Showing papers on "Central limit theorem published in 1980"
TL;DR: In this article, the authors considered the problem of finding conditions on the quadratic variations of a local martingale sequence to insure its convergence towards a continuous Gaussian limit, where convergence in distribution is defined as the weak convergence of the (s sequence on the space D of right continuous and left hand limited functions.
Abstract: This paper is involved with the following problem. Given a sequence of local martingales, say (Mn), under which conditions on the quadratic variations ([M,]), can we state the convergence in distribution of the (M,) sequence towards a continuous gaussian martingale limit? "Convergence in distribution" means here the weak convergence of the (s sequence on the space D of right continuous and left hand limited functions, f ( M , ) being the probability measure induced on D by M n (i.e. the distribution or law of the M, process). In preceding works, the author has investigated an analogous problem for locally square integrable local martingales (in short, "locally square integrable martingales"). In such case we were interested in finding out conditions on the ( (M,)) sequence of associated increasing processes to insure the (M.)'s convergence in distribution. It is a well known fact (c.f. [-9]) that for a local martingale M the associated increasing process ( M ) exists if and only if M is locally square integrable. On the contrary, [M] always exists and, furthermore, [M] is easier to calculate than ( M ) when both processes exist. Thus the problem with which we will deal below is a very natural one. In the first paragraph, we will explain some notations. Paragraph two is devoted to state the main results of this paper. Proofs of these results are given in paragraph three. Paragraph four contains some particular cases of the main theorems. The last paragraph gives a complementary result for locally square integrabte martingales. The Appendix contains the recall of a classical Tightness Criterion used in the paper.
233 citations
TL;DR: In this paper, the asymptotic properties of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variable and innovations to the output vector.
92 citations
TL;DR: In particular, it is shown in this paper that it is sufficient for the chain to have recurrence times with third moments on order to obtain O(1/√n) error bound.
Abstract: The error bound O(1/√n) is derived in the central limit theorem for partial sums
$$\sum\limits_{j = 1}^n {f(\xi _j )} $$
where ξj is a recurrent discrete Markov chain and f is a real valued function on the state space. In particular it is shown that for bounded f and starting distribution dominated by some multiple of the stationary one, it is sufficient for the chain to have recurrence times with third moments on order to get this bound.
77 citations
TL;DR: In this paper, a two-stage regression procedure is used to estimate the unknown parameters of a class of random coefficient autoregressive models, and the estimates are shown to satisfy a central limit theorem.
Abstract: . This paper is concerned with autoregressive models in which the coefficients are assumed to be not constant but subject to random perturbations so that we are considering a class of random coefficient autoregressive models. By means of a two stage regression procedure estimates of the unknown parameters of these models are obtained. The estimates are shown to be strongly consistent and to satisfy a central limit theorem. A number of Monte Carlo experiments was carried out to illustrate the estimation procedure and their results are reported.
74 citations
TL;DR: In this paper, a new numerical method is presented, which allows for the calculation of the conductivity and the localisation properties of the eigenstates in a one-dimensional disordered system described by the Anderson Hamiltonian.
Abstract: A new numerical method is presented, which allows for the calculation of the conductivity and the localisation properties of the eigenstates in a one-dimensional disordered system described by the Anderson Hamiltonian. The computer storage requirements of the procedure are independent of the length of the system, the maximum system length treatable being only limited by the computer time available. First results obtained for a system containing up to 108 sites indicate that the DC conductivity obeys the central limit theorem. It vanishes in the limit of zero temperature and fulfils a scaling law.
50 citations
TL;DR: In this article, a numerical study of transmission through a one-dimensional lattice of random delta functions is presented, and results verify analytical predictions about the statistics of transport properties: averages of resistance and conductance are unrepresentative, but averages of the inverse localisation length obey the central limit theorem.
Abstract: A numerical study of transmission through a one-dimensional lattice of random delta functions is presented. Results verify analytical predictions about the statistics of transport properties: averages of resistance and conductance are unrepresentative, but averages of the inverse localisation length obey the central limit theorem. Dependence of the inverse localisation length on disorder is discussed.
