Showing papers on "Central limit theorem published in 1993"
TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
1,008 citations
Book•
01 Jan 1993TL;DR: The second edition of Stroock's text as mentioned in this paper is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis and includes more than 750 exercises.
Abstract: This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.
509 citations
Book•
01 Jan 1993TL;DR: In this paper, the asymptotic distribution of the QMLE and the information matrix equality is analyzed. And the Radon-Nikodym theorem and central limit theorem are discussed.
Abstract: 1. Introductory remarks 2. Probability densities, likelihood functions and the quasi-maximum likelihood estimator 3. Consistency of the QMLE 4. Correctly specified models of density 5. Correctly specified models of conditional expectation 6. The asymptotic distribution of the QMLE and the information matrix equality 7. Asymptotic efficiency 8. Hypothesis testing and asymptotic covariance matrix estimation 9. Specification testing via m-tests 10. Applications of m-testing 11. Information matrix testing 12. Conclusion Appendix 1. Elementary concepts of measure theory and the Radon-Nikodym theorem Appendix 2. Uniform laws of large numbers Appendix 3. Central limit theorems.
387 citations
TL;DR: The U-process theory as discussed by the authors is a collection of U-statistics over a family H of kernels h of m variables, based on a probability measure P on (S,S).
Abstract: A U-process is a collection of U-statistics. Concretely, a U-process over a family H of kernels h of m variables, based on a probability measure P on (S,S), is the collection \(\{ U_{n}^{{(m)}}(h,P):h \in \mathcal{H}\} \) of U-statistics. This chapter is devoted to the asymptotic theory of U-processes: we are interested in finding conditions on H and P ensuring that the law of large numbers, or the central limit theorem, or the law of the iterated logarithm, hold for \( U_n^{(m)} (h,P), \)uniformly in h ∈ H. The theory of empirical processes deals with the same questions for the case m = 1, and U-process theory is patterned after it. This is a relatively new subject, at least in the generality presented here (H being an arbitrary collection of kernels which are defined on a general measurable space). There is therefore a need to indicate its usefulness. To this effect, a section on applications is added at the end of the chapter (Section 5.5); there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median, and in general, of M-estimators, and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data.
308 citations
TL;DR: In this paper, an exchangeably weighted bootstrap of the general function-indexed empirical process was considered and sufficient conditions on the bootstrap weights for the central limit theorem to hold for the bootstrapped empirical process, almost surely and in probability.
Abstract: We consider an exchangeably weighted bootstrap of the general function-indexed empirical process. We find sufficient conditions on the bootstrap weights for the central limit theorem to hold for the bootstrapped empirical process, almost surely and in probability. The results resemble those of Gine and Zinn for Efron's bootstrap. As a corollary we obtain a result on the almost sure convergence in distribution of the Efron-bootstrapped empirical process with arbitrary sample size. A large number of bootstrap resampling schemes emerge as special cases of our results.
273 citations
TL;DR: New techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs.
Abstract: New techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs. Consistency is shown to follow from a certain local invertibility property, and asymptotic distributions are derived from a generalized implicit function theorem that characterizes asymptotic behavior in situations where estimates are subjected to constraints and estimation functionals are nonsmooth.
205 citations
TL;DR: In this article, random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability, and the focus is on limit theorems as the number being partitioned approaches ∞.
Abstract: Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches ∞. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct
179 citations
TL;DR: In this paper, it was shown that under a suitable regularity condition the central limit theorem can be obtained as a consequence of the large deviation principle, under the condition that the large deviations are bounded.
127 citations
TL;DR: In this article, the authors established central limit theorems for the smoothed unbiased periodogram ∫ π −π ⋯∫π −π g ( ω,θ ){I ∗ T,X (ω )−EI ∆ ∆,X( ω )}dω 1⋯dω r, where { X t } is a stationary r -dimensional random process or random field, possibly with long-range dependence, which is not necessarily Gaussian.
103 citations
TL;DR: The proofs of central limit theorems for strictly stationary sequences of random variables are based on approximating the partial sums of the process by martingales (cf. as discussed by the authors ).
90 citations
TL;DR: In this paper, the authors studied log-average versions of classical limit theorems and gave necessary and sufficient conditions for the law of large numbers in log density, which they called the stable limit theorem.
Abstract: Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums S(k) of independent r.v.'s we prove under mild technical conditions that (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)I{S(k)/a(k) is-an-element-of .} --> G(.) (a.s.) if and only if (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)P(S(k)/a(k) is-an-element-of .) --> G(.). A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.
TL;DR: In this paper, various statistical tests to determine whether an observed time series exhibits the so-called Hurst effect are presented, based on the fact that for the family of processes in the Brownian domain of attraction, R*n/((0n))1/2 converges in distribution to a nondegenerate random variable with known distribution.
