Showing papers on "Central limit theorem published in 1973"

The asymptotic theory of linear time series models

Abstract: A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

A theorem about random fields

Abstract: Averintsev [1] and Spitzer [2] proved that the class of Markov fields is identical to the class of Gibbs ensembles when the domain is a finite subset of the cubic lattice and each site may be in either of two given states. Hammersley and Clifford [3] proved the same result for the more general case when the domain is the set of sites of an arbitrary finite graph and the number of possible states for each site is finite. In order to show this, they extended the notion of a Gibbs ensemble to embrace more complex interactions than occur on the cubic lattice. Their method was circuitous and showed merely the existence of a potential function for a Markov field with little indication of its form. In [4], Preston gives a more direct approach to the two-state problem and presents an explicit formula for the potential. We show here that the equivalence of Markov fields and Gibbs ensembles follows immediately from a very simple application of the Mobius inversion theorem of [5] which allows us to construct a natural expression for the potential function of a Markov field. We confine our attention to the set of sites of an arbitrary finite graph and allow each site to be in any one of a countable set of states. The two-state solution of Preston emerges as a corollary.

The estimation of frequency

Abstract: Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency 0o, is in fact 0 or nt it is shown that the estimate, 0N, satisfies 0N = 0o for N ? No (w) where No (w) is an integer, determined by the realisation, w, of the process, that is almost surely finite.

Central limit theorems for martingales and for processes with stationary increments using a skorokhod representation approach

Abstract: The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

The asymptotic strength distribution of a general fiber bundle

Abstract: The asymptotic statistical distribution of tensile strength of a bundle of parallel fibers is determined in terms of the statistical characteristics of the individual fibers as the number of fibers in the bundle grows indefinitely large.

A Central Limit Theorem for $m$-Dependent Random Variables with Unbounded $m$

Abstract: For each $k = 1, 2, \cdots$ let $n = n(k)$, let $m = m(k)$, and suppose $y_1^k, \cdots, y_n^k$ is an $m$-dependent sequence of random variables. We assume the random variables have $(2 + \delta)$th moments, that $m^{2 + 2/\delta}/n \rightarrow 0$, and other regularity conditions, and prove that $n^{-\frac{1}{2}}(y_1^k + \cdots + y_n^k)$ is asymptotically normal. An example showing sharpness is given.

Convergence Rates for $U$-Statistics and Related Statistics

Abstract: Bounds are provided for the rates of convergence in the central limit theorem and the strong law of large numbers for $U$-statistics. The results are obtained by establishing suitable bounds upon the moments of the difference between a $U$-statistic and its projection. Analogous conclusions for the associated von Mises statistical functions are indicated. Statistics considered for exemplification are the sample variance and the Wilcoxon two-sample statistic.

An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas

Abstract: where the infimum is taken over all pairs of random variables X and Y defined on (f2, P) and distributed according to f and g respectively; here g is the Gaussian distribution with mean 0 and variance a 2 =~2 (f)e I-f] is sometimes denoted by e IX] when X is a random variable with distribution f. It should be noticed that the value of e [ f ] does not depend upon a choice of the probability space (f2, P). The purpose of this paper is to present some basic properties of e (especially, the inequality (2.2)) together with an application to the central limit theorem and then to show that the functional e is monotone decreasing along Boltzmann solutions of Kac's one-dimensional model of a Maxwellian gas. Some of our results can be generalized to the case of R 3; for example, the functional e similarly defined in R 3 decreases along solutions of Boltzmann's problem for the 3-dimensional Maxwellian gas, but this will be discussed in another occasion.

Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature

Abstract: For the Ising model with nearest neighbour interaction it is shown that the spin correlations 〈σAσB〉 -〈σA〉〈σB〉decrease exponentially asd(A, B) → ∞ in a pure phase when the temperature is well belowTc. This is used to prove that the free energyF(β,h) is infinitely differentiable in β and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(β,h).

