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Showing papers on "Central limit theorem published in 2011"


Book
23 Dec 2011
TL;DR: In this article, the traditional diffusion model was extended to the vector fractional diffusion model, which is the state-of-the-art diffusion model for the problem of diffusion.
Abstract: Preface 1 Introduction 1.1 The traditional diffusion model 1.2 Fractional diffusion 2 Fractional Derivatives 2.1 The Grunwald formula 2.2 More fractional derivatives 2.3 The Caputo derivative 2.4 Time-fractional diffusion 3 Stable Limit Distributions 3.1 Infinitely divisible laws 3.2 Stable characteristic functions 3.3 Semigroups 3.4 Poisson approximation 3.5 Shifted Poisson approximation 3.6 Triangular arrays 3.7 One-sided stable limits 3.8 Two-sided stable limits 4 Continuous Time Random Walks 4.1 Regular variation 4.2 Stable Central Limit Theorem 4.3 Continuous time random walks 4.4 Convergence in Skorokhod space 4.5 CTRW governing equations 5 Computations in R 5.1 R codes for fractional diffusion 5.2 Sample path simulations 6 Vector Fractional Diffusion 6.1 Vector random walks 6.2 Vector random walks with heavy tails 6.3 Triangular arrays of random vectors 6.4 Stable random vectors 6.5 Vector fractional diffusion equation 6.6 Operator stable laws 6.7 Operator regular variation 6.8 Generalized domains of attraction 7 Applications and Extensions 7.1 LePage Series Representation 7.2 Tempered stable laws 7.3 Tempered fractional derivatives 7.4 Pearson Diffusion 7.5 Classification of Pearson diffusions 7.6 Spectral representations of the solutions of Kolmogorov equations 7.7 Fractional Brownian motion 7.8 Fractional random fields 7.9 Applications of fractional diffusion 7.10 Applications of vector fractional diffusion Bibliography Index

647 citations


Book
23 Oct 2011
TL;DR: The Central Limit Theorem for Functions of a Finite Number of Increments (CILT) as discussed by the authors is a generalization of the Central Limit theorem for functions of an increasing number of incrementals.
Abstract: Part I Introduction and Preliminary Material.- 1.Introduction .- 2.Some Prerequisites.- Part II The Basic Results.- 3.Laws of Large Numbers: the Basic Results.- 4.Central Limit Theorems: Technical Tools.- 5.Central Limit Theorems: the Basic Results.- 6.Integrated Discretization Error.- Part III More Laws of Large Numbers.- 7.First Extension: Random Weights.- 8.Second Extension: Functions of Several Increments.- 9.Third Extension: Truncated Functionals.- Part IV Extensions of the Central Limit Theorems.- 10.The Central Limit Theorem for Random Weights.- 11.The Central Limit Theorem for Functions of a Finite Number of Increments.- 12.The Central Limit Theorem for Functions of an Increasing Number of Increments.- 13.The Central Limit Theorem for Truncated Functionals.- Part V Various Extensions.- 14.Irregular Discretization Schemes. 15.Higher Order Limit Theorems.- 16.Semimartingales Contaminated by Noise.- Appendix.- References.- Assumptions.- Index of Functionals.- Index.

409 citations


Proceedings ArticleDOI
06 Jun 2011
TL;DR: A new approach to characterizing the unobserved portion of a distribution is introduced, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size.
Abstract: We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n/log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1/n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used.In the second half of the paper, we provide a matching lower bound of Ω(n/log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n/2O(√log n).To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community.

334 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigen values of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line.
Abstract: We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line, where b ′ x is white noise. In doing so we extend the definition of the Tracy-Widom(β) distributions to all β > 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

279 citations


Journal ArticleDOI
TL;DR: General convergence results, including exponential deviation inequalities and central limit theorems, are established and time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain.
Abstract: Computing smoothing distributions, the distributions of one or more states conditional on past, present, and future observations is a recurring problem when operating on general hidden Markov models The aim of this paper is to provide a foundation of particle-based approximation of such distributions and to analyze, in a common unifying framework, different schemes producing such approximations In this setting, general convergence results, including exponential deviation inequalities and central limit theorems, are established In particular, time uniform bounds on the marginal smoothing error are obtained under appropriate mixing conditions on the transition kernel of the latent chain In addition, we propose an algorithm approximating the joint smoothing distribution at a cost that grows only linearly with the number of particles

172 citations


Proceedings ArticleDOI
01 Jun 2011
TL;DR: A survey of the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions' existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market is given in this article.
Abstract: We give a survey of the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions’ existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market. We also present our new framework of nonlinear expectation and its applications to financial risk measures under uncertainty of probability distributions. The generalized form of law of large numbers and central limit theorem under sublinear expectation shows that the limit distribution is a sublinear Gnormal distribution. A new type of Brownian motion, G-Brownian motion, is constructed which is a continuous stochastic process with independent and stationary increments under a sublinear expectation (or a nonlinear expectation).