Abstract: After more than 40 years the so-called Hurst effect remains an open problem in stochastic hydrology. Historically, its existence has been explained either by preasymptotic behavior of the rescaled adjusted range R*n, certain classes of nonstationari ty in time series, infinite memory, or erroneous estimation of the Hurst exponent. Various statistical tests to determine whether an observed time series exhibits the Hurst effect are presented. The tests are based on the fact that for the family of processes in the Brownian domain of attraction, R*n/((0n))1/2 converges in distribution to a nondegenerate random variable with known distribution (functional central limit theorem). The scale of fluctuation 0, defined as the sum of the correlation function, plays a key role. Application of the tests to several geophysical time series seems to indicate that they do not exhibit the Hurst effect, although those series have been used as examples of its existence, and furthermore the traditional pox diagram method to estimate the Hurst exponent gives values larger than 0.5. It turned out that the coefficient in the relation of/?* versus n, which is directly proportional to the scale of fluctuation, was more important than the exponent. The Hurst effect motivated the popularization of I//noises and related ideas of fractals and scaling. This work illustrates how delicate the procedures to deal with infinity must be.
TL;DR: In this article, a Berry-Esseen theorem for convergence of general nonlinear multivariate sampling statistics with normal limit distribution is derived via a multivariate extension of Stein's method, which generalizes in particular previous results of Bolthausen for one-dimensional linear rank statistics and van Zwet and Friedrich for general functions of independent random elements.
Abstract: A Berry-Esseen theorem for the rate of convergence of general nonlinear multivariate sampling statistics with normal limit distribution is derived via a multivariate extension of Stein's method. The result generalizes in particular previous results of Bolthausen for one-dimensional linear rank statistics, one-dimensional results of van Zwet and Friedrich for general functions of independent random elements and provides convergence bounds for general multivariate sampling statistics without restrictions on the sampling proportions.
TL;DR: The set of all n-long words from a finite alphabet into a probability space with a Bernoulli distribution is made and errata in Mood's covariance matrices for runs count statistics are correct.
Abstract: Make the set of all n-long words from a finite alphabet into a probability space with a Bernoulli distribution. The joint probability distribution for `independent' counts of subwords from a finite set usually satisfies a central limit theorem, with means and covariances growing asymptotically with n. This usually remains true even when we condition on the values of other word counts, including the possibility of excluding certain words entirely. A local limit theorem also often holds. Practical formulas are given for computing the parameters when there is no conditioning. Impractical formulas are given for the general case. We correct errata in Mood's covariance matrices for runs count statistics.
Journal Article•
TL;DR: In this paper, the authors investigated the asymptotic behavior of the variance function V of a natural exponential family with support S c R and showed that V(O) = 0 and that the right derivative at zero is V'(O+) = inf {S\{0}}.
Abstract: We investigate the asymptotic behaviour of the variance function V of a natural exponential family with support S c R. If inf S = 0, we show that V(O) = 0 and that the right derivative at zero is V'(O+) = inf {S\{0}}. Using a theorem by Mora (1990) we show that if lim c -P V(cp) = uP uniformly on compact subsets in p for either c -+ oo or c -+0, then p 0 (0, 1), and the corresponding exponential dispersion model, suitably scaled, converges to a member of the Tweedie family of exponential dispersion models, corresponding to the variance function V(p) = pP. This gives a kind of central limit theory for exponential dispersion models. In the case p = 2, the limiting family is gamma, and the result essentially follows from Tauber theory. For p = 1, we obtain a version of the Poisson law of small numbers, generalizing a result for discrete models due to Jorgensen (1986). For 1 2 or p < 0 the limiting families are generated by respectively positive stable distributions or extreme stable distributions, in the latter case inf S =-oo. A number of illustrative examples are considered.
01 Sep 1993
TL;DR: In this article, the authors relax the restriction that q be a prime power, and consider a multiset construction in which the total number of possibilities of weight n is qn.
Abstract: We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.
TL;DR: In this article, necessary and sufficient conditions are established for cumulative process (associated with regenerative processes) to obey several classical limit theorems; e.g., a strong law of large numbers, a law of the iterated logarithm and a functional central limit theorem.
TL;DR: The main result of as discussed by the authors provides necessary and sufficient conditions for the complete convergence of the Cesaro means of i.i.d random variables for divergent series of real numbers.
Abstract: Various results generalizing summation methods for divergent series of real numbers to analogous results for independent, identically distributed random variables have appeared during the last two decades. The main result of this paper provides necessary and sufficient conditions for the complete convergence of the Cesaro means of i.i.d random variables.
TL;DR: In this paper, a functional central limit theorem for LPQD processes is proved for covariances, satisfying some assumptions on the covariance and the moment condition sup j ≥ 1 E | X j | 2+ρ 0.
TL;DR: In this article, the central limit theorem in Davidson [2] is extended to allow cases where the variances of sequence coordinates can be tending to zero, and a trade-off is demonstrated between the degree of dependence and the rate of degeneration.