48 citations
TL;DR: In this article, necessary and sufficient conditions are given for the existence of matrices An such that the transform of Sn by An is asymptotically multivariate normal with identity covariance matrix.
Abstract: Let S, be a sequence of partial sums of mean zero purely d-dimensional i.i.d. random vectors. Necessary and sufficient conditions are given for the existence of matrices An such that the transform of Sn by An is asymptotically multivariate normal with identity covariance matrix. This is more general than previous d-dimensional results. Examples are given to illustrate the need for the present approach. The matrices An take a particularly simple form because of a degree of uncorrelatedness between certain pairs of 1-dimensional random variables obtained by projection. 0. Introduction. For a sequence of independent and identically distributed (iLi.d.) 1-dimensional random variables Zi, not assumed to possess finite second moments, Levy and Feller obtained necessary and sufficient conditions for the existence of constants an such that i7 Zi/an converges weakly to the standard normal random variable. We are interested in the central limiting behavior of
47 citations
TL;DR: In this paper, a functional limit theorem concerning the deviation of the empirical distribution from the 'true' one is established. But the main result is not related to our work, but is related to a class of statistics of planar and spatial data closely connected with empirical distributions.
Abstract: A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate 'ergodic' distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the 'true' one. Examples in mathematical morphology are given. EMPIRICAL DISTRIBUTIONS; MATHEMATICAL MORPHOLOGY; RANDOM SETS; RANDOM FIELDS; FUNCTIONAL CENTRAL LIMIT THEOREM
46 citations
39 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of families of dependent random variables called "block spins", which are associated with random fields arising in statistical mechanics, and gave sufficient conditions for these families to converge weakly to products of independent Gaussian random variables.
Abstract: We study the asymptotic behavior of families of dependent random variables called “block spins,” which are associated with random fields arising in statistical mechanics. We give sufficient conditions for these families to converge weakly to products of independent Gaussian random variables. We also estimate the error terms involved. In addition we give some conditions which imply that the block spins can converge weakly only to families of normal or degenerate random variables. Central to our proofs is a mixing property which is weaker than strong mixing and which holds for many random fields studied in statistical mechanics. Finally we give a simple method for determining when a stationary random field does not satisfy a strong mixing property. This method implies that the two-dimensional Ising model at the critical temperature is not strong mixing, a result obtained by a different method by M. Cassandro and G. Jona-Lasinio. The method also shows that a stationary, mean-zero, positively correlated Gaussian process indexed by ℝ is not strong mixing if its covariance function decreases liket
−α
, 0 <α < 1.
TL;DR: In this article, it was shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
Abstract: In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
01 Jan 1980
TL;DR: The weak convergence theory for dependent random variables was initiated by Brown [2] and Dvoretzky [4], [5] and has been studied in this paper for row-wise independent random variables.
Abstract: Let {Xnk}, k = 1,2 kn; n = 1,2,..., be an array of random variables defined on a common probability space (Ω, F, P). If {Xnk} are row-wise independent, then there exists a quite satisfactory theory of the weak convergence of sums \(S_n ^\prime = \sum\limits_{k = 1}^{k_n } {X_{nk} .} \) One of the most reasonable trends in the analogous theory for dependent random variables is initiated by papers of Brown [2] and Dvoretzky [4], [5].
TL;DR: It is shown how, by sequential minimization of an average prediction error, it is possible to construct recursive algorithms (Predict ion Error Recursions, I'KRs) for almost any lumped parametric model.
Abstract: The aim of this paper is to develop a unified view of the nature of the recursive, Stochastic Approximation (SA) and Model Reference methods for estimating the parameters of a lumped model of a dynamic system, Thus it is shown how, by sequential minimization of an average prediction error, it is possible to construct recursive algorithms (Predict ion Error Recursions, I'KRs) for almost any lumped parametric model. For a general class of recursions (including I'ERs and SAs) an informal analysis of convergence is given by considering the first order moment behaviour of the recursion (viewed as a stochastic difference equation). This leads to equations given by Ljung. Continuing however yields a second order analysis that provides the asymptotic variance behaviour (Central Limit Theorem) of the algorithms.