Central Limit Analogues for Markov Population Processes

Abstract: 1 SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞ A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities The paper concludes with a discussion of various related results It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman Transient processes such as epidemics are also shown to exhibit a “central limit” behavior

On Determining the Irrationality of the Mean of a Random Variable

Abstract: A complexity approach is used to decide whether or not the mean of a sequence of independent identically distributed random variables lies in an arbitrary specified countable subset of the real line. A procedure is described that makes only a finite number of mistakes with probability one. This leads to some speculations on inference of the laws of physics and the computability of the physical constants.

Some Characterizations of the Exponential Distribution by Geometric Compounding

Abstract: If, for every $p \in ( {0,1} ),p$ times a geometric $( p )$ sum of independent identically distributed nonnegative random variables has the same distribution as the individual random variables, then the random variables are exponentially distributed. The phase “for every $p \in ( {0,1} )$ ” can be weakenedsomewhat, and if first moments are assumed to exist, it can be replaced by “for some $p \in ( {0,1} )$.” Related characterizations of the power function distribution and of the Poisson process are discussed, as are the effects of dropping the assumption of nonnegativity of the random variables.

Limiting Behavior of the Extremum of Certain Sample Functions

Abstract: For a sequence of random variables forming an $m$-dependent stochastic process (not necessarily stationary), asymptotic distribution and other convergence properties of the extremum of certain functions of the empirical distribution are studied. In this context, it is shown that the asymptotic probability of the classical Kolmogorov-Smirnov statistic exceeding any positive real number provides an upper bound for the corresponding probability when the underlying random variables are not necessarily identically distributed. The theory is specifically applied to the study of the limiting distribution, strong convergence and convergence of the first moment of the strength of a bundle of parallel filaments (which is shown to be the extremum of a function of the empirical distribution).

Proceedings of the second Japan-USSR Symposium on Probability Theory

Abstract: Some limit theorems for a queueing system with absolute priority in heavy traffic.- On certain problems of uniform distribution of real sequences.- Norms of Gaussian sample functions.- On a new approach to Markov processes.- Limit theorems for linear combinations of order statistics.- Some estimates of the rate of convergence in multidimensional limit theorems for homogeneous Markov processes.- Expectation semigroup of a cascade process and a limit theorem.- Potential theory of symmetric markov processes and its applications.- Hilbert space methods in classical problems of mathematical statistics.- On the martingale aproach to statistical problems for stochastic processes with boundary conditions.- Probabilities of the first exit for continuous processes with independent increments on a markov chain.- Noncommutative analogues of the Cramer-Rao inequality in the quantum measurement theory.- Test of hypotheses for distributions with monotone likelihood ratio: case of vector valued parameter.- Criteria of absolute continuity of measures corresponding to multivariate point processes.- Normal numbers and ergodic theory.- On multitype branching processes with immigration.- Statistics of stochastic processes with jumps.- Evolution asymptotique des temps d'arret et des temps de sejour lies aux trajectoires de certaines fonctions aleatoires gaussiennes.- Asymptotic enlarging of semi-markov processes with an arbitrary state space.- The method of accompanying infinitely divisible distributions.- Optimal stopping of controlled diffusion process.- Additive arithmetic functions and Brownian motion.- Asymptotic behavior of the fisher information contained in additive statistics.- Nonlinear functionals of gaussian stationary processes and their applications.- Stationary matrices of probabilities for stochastic supermatrix.- An estimate of the remainder term in the multidimensional central limit theorem.- A remark on the non-linear Dirichlet problem of branching markov processes.- Some remarks on stochastic optimal controls.- On stationary linear processes with Markovian property.- Some limit theorems for the maximum of normalized sums of weakly dependent random variables.- Non-uniform estimate in the central limit theorem in a separable Hilbert space.- Generalized diffusion processes.- Semifields and probability theory.- Convergence to diffusion processes for a class of Markov chains related to population genetics.- Random operators in a Hilbert space.- Bernoulli shifts on groups and decreasing sequences of partitions.- On the second order asymptotic efficiencies of estimators.- On the relaxed solutions of a certain stochastic differential equation.- On limit theorems for non-critical Galton-Watson processes with EZ1logZ1=?.- Construction of diffusion processes by means of poisson Point process of Brownian excursions.- Non-anticipating solutions of stochastic equations.- A stochastic maximum principle in control problems with discrete time.- Selection of variables in multiple regression analysis.