Abstract: The central limit theorem in Davidson [2] is extended to allow cases where the variances of sequence coordinates can be tending to zero. A trade-off is demonstrated between the degree of dependence and the rate of degeneration. For the martingale difference case, it is sufficient for the sum of the variances to diverge at the rate of log n .
TL;DR: In this article, the authors considered a random field of real-valued positively or negatively associated random variables over the lattice of points in the d-dimensional Euclidean space with integer numbers as their coordinates, and assumed that the random variables are identically distributed with distribution function F and probability density function f.
Abstract: Consider a random field of real-valued positively or negatively associated random variables over the lattice of points in the d-dimensional Euclidean space with integer numbers as their coordinates, and suppose that the random variables are identically distributed with distribution function F and probability density function f. On the basis of an expanding segment of the underlying random field, the empirical distribution function is constructed, as well as kernel estimates for f, its derivatives, and the associated hazard rate. Under some additional weak conditions, it is shown that the empirical distribution function converges almost surely and uniformly to F, with rates. Similar results are established for the estimates of f, its derivatives, and the hazard rate, provided the support of F is a finite interval in the real line.
TL;DR: In this article, the authors give a short derivation of (uniform and non-uniform) quantitative results on the speed of convergence in the central limit theorem for the compound Poisson distribution.
Abstract: We give a short derivation of (uniform and nonuniform) quantitative results on the speed of convergence in the central limit theorem for the compound Poisson distribution.
TL;DR: In this paper, the central limit theorem for perturbed sample quantiles based on a kernel k and a sequence of window-width an > 0 was studied and necessary and sufficient conditions for these quantiles to hold for the sequence {an}.
TL;DR: The constructions presented in the above paper use a finite field which is either GF(2") or GF( p) for some prime p, and assume that one has a representation of the field (i.e., an irreducible polynomial of degree m or the prime p), and use the known pairwise independent constructions in a slightly less straightforward manner.
Abstract: The constructions presented in the above paper use a finite field which is either GF(2") or GF( p) for some prime p. The constructions are presented assuming that one has a representation of the field (i.e., an irreducible polynomial of degree m or the prime p , respectively). Such representations could be found, with overwhelmingly high probability, in probabilistic polynomial-time (in m or I P I , respectively). The paper contained some remarks indicating how to achieve this goal using only a linear number of unbiased coin tosses. However, in retrospect we feel that some more details should be given. For uniformity of exposition, we denote by m the logarithm (to base 2) of the size of the required field. The field representations in both cases can be encoded by strings of length m. Furthermore, in both cases about a fraction of all m-bit long strings are valid representations, and one can efficiently determine whether a string is a valid representation. Hence, selecting a valid representation can be done by selecting candidates at random until a valid one is found. As indicated in the paper, to save on randomness, we use an efficient sampling which in turn uses a construction of a sequence of pairwise independent variables, each uniformly distributed in (0, l}". The problem which arises is that the standard constructions of such pairwise independent sequences use a field of similar cardinality (i.e., with at least 2"' elements), and hence we need a representation for this field, which brings us to a circular argument. The solution is to use the known pairwise independent constructions in a slightly less straightforward manner. Specifically, suppose we need to generate a t-long sequence of pairwise independent m-bit strings (e.g., in the above application t = O(rn)). The idea is to 1 m
TL;DR: This paper determines the variance constant associated with the central limit theorem for the difference between the two averages when PASTA holds, which helps determine which finite average is a more asymptotically efficient estimator of its limit.
Abstract: In this paper we establish a joint central limit theorem for customer and time averages by applying a martingale central limit theorem in a Markov framework. The limiting values of the two averages appear in the translation terms. This central limit theorem helps to construct confidence intervals for estimators and perform statistical tests. It thus helps determine which finite average is a more asymptotically efficient estimator of its limit. As a basis for testing for PASTA Poisson arrivals see time averages, we determine the variance constant associated with the central limit theorem for the difference between the two averages when PASTA holds.
TL;DR: In this article, a space-time jump Markov process is compared with the solution of a nonlinear reaction-diffusion equation, and the central limit theorem and law of large numbers are shown to hold in the nicest possible state spaces.
TL;DR: In this article, the authors considered the problem of testing symmetry of a distribution in R m − 1 − m − 2 based on the empirical distribution function and obtained limit theorems which play important roles for investigating asymptotic behavior of such tests.
Abstract: This paper considers the problem for testing symmetry of a distribution inR
m
based on the empirical distribution function. Limit theorems which play important roles for investigating asymptotic behavior of such tests are obtained. The limit processes of the theorems are multiparameter Wiener process. Based on the limit theorems, nonparametric tests are proposed whose asymptotic distributions are functionals of a multiparameter standard Wiener process. The tests are compared asymptotically with each other in the sense of Bahadur.