TL;DR: In this article, the authors compare the corrected inversion method, the log(n)/n-tail method and the record time method for the normal, exponential and gamma densities.
Abstract: Frequently the need arises for the computer generation of variates that are exactly distributed as Z = max(X1, …, Xn) where X1, …, Xn form a sequence of independent identically distributed random variables. For large n the individual generation of the Xi's is unfeasible, and the inversion-of-a-beta-variate is potentially inaccurate. In this paper, we discuss and compare the corrected inversion method, the log(n)/n-tail method and the record time method. The latter two methods have an average complexity 0(log(n)), are very accurate and do not require the inversion of a distribution function. The normal, exponential and gamma densities are treated in detail. The existence of fast and accurate inversion methods for the error function makes the corrected inversion method faster than the other ones for n large enough when the Xi's are normal random variables.
01 Jan 1980
TL;DR: The classic condition of negligibility in the theory of sums of independent r.v. requires X ni to be uniformly asymptotically negligible as mentioned in this paper, which is generally weaker than the summation condition.
Abstract: This chapter focuses on the central limit theorem. A large variety of negligibility assumptions have been made about the differences X ni during the formulation of martingale central limit theorems. The classic condition of negligibility in the theory of sums of independent r.v. requires X ni to be uniformly asymptotically negligible. This is generally a little weaker than the summation condition. The conditional variance V n 2 is one of several estimates of the variance ES n 2 . It is an intrinsic measure of time for the martingale. For many purposes, the time taken for a martingale to cross a level is best represented through its conditional variance rather than the number of increments up to the crossing. The duality between the definitions of forward and reverse martingales suggests that forward martingale limit theorems should have reverse martingale duals. In the case of the central limit theorem, the analog is perhaps best presented by considering infinite martingale arrays.
01 Jul 1980
TL;DR: In this article, the authors give conditions under which there exist constants an > 0 and bn such that where G(x) is taken to be one of the extreme value distributions G1(x), G2, G3, G4, G5, G6, G7, G8, G10, G11, G12, G13, G14, G15, G16, G17, G18, G20, G21, G22, G23, G24, G25, G26, G27, G28
Abstract: Let , where the Xi, i = 1, 2, … are independent identically distributed random variables. Classical extreme value theory, described for example in the books of do Haan(6) and Galambos(3) gives conditions under which there exist constants an > 0 and bn such thatwhere G(x) is taken to be one of the extreme value distributions G1(x) = exp (− e−x), G2(x) = exp (− x−a) (x > 0, α > 0) and G3(x) = exp (−(− x)α) (x 0).
TL;DR: The functional central limit theorem as discussed by the authors states that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index π.
Abstract: Suppose that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index $\alpha$. Then without changing its distribution, one can redefine the sequence on a new probability space such that these partial sums converge in probability and consequently even in $L^p (p < \alpha)$ to the corresponding stable process. This provides a new method to prove functional central limit theorems and related results. A similar theorem holds for stationary $\phi$-mixing sequences of random variables.
TL;DR: Asymptotic upper and lower bounds for the uniform measure of the rate of convergence in the central limit theorem using a variety of norming constants were obtained in this article, and extensive generalizations of the classical characterizations of convergence were deduced in terms of series and order of magnitude conditions.
Abstract: Asymptotic upper and lower bounds are obtained for the uniform measure of the rate of convergence in the central limit theorem using a variety of norming constants. For many distributions the upper and lower bounds are of the same order of magnitude. As easy corollaries we deduce extensive generalizations of the classical characterizations of the rate of convergence in terms of series and order of magnitude conditions.
TL;DR: In this article, a strictly stationary sequence with finite second moments is given, for which Var(X 1 + … + Xn )→ ∞ and ρ n → 0 as n→∞, but there is partial attraction to non-stable limit laws.