164 citations


Posted Content
TL;DR: In this article, the Central Limit Theorem for linear eigenvalue statistics under weak conditions on the number of derivatives of the test functions and also on the entries moments was proved for Wigner matrices and sample covariance matrices.
Abstract: We consider two classical ensembles of the random matrix theory: the Wigner matrices and sample covariance matrices, and prove Central Limit Theorem for linear eigenvalue statistics under rather weak (comparing with results known before) conditions on the number of derivatives of the test functions and also on the number of the entries moments Moreover, we develop a universal method which allows one to obtain automatically the bounds for the variance of differentiable test functions, if there is a bound for the variance of the trace of the resolvent of random matrix The method is applicable not only to the Wigner and sample covariance matrices, but to any ensemble of random matrices

155 citations


Book ChapterDOI
01 Jan 2011
TL;DR: In this paper, the central limit theorem for a sequence of dependent random variables X 1, X 2, X 3, X 4, X 5, X 6, X 7, X 8, X 9
Abstract: This paper presents a central limit theorem for a sequence of dependent random variables X 1 , X 2

128 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems, and they focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure.
Abstract: We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis-Hastings algorithm that adds an accept-reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein-Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an $L^2$-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

120 citations


BookDOI
01 Jan 2011

116 citations


Journal ArticleDOI
TL;DR: In this article, a nonparametric regression with spatial, or spatio-temporal, data is considered and sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates.

Journal ArticleDOI
TL;DR: Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable and the length of a random geometric graph are investigated.
Abstract: A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-Ito chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.

Book
18 May 2011
TL;DR: Probability Avoiding being a sure loser Disjoint events Events not necessarily disjoint Random variables, also known as uncertain quantities Finite number of values Other properties of expectation Coherence implies not a sure winner Expectations and limits Conditional Probability and Bayes Theorem Conditional probability The Birthday Problem Simpson's Paradox Bayes theorem Independence of events The Monty Hall problem Gambler's Ruin problem Iterated Expectation and Independence The binomial and multinomial distributions Sampling without replacement Variance and covariance A short introduction to multivariate thinking Tchebychev's
Abstract: Probability Avoiding being a sure loser Disjoint events Events not necessarily disjoint Random variables, also known as uncertain quantities Finite number of values Other properties of expectation Coherence implies not a sure loser Expectations and limits Conditional Probability and Bayes Theorem Conditional probability The Birthday Problem Simpson's Paradox Bayes Theorem Independence of events The Monty Hall problem Gambler's Ruin problem Iterated Expectations and Independence The binomial and multinomial distributions Sampling without replacement Variance and covariance A short introduction to multivariate thinking Tchebychev's inequality Discrete Random Variables Countably many possible values Finite additivity Countable Additivity Properties of countable additivity Dynamic sure loss Probability generating functions Geometric random variables The negative binomial random variable The Poisson random variable Cumulative distribution function Dominated and bounded convergence Continuous Random Variables Introduction Joint distributions Conditional distributions and independence Existence and properties of expectations Extensions An interesting relationship between cdf's and expectations of continuous random variables Chapter retrospective so far Bounded and dominated convergence The Riemann-Stieltjes integral The McShane-Stieltjes Integral The road from here The strong law of large numbers Transformations Introduction Discrete Random Variables Univariate Continuous Distributions Linear spaces Permutations Number systems DeMoivre's formula Determinants Eigenvalues, eigenvectors and decompositions Non-linear transformations The Borel-Kolmogorov paradox Normal Distribution Introduction Moment generating functions Characteristic functions Trigonometric Polynomials A Weierstrass approximation theorem Uniqueness of characteristic functions Characteristic function and moments Continuity Theorem The Normal distribution Multivariate normal distributions Limit theorems Making Decisions Introduction An example In greater generality The St. Petersburg Paradox Risk aversion Log (fortune) as utility Decisions after seeing data The expected value of sample information An example Randomized decisions Sequential decisions Conjugate Analysis A simple normal-normal case A multivariate normal case, known precision The normal linear model with known precision The gamma distribution Uncertain Mean and Precision The normal linear model, uncertain precision The Wishart distribution Both mean and precision matrix uncertain The beta and Dirichlet distributions The exponential family Large sample theory for Bayesians Some general perspective Hierarchical Structuring of a Model Introduction Missing data Meta-analysis Model uncertainty/model choice Graphical Hierarchical Models Causation Markov Chain Monte Carlo Introduction Simulation The Metropolis Hasting Algorithm Extensions and special cases Practical considerations Variable dimensions: Reversible jumps Multiparty Problems A simple three-stage game Private information Design for another's analysis Optimal Bayesian Randomization Simultaneous moves The Allais and Ellsberg paradoxes Forming a Bayesian group Exploration of Old Ideas Introduction Testing Confidence intervals and sets Estimation Choosing among models Goodness of fit Sampling theory statistics Objective" Bayesian Methods Epilogue: Applications Computation A final thought

