combinatorial identities much in the spirit of works by Kac and Spohn.

Slowdown estimates and central limit theorem for random walks in random environment

Abstract: This work is concerned with asymptotic properties of multi-dimensional random walks in random environment. Under Kalikow’s condition, we show a central limit theorem for random walks in random environment on ℤd, when d≥2. We also derive tail estimates on the probability of slowdowns. These latter estimates are of special interest due to the natural interplay between slowdowns and the presence of traps in the medium. The tail behavior of the renewal time constructed in [25] plays an important role in the investigation of both problems. This article also improves the previous work of the author [24], concerning estimates of probabilities of slowdowns for walks which are neutral or biased to the right.

The functional central limit theorem and weak convergence to stochastic integrals I: weakly dependent processes

Abstract: This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

Homogenization for¶Stochastic Hamilton-Jacobi Equations

Abstract: Homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high-frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales. We prove homogenization for certain stochastic Hamilton-Jacobi partial differential equations; the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale-separation limit. In some cases, we also establish a central limit theorem.

Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

Abstract: We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin–Lebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.

The functional central limit theorem and weak convergence to stochastic integrals II: fractionally integrated processes

Abstract: This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

Statistical estimation for multiplicative cascades

Abstract: The probability distribution of the cascade generators in a random multiplicative cascade represents a hidden parameter which is reflected in the fine scale limiting behavior of the scaling exponents (sample moments) of a single sample cascade realization as a.s. constants. We identify a large class of cascade generators uniquely determined by these scaling exponents. For this class we provide both asymptotic consistency and confidence intervals for two different estimators of the cumulant generating function (log Laplace transform) of the cascade generator distribution. These results are derived from investigation of the convergence properties of the fine scale sample moments of a single cascade realization.

Elements of Stochastic Processes

Abstract: Much of statistical analysis is concerned with models in which the observations are assumed to vary independently. However, a great deal of data in economics, engineering, and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest in itself. A model which describes the probability structure of a series of observations X t , t = 1,…, n, is called a stochastic process. An Xt might be the value of a stock price at time point t, the water level in a lake at time point t, and so on. The primary purpose of this book is to provide statistical inference for stochastic processes, which is based on the probability theory for them. In this chapter some of the elements of stochastic processes will be reviewed. Because the statistical analysis for stochastic processes largely relies on the asymptotic theory, we also explain some useful limit theorems and central limit theorems. We have placed some fundamental results of mathematics, probability, and statistics in the Appendix.

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

Abstract: We consider a one-dimensional diffusion process X, with ergodic property, with drift b(x, 0) and diffusion coefficient a(x, 0) depending on an unknown parameter 0 that may be multidimensional. We are interested in the estimation of 0 and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times ti = iA, i = 0, . . ., n. We study a particular class of estimating functions of the form ZEft, Xt,_.) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.

Central limit theorem for Maxwellian molecules and truncation of the wild expansion

Abstract: We prove an L bound on the error made when the Wild summation for solutions of the Boltzmann equation for a gas of Maxwellian molecules is truncated at the nth stage. This gives quantitative control over the only constructive method known for solving the Boltzmann equation. As such, it has been recently applied to numerical computation, but without control on the approximation made in truncation. We also show that our bound is qualitatively sharp, and that it leads to a simple proof of the exponentially fast rate of relaxation to equilibrium for Maxwellian molecules, along lines originally suggested by McKean.

Albanese maps and off-diagonal long time asymptotics for the heat kernel

Abstract: We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque–Bloch theory on periodic Schrodinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones [4]. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role.

Generalized spectral tests for serial dependence

Abstract: Summary. Two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function. The tests are generalizations of the Cramer-von Mises and Kolmogorov-Smirnov tests based on the standardized spectral distribution function. They do not involve the choice of a lag order, and they are consistent against all types of pairwise serial dependence, including those with zero autocorrelation. They also require no moment condition and are distribution free under serial independence. A simulation study compares the finite sample performances of the new tests and some closely related tests. The asymptotic distribution theory works well in finite samples. The generalized Cramer-von Mises test has good power against a variety of dependent alternatives and dominates the generalized Kolmogorov-Smirnov test. A local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function.

Asymptotics for Analysis of Variance When the Number of Levels is Large

Abstract: We study asymptotic results for F tests in analysis of variance models as the number of factor levels goes to ∞ but the number of observations for each factor combination is fixed. Asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to ∞ (at the same rate). We consider null and alternative distributions of F, the usual F statistic, for fixed-effects and random-effects, balanced and unbalanced, one-way and two-way, and normal and nonnormal analysis of variance (ANOVA) models. The results may be most relevant for random-effects and mixed models. For example, we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small. The results would also be relevant for fixed-effects models in which there are many factor levels but not many observations for each factor level.

On stability of nonlinear AR processes with Markov switching

Abstract: We investigate the stability problem for a nonlinear autoregressive model with Markov switching. First we give conditions for the existence and the uniqueness of a stationary ergodic solution. The existence of moments of such a solution is then examined and we establish a strong law of large numbers for a wide class of unbounded functions, as well as a central limit theorem under an irreducibility condition.