Showing papers on "Central limit theorem published in 2006"

Lectures on the Combinatorics of Free Probability

Abstract: Part I. Basic Concepts: 1. Non-commutative probability spaces and distributions 2. A case study of non-normal distribution 3. C*-probability spaces 4. Non-commutative joint distributions 5. Definition and basic properties of free independence 6. Free product of *-probability spaces 7. Free product of C*-probability spaces Part II. Cumulants: 8. Motivation: free central limit theorem 9. Basic combinatorics I: non-crossing partitions 10. Basic Combinatorics II: Mobius inversion 11. Free cumulants: definition and basic properties 12. Sums of free random variables 13. More about limit theorems and infinitely divisible distributions 14. Products of free random variables 15. R-diagonal elements Part III. Transforms and Models: 16. The R-transform 17. The operation of boxed convolution 18. More on the 1-dimensional boxed convolution 19. The free commutator 20. R-cyclic matrices 21. The full Fock space model for the R-transform 22. Gaussian Random Matrices 23. Unitary Random Matrices Notes and Comments Bibliography Index.

Determinantal Processes and Independence

Abstract: We give a probabilistic introduction to determinantal and per- manental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L 2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental pro- cesses, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

On the ergodicity properties of some adaptive MCMC algorithms

Abstract: In this paper we study the ergodicity properties of some adaptive Markov chain Monte Carlo algorithms (MCMC) that have been recently proposed in the literature. We prove that under a set of verifiable conditions, ergodic averages calculated from the output of a so-called adaptive MCMC sampler converge to the required value and can even, under more stringent assumptions, satisfy a central limit theorem. We prove that the conditions required are satisfied for the independent Metropolis–Hastings algorithm and the random walk Metropolis algorithm with symmetric increments. Finally, we propose an application of these results to the case where the proposal distribution of the Metropolis–Hastings update is a mixture of distributions from a curved exponential family.

A CLT for a band matrix model

Abstract: A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincare type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.

Asymptotic properties of realized power variations and related functionals of semimartingales

Abstract: This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is 2 we get the realized quadratic variation) . We prove a variety of ``laws of large numbers'', that is convergence in probability of these sums, sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems.

A generalization of the Lindeberg principle

Abstract: We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

Cosmology from random multifield potentials

Abstract: We consider the statistical properties of vacua and inflationary trajectories associated with a random multifield potential. Our underlying motivation is the string landscape, but our calculations apply to general potentials. Using random matrix theory, we analyse the Hessian matrices associated with the extrema of this potential. These potentials generically have a vast number of extrema. We show that if the cross-couplings (off-diagonal terms) are of the same order as the self-couplings (diagonal terms), essentially all extrema are saddles, and the number of minima is effectively zero. Avoiding this requires the same separation of scales as is needed to ensure that Newton's constant is stable against radiative corrections in a string landscape. Using the central limit theorem we find that even if the number of extrema is enormous, the typical distance between extrema is still substantial—with challenging implications for inflationary models that depend on the existence of a complicated path inside the landscape.

Recent advances in invariance principles for stationary sequences

Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.

Recent advances in invariance principles for stationary sequences

Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.

Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing

Abstract: We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm, where $X$ is the solution of a stochastic differential equation and $f$ is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps $\delta$ and $\delta^{\beta},0<\beta<1$. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal $\beta$ (which turns out to be $\beta=1/2$), we establish a central limit type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.

Generalized extreme value statistics and sum of correlated variables

Abstract: We show that generalized extreme value statistics—the statistics of the kth largest value among a large set of random variables—can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalizations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

Limit Theorems in the Stadium Billiard

Abstract: We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a Open image in new window normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.

Monotonic Decrease of the Non-Gaussianness of the Sum of Independent Random Variables: A Simple Proof

Abstract: Artstein, Ball, Barthe, and Naor have recently shown that the non-Gaussianness (divergence with respect to a Gaussian random variable with identical first and second moments) of the sum of independent and identically distributed (i.i.d.) random variables is monotonically nonincreasing. We give a simplified proof using the relationship between non-Gaussianness and minimum mean-square error (MMSE) in Gaussian channels. As Artstein , we also deal with the more general setting of nonidentically distributed random variables

Central limit theorems for Gaussian polytopes

Abstract: Choose $n$ random, independent points in $\R^d$ according to the standard normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random polytope}. We prove that the volume and the number of faces of $K_n$ satisfy the central limit theorem, settling a well known conjecture in the field.

Advanced Statistical Properties of Dispersing Billiards

Abstract: A new approach to statistical properties of hyperbolic dynamical systems emerged recently; it was introduced by L.-S. Young and modified by D. Dolgopyat. It is based on coupling method borrowed from probability theory. We apply it here to one of the most physically interesting models—Sinai billiards. It allows us to derive a series of new results, as well as make significant improvements in the existing results. First we establish sharp bounds on correlations (including multiple correlations). Then we use our correlation bounds to obtain the central limit theorem (CLT), the almost sure invariance principle (ASIP), the law of iterated logarithms, and integral tests.