Journal ArticleDOI
TL;DR: In this article, the authors derived a functional central limit theorem for the joint dynamics of the bid and ask queues and showed that, when the frequency of order arrivals is large, the intraday dynamics of a limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant.
Abstract: We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

Journal ArticleDOI
TL;DR: A weighted version of the k-nearest neighbor density estimate is introduced and a general central limit theorem under the lightest possible conditions is presented.
Abstract: Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the k-nearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical k-nearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and real-life data.

Journal ArticleDOI
TL;DR: The hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n → 1 in probability and, more precisely, (Hn − βn log n)/, where βn = λn/(λn − 1), has a limiting standard normal distribution.
Abstract: In this paper we explore first passage percolation (FPP) on the ErdA‘s-Renyi random graph Gn(pn), where we assign independent random weights, having an exponential distribution with rate 1, to the edges. In the sparse regime, i.e., when npn → λ > 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to (λ log n)/(λ-1). Furthermore, we prove that the minimal weight centred by (log n)/(λ-1) converges in distribution. We also investigate the dense regime, where npn → ∞. We find that although the base graph is ultra-small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n → 1 in probability and, more precisely, (Hn-βn log n)/ , where βn = λn/(λn-1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ , a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount. This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the ErdA‘s-Renyi random graph and thinned continuous-time branching processes.

Posted Content
TL;DR: In this paper, a nonparametric estimation of the quadratic covariation of non-synchronously observed It\^o processes in an additive microstructure noise model is presented.
Abstract: The article is devoted to the nonparametric estimation of the quadratic covariation of non-synchronously observed It\^o processes in an additive microstructure noise model. In a high-frequency setting, we aim at establishing an asymptotic distribution theory for a generalized multiscale estimator including a feasible central limit theorem with optimal convergence rate on convenient regularity assumptions. The inevitably remaining impact of asynchronous deterministic sampling schemes and noise corruption on the asymptotic distribution is precisely elucidated. A case study for various important examples, several generalizations of the model and an algorithm for the implementation warrant the utility of the estimation method in applications.

Journal ArticleDOI
TL;DR: In this paper, the asymptotic behavior of power and multipower variations of processes with spot intermittency was studied and the central limit theorem for triangular Gaussian schemes was proved.
Abstract: In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$: $$Y_t = ∫_{−∞}^tg(t − s)σ_sW(\mathrm{d}s) + Z_t,$$ where $g : (0, ∞) → ℝ$ is deterministic, $σ > 0$ is a random process, $W$ is the stochastic Wiener measure and $Z$ is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency $σ$. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of $Y$ as a basis for studying properties of the intermittency process $σ$. Notably the processes $Y$ are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.

Journal ArticleDOI
TL;DR: In this paper, the authors revisited the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed entries with mean zero and unit variance.
Abstract: In this note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed (i.i.d.) entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere.

Journal ArticleDOI
TL;DR: In this article, the authors study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree, and prove a law of large numbers for the empirical measure of individuals alive at time $t.
Abstract: We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton-Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time $t$. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. This latter has the same generator as the Markov process along the branches plus additional branching events, associated with jumps of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time $t$ favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching Levy processes and ancestral lineages.

Journal ArticleDOI
TL;DR: An Edgeworth-type expansion for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions, was established in this paper.
Abstract: An Edgeworth-type expansion is established for the entropy distance to the class of normal distributions of sums of i.i.d. random variables or vectors, satisfying minimal moment conditions.