Abstract: Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk } is given with finite second moments, for which Var(X 1 + … + Xn )→∞ and ρ n → 0 as n→∞, but (X 1 + … + Xn ) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.
TL;DR: In this article, the branching random field is studied under general branching and diffusion laws, and it is shown that at finite fixed time, it converges in law to a generalized Gaussian random field with independent increments.
Abstract: The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.
TL;DR: In this article, a functional central limit theorem for a martingale difference sequence or array of C(ÄT)-valued random variables (K compact metric space) is proved under a continuity assumption which in the case K=[Q, l] is of log log type.
Abstract: In this paper a functional central limit theorem (invariance principle) for a martingale difference sequence or, more generally, array of C(ÄT)-valued random variables (K compact metric space) is proved under a continuity assumption which in the case K=[Q, l] is of log log type. Further there is established a connection between an invariance principle for a sequence of partial sums of random elements in a real separable Banach space L and an invariance principle for the more general Robbins-Monro process in L and a modification of it; the latter process concerns a strongly consistent sequential estimation of the root of a linear equation in L, where the operator which satisfies a certain spectral condition, is only empirically available, like the absolute element, and is estimated in the procedure. The results are applied to the case L = C(K). For K=[0, l] an integral equation of the above type appears in filtering theory.
01 Feb 1980
TL;DR: In this article, an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure is presented, and it is shown that under reasonable conditions, the normalized sums converge in distribution to "higher-order" analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation.
Abstract: ABsTRAcr. This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to "higher-order" analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.
TL;DR: In this paper, it was shown that at most pairwise independent random variables, all uniform on $\mathscr{M}$ and all functions of $(Y_1, \cdots, Y_r) can be defined.
Abstract: Let $Y_1, \cdots, Y_r$ be independent random variables, each uniformly distributed on $\mathscr{M} = \{1,2, \cdots, M\}$. It is shown that at most $N = 1 + M + \cdots + M^{r-1}$ pairwise independent random variables, all uniform on $\mathscr{M}$ and all functions of $(Y_1, \cdots, Y_r)$, can be defined. If $M = p^k$ for some prime $p$, the maximum can be attained by a strictly stationary sequence $X_1, \cdots, X_N$, for which any $r$ successive random variables are independent.
TL;DR: In this paper, strong laws of large numbers, obtained for positive, independent random variables, are utilized to prove iterated logarithm laws (with a nonrandom normalizing sequence) for a class of martingales.
Abstract: Strong laws of large numbers, obtained for positive, independent random variables, are utilized to prove iterated logarithm laws (with a nonrandom normalizing sequence) for a class of martingales. A law of the iterated logarithm is also established for certain random quadratic forms.
TL;DR: In this article, the authors studied the branching diffusion with immigration and showed that the diffusion process is mixing under general branching and diffusion laws, according to Brillinger's definition, and they proved that the steady state random field also obeys a spatial central limit theorem for spatially homogeneous mixing processes.
Abstract: The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem. RANDOM FIELD; RANDOM MEASURE; PROBABILITY GENERATING FUNCTIONAL; CUMULANT AND MOMENT DENSITY FUNCTIONS; BRILLINGER'S MIXING CONDITION; SIMPLE BRANCHING DIFFUSION PROCESS; BRANCHING DIFFUSION WITH IMMIGRATION; STEADY-STATE RANDOM FIELD
TL;DR: In this paper, the Skorokhod topology is extended to the function space D([0, ∞)ρ, E) of functions, from [0, ǫ)ρ to a complete separable metric space E, which are continuous from above with limits from below.
Abstract: The Skorokhod topology is extended to the function space D([0, ∞)ρ, E) of functions, from [0, ∞)ρ to a complete separable metric space E, which are “continuous from above with limits from below. Criteria for tightness are developed. The case in which E is a product space is considered, and conditions under which tightness may be proven componentwise are given. Various applications are studied, including a multidimensional version of Donsker's Theorem, and a functional Central Limit Theorem for a multitype Poisson cluster process.