Capacity of Finite State Channels Based on Lyapunov Exponents of Random Matrices

Abstract: The finite-state Markov channel (FSMC) is a time-varying channel having states that are characterized by a finite-state Markov chain. These channels have infinite memory, which complicates their capacity analysis. We develop a new method to characterize the capacity of these channels based on Lyapunov exponents. Specifically, we show that the input, output, and conditional entropies for this channel are equivalent to the largest Lyapunov exponents for a particular class of random matrix products. We then show that the Lyapunov exponents can be expressed as expectations with respect to the stationary distributions of a class of continuous-state space Markov chains. This class of Markov chains, which is closely related to the prediction filter in hidden Markov models, is shown to be nonirreducible. Hence, much of the standard theory for continuous state-space Markov chains cannot be applied to establish the existence and uniqueness of stationary distributions, nor do we have direct access to a central limit theorem (CLT). In order to address these shortcomings, we utilize several results from the theory of random matrix products and Lyapunov exponents. The stationary distributions for this class of Markov chains are shown to be unique and continuous functions of the input symbol probabilities, provided that the input sequence has finite memory. These properties allow us to express mutual information and channel capacity in terms of Lyapunov exponents. We then leverage this connection between entropy and Lyapunov exponents to develop a rigorous theory for computing or approximating entropy and mutual information for finite-state channels with dependent inputs. We develop a method for directly computing entropy of finite-state channels that does not rely on simulation and establish its convergence. We also obtain a new asymptotically tight lower bound for entropy based on norms of random matrix products. In addition, we prove a new functional CLT for sample entropy and apply this theorem to characterize the error in simulated estimates of entropy. Finally, we present numerical examples of mutual information computation for intersymbol interference (ISI) channels and observe the capacity benefits of adding memory to the input sequence for such channels

Numerical indications of a q-generalised central limit theorem

Abstract: We provide numerical indications of the q-generalised central limit theorem that has been conjectured ( TSALLIS C., Milan J. Math., 73 (2005) 145) in nonextensive statistical mechanics. We focus on N binary random variables correlated in a scale-invariant way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called q-product with q ≤ 1. We show that, in the large-N limit (and after appropriate centering, rescaling, and symmetrisation), the emerging distributions are qe-Gaussians, i.e., p(x) ∝ [1-(1-q_e), β(N)x^2]^{1/(1-q_e)}, with q_e=2-(1/q), and with coefficients β(N) approaching finite values β(∞). The particular case q=q_e=1 recovers the celebrated de Moivre-Laplace theorem.

Central limit theorem for stationary linear processes

Abstract: We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeish [Ann. Probab. 5 (1977) 616-621] and motivated by Gordin [Soviet Math. Dokl. 10 (1969) 1174-1176]. In doing so we shall preserve the generality of the coefficients, including the long range dependence case, and we shall express the variance of partial sums in a form easy to apply. Ergodicity is not required.

Limiting laws of linear eigenvalue statistics for Hermitian matrix models

Abstract: We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n×n Hermitian matrices as n→∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al. [ Commun. Pure Appl. Math.52, 1325–1425 (1999)] for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of q≥2 intervals, then in the global regime the variance of statistics is a quasiperiodic function of n as n→∞ generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not, in general, 1∕2× variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion

Abstract: We prove a nonequilibrium central limit theorem for the position of a tagged particle in the one-dimensional nearest neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile ρ 0 : R → [ 0 , 1 ] .

Generalised extreme value statistics and sum of correlated variables

Abstract: We show that generalised extreme value statistics -the statistics of the k-th largest value among a large set of random variables- can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalisations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

Pseudo-maximum likelihood estimation of ARCH(∞) models

Abstract: Strong consistency and asymptotic normality of the Gaussian pseudo-maximum likelihood estimate of the parameters in a wide class of ARCH(1) processes are established. We require the ARCH weights to decay at least hyperbolically, with a faster rate needed for the central limit theorem than for the law of large numbers. Various rates are illustrated in examples of particular parameteriza- tions in which our conditions are shown to be satis ed.

Gaussian fluctuations of characters of symmetric groups and of Young diagrams

Abstract: We study asymptotics of reducible representations of the symmetric groups Sq for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian; in this way we generalize Kerov's central limit theorem. The considered class consists of representations for which the characters almost factorize and this class includes, for example, the left-regular representation (Plancherel measure), irreducible representations and tensor representations. This class is also closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.

Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

Abstract: The aim of this paper is to prove a stretched-exponential bound for the decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations (Theorem 4). A corollary is the Central Limit Theorem for the Teichm?ller flow on the moduli space of abelian differentials with prescribed singularities (Theorem 10). The proof of Theorem 4 proceeds by the method of Sinai [13] and Bunimovich Sinai [14]: the induction map is approximated by a sequence of Markov chains satisfying the Doeblin condition. The main "loss of memory" estimate is Lemma 4.

Consistent estimation of the memory parameter for nonlinear time series

Abstract: For linear processes, semiparametric estimation of the memory parameter, based on the log-periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξt and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite-sample performance of the estimator is investigated in a small Monte Carlo study.

Gaussian fluctuations for non-Hermitian random matrix ensembles

Abstract: Consider an ensemble of N×N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4+ɛ) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494–529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting N→∞, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type XN(f)=∑k=1Nf(λk) where λ1, λ2, …, λN denote the ensemble eigenvalues and the test function f is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533–605], where the analogous result for random sample covariance matrices is established.

Convergence of the SMC Implementation of the PHD Filte

Abstract: The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order $$p \geq 1$$ , and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.