Journal ArticleDOI
TL;DR: In this paper, a class of kinetic-type equations on the real line is introduced, which constitute extensions of the classical Kac caricature, and the collisional gain operators are defined by smoothing transformations with rather general properties.
Abstract: We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law να, then the limit is a scale mixture of να. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

Journal ArticleDOI
TL;DR: The Tweedie Poisson-gamma model for Taylor's law was proposed by as mentioned in this paper, where the authors show that the cumulant generating functions derived from this thermodynamic model correspond to those derived over a quarter century earlier for a class of probabilistic models known as the TweedIE exponential dispersion models, characterized by variance-to-mean power functions.
Abstract: A power function relationship observed between the variance and the mean of many types of biological and physical systems has generated much debate as to its origins This Taylor's law (or fluctuation scaling) has been recently hypothesized to result from the second law of thermodynamics and the behavior of the density of states This hypothesis is predicated on physical quantities like free energy and an external field; the correspondence of these quantities with biological systems, though, remains unproven Questions can be posed as to the applicability of this hypothesis to the diversity of observed phenomena as well as the range of spatial and temporal scales observed with Taylor's law We note that the cumulant generating functions derived from this thermodynamic model correspond to those derived over a quarter century earlier for a class of probabilistic models known as the Tweedie exponential dispersion models These latter models are characterized by variance-to-mean power functions; their phenomenological basis rests with a central-limit-theorem-like property that causes many statistical systems to converge mathematically toward a Tweedie form We review evaluations of the Tweedie Poisson-gamma model for Taylor's law and provide three further cases to test: the clustering of single nucleotide polymorphisms (SNPs) within the horse chromosome 1, the clustering of genes within human chromosome 8, and the Mertens function This latter case is a number theoretic function for which a thermodynamic model cannot explain Taylor's law, but where Tweedie convergence remains applicable The Tweedie models are applicable to diverse biological, physical, and mathematical phenomena that express power variance functions over a wide range of measurement scales; they provide a probabilistic description for Taylor's law that allows mechanistic insight into complex systems without the assumption of a thermodynamic mechanism

Journal ArticleDOI
TL;DR: In this article, the authors consider the problem of convergence of a Levy-driven S.D. by a Brownian S.E. and show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them.
Abstract: We consider the approximate Euler scheme for Levy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Levy measure of the driving process behaves like |z |−1−α dz near 0 , for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα . For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Levy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of order n α /(2−α ) , which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady [J. Komlos, P. Major and G. Tusnady, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].

Journal ArticleDOI
TL;DR: In this article, the authors deal with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants.

Journal ArticleDOI
TL;DR: In this article, weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Frechet means) of independent non-identically distributed random variables taking values in Riemannian manifolds are proved.
Abstract: We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Frechet means) of independent nonidentically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg–Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Frechet means, while extending its scope.

Journal ArticleDOI
TL;DR: In this paper, a central limit theorem for the integrated squared error is derived, and a hypothesis-testing procedure is proposed to assess the mean pattern of lifetime-maximum wind speeds of global tropical cyclones from 1981 to 2006.
Abstract: The paper considers testing whether the mean trend of a nonstationary time series is of certain parametric forms. A central limit theorem for the integrated squared error is derived, and a hypothesis-testing procedure is proposed. The method is illustrated in a simulation study, and is applied to assess the mean pattern of lifetime-maximum wind speeds of global tropical cyclones from 1981 to 2006. We also revisit the trend pattern in the central England temperature series. Copyright 2011, Oxford University Press.

Journal ArticleDOI
TL;DR: In this article, the authors considered excited random walks on Z with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies.
Abstract: We consider excited random walks (ERWs) on Z with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta > 4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta is an element of (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [2] for delta is an element of (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.

Journal ArticleDOI
TL;DR: In this article, a statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths, and three particular cases where the IR-roughness exists and is explicitly computed are considered.
Abstract: A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a Levy process with $\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\alpha \in (0,2)$ following a central limit theorem.

Journal ArticleDOI
TL;DR: An analysis of the joint impact of path-loss, shadowing and fast fading on cellular networks and two analytical methods are developed to express the outage probability, based on the Fenton-Wilkinson approach and the central limit theorem for causal functions.
Abstract: In this paper, we propose an analysis of the joint impact of path-loss, shadowing and fast fading on cellular networks. Two analytical methods are developed to express the outage probability. The first one based on the Fenton-Wilkinson approach, approximates a sum of log-normal random variables by a log-normal random variable and approximates fast fading coefficients in interference terms by their average value. We denote it FWBM for Fenton-Wilkinson based method. The second one is based on the central limit theorem for causal functions. It allows to approximate a sum of positive random variables by a Gamma distribution. We denote it CLCFM for central limit theorem for causal functions method. Each method allows to establish a simple and easily computable outage probability formula, which jointly takes into account path-loss, shadowing and fast fading. We compute the outage probability, for mobile stations located at any distance from their serving BS, by using a fluid model network that considers the cellular network as a continuum of BS. We validate our approach by comparing all results to extensive Monte Carlo simulations performed in a traditional hexagonal network and we provide the limits of the two methods in terms of system parameters. The proposed framework is a powerful tool to study performances of cellular networks, e.g., OFDMA systems (WiMAX, LTE).