Showing papers on "Central limit theorem published in 1996"

Weak Convergence and Empirical Processes: With Applications to Statistics

Abstract: 1.1. Introduction.- 1.2. Outer Integrals and Measurable Majorants.- 1.3. Weak Convergence.- 1.4. Product Spaces.- 1.5. Spaces of Bounded Functions.- 1.6. Spaces of Locally Bounded Functions.- 1.7. The Ball Sigma-Field and Measurability of Suprema.- 1.8. Hilbert Spaces.- 1.9. Convergence: Almost Surely and in Probability.- 1.10. Convergence: Weak, Almost Uniform, and in Probability.- 1.11. Refinements.- 1.12. Uniformity and Metrization.- 2.1. Introduction.- 2.2. Maximal Inequalities and Covering Numbers.- 2.3. Symmetrization and Measurability.- 2.4. Glivenko-Cantelli Theorems.- 2.5. Donsker Theorems.- 2.6. Uniform Entropy Numbers.- 2.7. Bracketing Numbers.- 2.8. Uniformity in the Underlying Distribution.- 2.9. Multiplier Central Limit Theorems.- 2.10. Permanence of the Donsker Property.- 2.11. The Central Limit Theorem for Processes.- 2.12. Partial-Sum Processes.- 2.13. Other Donsker Classes.- 2.14. Tail Bounds.- 3.1. Introduction.- 3.2. M-Estimators.- 3.3. Z-Estimators.- 3.4. Rates of Convergence.- 3.5. Random Sample Size, Poissonization and Kac Processes.- 3.6. The Bootstrap.- 3.7. The Two-Sample Problem.- 3.8. Independence Empirical Processes.- 3.9. The Delta-Method.- 3.10. Contiguity.- 3.11. Convolution and Minimax Theorems.- A. Appendix.- A.1. Inequalities.- A.2. Gaussian Processes.- A.2.1. Inequalities and Gaussian Comparison.- A.2.2. Exponential Bounds.- A.2.3. Majorizing Measures.- A.2.4. Further Results.- A.3. Rademacher Processes.- A.4. Isoperimetric Inequalities for Product Measures.- A.5. Some Limit Theorems.- A.6. More Inequalities.- A.6.1. Binomial Random Variables.- A.6.2. Multinomial Random Vectors.- A.6.3. Rademacher Sums.- Notes.- References.- Author Index.- List of Symbols.

Fisher information and stochastic complexity

Abstract: By taking into account the Fisher information and removing an inherent redundancy in earlier two-part codes, a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes. The main condition required is that the maximum-likelihood estimates satisfy the central limit theorem. The same code length is also obtained from the so-called maximum-likelihood code.

Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms

Abstract: We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given ; these show that the negative-definiteness property is not redundant.

Nonparametric testing of closeness between two unknown distribution functions

Abstract: Based on the kernel integrated square difference and applying a central limit theorem for degenerate V-statistic proposed by Hall (1984), this paper proposes a consistent nonparametric test of closeness between two unknown density functions under quite mild conditions. We only require the unknown density functions to be bounded and continuous. Monte Carlo simulations show that the proposed tests perform well for moderate sample sizes.

13 Financial applications of stable distributions

Abstract: Publisher Summary Financial asset returns are the cumulative outcome of a vast number of pieces of information and individual decisions arriving continuously in time. According to the Central Limit Theorem, if the sum of a large number of iid random variates has a limiting distribution after appropriate shifting and scaling, the limiting distribution must be a member of the stable class. It is therefore natural to assume that asset returns are at least approximately governed by a stable distribution if the accumulation is additive, or by a log-stable distribution if the accumulation is multiplicative. The Gaussian is the most familiar and tractable stable distribution, and therefore either it or the log-normal has routinely been postulated to govern asset returns. However, returns are often much more leptokurtic than is consistent with normality. This naturally leads one to consider also the non-Gaussian stable distributions as a model of financial returns.

Multivariate normal approximations by Stein's method and size bias couplings

Abstract: Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

Convergent multiplicative processes repelled from zero: power laws and truncated power laws

Abstract: Random multiplicative processes $w_t =\lambda_1 \lambda_2 ... \lambda_t$ (with 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+\mu)}$. We provide a simple and physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic ($t \to \infty$) distribution of $w_t$ and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable ${1 \over \sqrt{t}}(log w_t - )$; 2) the necessary and sufficient conditions for $P(w_t)$ to be a power law are that < 0 (corresponding to a drift $w_t \to 0$) and that $w_t$ not be allowed to become too small. We discuss several models, previously unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable $\log w_t$ undergoes a random walk biased to the left but is bounded by a repulsive ''force''. We give an approximate treatment, which becomes exact for narrow or log-normal distributions of $\lambda$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $\mu$ is shown exactly to be the solution of $\langle \lambda^{\mu} \rangle = 1$ and is therefore non-universal and depends on the distribution of $\lambda$.