The Central Limit Theorem for Random Motions of $d$-Dimensional Euclidean Space

Abstract: Let $g_1, g_2,\cdots$ be random elements of the Euclidean group of motions of $d$-dimensional Euclidean space $R^d (d \geqq 1)$, that are independent and identically distributed. The product $g_1\cdots g_n$ is represented in the form $t(n)r(n)$, where $t(n)$ is a translation and $r(n)$ is a rotation. In this paper it is shown that under natural conditions $r(n)$ and $n^{-\frac{1}{2}}t(n)$ jointly converge weakly as $n \rightarrow \infty$ to the product distribution of the Haar measure on a certain closed subgroup of the rotations group, and a normal distribution on $R^d$, with mean zero and covariance matrix $\sigma^2\mathbf{I}$ ($\mathbf{I}$ is the identity matrix), and the value of $\sigma^2$ is identified.

The distribution of the maximum of a random number of random variables with applications

Abstract: The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group. ASYMPTOTIC DISTRIBUTION; RANDOM NUMBER OF RANDOM VARIABLES; SERVICE TIME; SYSTEM OF A LARGE NUMBER OF COMPONENTS; DEPENDENT RANDOM VARIABLES 1. The main result and applications Let X1,X2,'"-,X, be random variables with the same distribution function F(x). Let nl 0, if t = io 0 and b, be real numbers such that, for n + o , (4) n[1 F(ax + b,)] -~ w(x). Let further v, be a sequence of positive integer valued random variables such that v,/n converges stochastically to a positive random variable v. Then, as tn -+ + c0, (5) lim P[(W, b,,)/a,, < x] = e-w(x) There are several practical situations when our model, described in (1)-(3) and the result of Theorem 1 are needed. We shall describe one through a concrete example; the similarity of several situations to this example is evident. Consider a system of n components which require regular servicing. The number of components to be serviced at a given time is a random variable, i.e., varies from time to time. If v, is the number of components to be serviced then the service is completed in a time period not exceeding a given number T, if, and only if, Wn = max{X1, X2, ...,X,,} does not exceed T, where X1 is the time period required for servicing the jth component. Thus for large n, the conclusion of Theorem 1 gives a good approximation for the time period needed to complete the service. Our assumptions (1)-(3) were made under the guidance of this specific problem. Namely, the assumption of previous models that the service times are stochastically independent is practically never satisfied, even if as many machines are available as there are components to be serviced. To make clear how our assumptions apply to a practical situation, let us specify our system to be an automobile car. Automatic equipment starts servicing all parts virtually at the same time. Because of the relations of the parts, however, it cannot be assumed that service can continue uninterrupted on all parts. As a matter of fact, some This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 05:47:31 UTC All use subject to http://about.jstor.org/terms

A large deviation theorem for weighted sums

Abstract: Asymptotic representations are derived for large deviation probabilities of weighted sums of independent, identically distributed random variables. The main theorem generalizes a 1952 theorem of Chernoff which asserts that n−1 log P(Sn>cn)→−log ρ, where Sn is the partial sum of a sequence of independent, identically distributed random variables X1, X2, ... and ρ is a constant depending on X1. The main result is similar in form to, but different in focus from, a particular case of Feller's (1969) theorem on large deviations for triangular arrays.

A Functional Central Limit Theorem Connected with Extended Renewal Theory

Abstract: If p = 0 and if ~l, ~2 . . . . , are assumed to be positive random variables, the theorem is contained in Billingsley [2], Theorem 17.3:, p. 148. If p = 0, the theorem has been proved by Basu [1] and Vervaat [11]. As is pointed out in [2], n may tend to infinity in a continuous manner. Since the projections from C to R k are continuous mappings, (see [2], 20), it follows that the finite-dimensional distributions converge to multidimensional normal distributions (cp. [2], 30). In particular, the following corollary holds.

An invariance principle for a class of $d$-dimensional polygonal random functions

Abstract: A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space (d > 1) undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if d > 2, and not being almost surely nonnegative if d = 1. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rose*n, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.