Testing for a change of the long-memory parameter

Abstract: SUMMARY Long-range dependence is often observed in long time series Correlations decay approximately like Ik I2"-2, with H E (0 5, 1), as the lag k tends to infinity The long-term features of the data are essentially characterised by the parameter H Small changes of H have strong implications for the long-term behaviour of the process In particular, rates of convergence of estimators for the mean, and for many other parameters of interest, differ for different values of H For some data sets, H appears to change with time In this paper we consider a simple test of the null hypothesis that H is constant The test is based on a functional central limit theorem for quadratic forms Critical values for the test statistic are given Simulations confirm the validity of the test A data example illustrates its practical application

Universal Hashing and k-Wise Independent Random Variables via Integer Arithmetic without Primes

Abstract: Let u, m≥1 be arbitrary integers and let k≥u. The central result of this paper is that the multiset H={itha,b¦0≤a, b

McKean-Vlasov limit for interacting random processes in random media

Abstract: We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.

The Berry-Esseen bound for student's statistic

Abstract: We prove the Berry-Esseen bound for the Student t-statistic. Under the assumption of a third moment this bound coincides (up to an absolute constant) with the classical Berry-Esseen bound for the mean. In general the distribution of the Student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean, and sometimes faster. For example, rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law.

The central limit theorem for weighted minimal spanning trees on random points

Abstract: Let ${X_i, 1 \leq i 0$, $$\frac{M(X_1, \dots, X_n; \alpha) - EM (X_1, \dots, X_n; \alpha)}{n^{(d-2 \alpha)/2d}} \to N(0, \sigma_{\alpha, d}^2)$$ in distribution for some $\sigma_{\alpha, d}^2 > 0$.

On the asymptotic normality of sequences of weak dependent random variables

Abstract: The aim of this paper is to investigate the asymptotic normality for strong mixing sequences of random variables in the absense of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences. The class of applications include filters of certain Gaussian sequences.

Central limit theorem for a class of random measures associated with germ-grain models

Abstract: We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.

The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees

Abstract: We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.

Large deviations for combinatorial distributions. I. Central limit theorems

Abstract: We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain natural analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius’ version of Cram er’s classical method based on analytic moment generating functions. We thus generalize Cram er’s and Kubilius’ theorems on large deviations.

Balls and Bins: A Study in Negative Dependence

Abstract: This paper investigates the notion of negative dependence amongst random variables and attempts to advocate its use as a simple and unifying paradigm for the analysis of random structures and algorithms. The assumption of independence between random variables is often very convenient for the several reasons. Firstly, it makes analyses and calculations much simpler. Secondly, one has at hand a whole array of powerful mathematical concepts and tools from classical probability theory for the analysis, such as laws of large numbers, central limit theorems and large deviation bounds which are usually derived under the assumption of independence. Unfortunately, the analysis of most randomized algorithms involves random variables that are not independent. In this case, classical tools from standard probability theory like large deviation theorems, that are valid under the assumption of independence between the random variables involved, cannot be used as such. It is then necessary to determine under what conditions of dependence one can still use the classical tools. It has been observed before [32, 33, 38, 8], that in some situations, even though the variables involved are not independent, one can still apply some of the standard tools that are valid for independent variables (directly or in suitably modified form), provided that the variables are dependent in specific ways. Unfortunately, it appears that in most cases somewhat ad hoc strategems have been devised, tailored to the specific situation at hand, and that a unifying underlying theory that delves deeper into the nature of dependence amongst the variables involved is lacking. A frequently occurring scenario underlying the analysis of many randomised algorithms and processes involves random variables that are, intuitively, dependent in the following negative way: if one subset of the variables is "high" then a disjoint subset of the variables is "low". In this paper, we bring to the forefront and systematize some precise notions of negative dependence in the literature, analyse their properties, compare them relative to each other, and illustrate them with several applications. One specific paradigm involving negative dependence is the classical "balls and bins" experiment. Suppose we throw m balls into n bins independently at random. For i in [n], let Bi be the random variable denoting the number of balls in the ith bin. We will often refer to these variables as occupancy numbers. This is a classical probabilistic paradigm [16, 22, 26] (see also [31, sec. 3.1]) that underlies the analysis of many probabilistic algorithms and processes. In the case when the balls are identical, this gives rise to the well-known multinomial distribution [16, sec VI.9]: there are m repeated independent trials (balls) where each trial (ball) can result in one of the outcomes E1, ..., En (bins). The probability of the realisation of event Ei is pi for i in [n] for each trial. (Of course the probabilities are subject to the condition Sum_i pi = 1.) Under the multinomial distribution, for any integers m1, ..., mn such that Sum_i mi = m the probability that for each i in [n], event Ei occurs mi times is m! m1! : : :mn!pm1 1 : : :pmn n : The balls and bins experiment is a generalisation of the multinomial distribution: in the general case, one can have an arbitrary set of probabilities for each ball: the probability that ball k goes into bin i is pi;k, subject only to the natural restriction that for each ball k, P i pi;k = 1. The joint distribution function correspondingly has a more complicated form. A fundamental natural question of interest is: how are these Bi related? Note that even though the balls are thrown independently of each other, the Bi variables are not independent; in particular, their sum is fixed to m. Intuitively, the Bi's are negatively dependent on each other in the manner described above: if one set of variables is "high", a disjoint set is "low". However, establishing such assertions precisely by a direct calculation from the joint distribution function, though possible in principle, appears to be quite a formidable task, even in the case where the balls are assumed to be identical. One of the major contributions of this paper is establishing that the the Bi are negatively dependent in a very strong sense. In particular, we show that the Bi variables satisfy negative association and negative regression, two strong notions of negative dependence that we define precisely below. All the intuitively obvious assertions of negative dependence in the balls and bins experiment follow as easy corollaries. We illustrate the usefulness of these results by showing how to streamline and simplify many existing probabilistic analyses in literature.

A Refinement of the Central Limit Theorem for Sums of Independent Random Indicators

Abstract: This paper introduces an estimate of approximation errors for the distribution function of a sum of random indicators. The approximation is demonstrated with the help of the problem of estimating the distribution function of the empty cells number in the equiprobable scheme for group distribution of particles.

Large deviation asymptotics for Anosov flows

Abstract: We derive precise asymptotic formulae for large deviation probabilities for suspensions of subshifts of finite type. As a corollary, we give a stronger version of the Central Limit Theorem. We apply our results to transitive Anosov flows, giving a result describing fluctuations in the volume of Bowen balls and an asymptotic large deviation formula of a homological nature involving Schwartzmann’s winding cycle.

Zipf's law, the central limit theorem, and the random division of the unit interval.

Abstract: It is shown that a version of Mandelbrot's monkey-at-the-typewriter model of Zipf's inverse power law is directly related to two classical areas in probability theory: the central limit theorem and the ``broken stick'' problem, i.e., the random division of the unit interval. The connection to the central limit theorem is proved using a theorem on randomly indexed sums of random variables [A. Gut, Stopped Random Walks: Limit Theorems and Applications (Springer, New York, 1987)]. This reveals an underlying log-normal structure of pseudoword probabilities with an inverse power upper tail that clarifies a point of confusion in Mandelbrot's work. An explicit asymptotic formula for the slope of the log-linear rank-size law in the upper tail of this distribution is also obtained. This formula relates to known asymptotic results concerning the random division of the unit interval that imply a slope value approaching -1 under quite general conditions. The role of size-biased sampling in obscuring the bottom part of the distribution is explained and connections to related work are noted. \textcopyright{} 1996 The American Physical Society.

Zariski closure and the dimension of the Gaussian low of the product of random matrices. I

Abstract: We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG.

Repeated Games, Duality and the Central Limit Theorem

Abstract: This paper is concerned with the repeated zero-sum games with one-sided information and standard signaling. We introduce here dual games that allow us to analyze the “Markovian” behavior of the uninformed player, and to explicitly compute his optimal strategies. We then apply our results on the dual games to explain the appearance of the normal density in the n-1/2-term of the asymptotic expansion of vn as a consequence of the Central Limit Theorem.

R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata

Abstract: We prove that certain (discrete time) probabilistic automata which can be absorbed in a "null state" have a normalized quasi-stationary distribution (when restricted to the states other than the null state). We also show that the conditional distribution of these systems, given that they are not absorbed before time n, converges to an honest probability distribution; this limit distribution is concentrated on the configurations with only finitely many "active or occupied" sites. A simple example to which our results apply is the discrete time version of the subcritical contact process on $\mathbb{Z}^d$ or oriented percolation on $\mathbb{Z}^d$ (for any $d \geq 1$) as seen from the "leftmost particle." For this and some related models we prove in addition a central limit theorem for $n^{-1/2}$ times the position of the leftmost particle (conditioned on survival until time n). The basic tool is to prove that our systems are R-positive-recurrent.

Mixing Properties and Central Limit Theorem for a Class of Non-Identical Piecewise Monotonic C2 — Transformations

Abstract: For a sequence T(1), T(2),…of piecewise monotonic C2 - transformations of the unit interval I onto itself, we prove exponential ψ- mixing, an almost Markov property and other higher-order mixing properties. Furthermore, we obtain optimal rates of convergence in the central limit Theorem and large deviation relations for the sequence fk oT(k−1)o…oT(1), k=1, 2, …, provided that the real-valued functions f1, f2,…on I are of bounded variation and the corresponding probability measure on I possesses a positive, Lipschitz-continuous Lebesgue density.

Optimal rates of convergence in the CLT for quadratic forms

Abstract: We prove optimal convergence rates in the central limit theorem for sums ${\bf R}^k.$ Assuming a fourth moment, we obtain a Berry-Esseen type bound of $O(N^{-1})$ for the probability of hitting a ball provided that $k\leq 5$. The proof still requires a technical assumption related to the independence of coordinate sums.

On extreme values in stationary sequences

Abstract: Given a stationary sequence of random variables, we show that O'Brien's condition is necessary and sufficient for the sample maximum to have the same limiting distribution as the sample maximum of independent and identically distributed random variables. The rate of convergence in Newell's limit theorem is shown to be O(1/n).