Showing papers on "Central limit theorem published in 1972"
01 Jan 1972
TL;DR: In this article, the probability density, Fourier transforms and characteristic functions, joint statistics and statistical independence, Correlation functions and spectra, the central limit theorem, and the relation functions are discussed.
Abstract: This chapter contains sections titled: The probability density, Fourier transforms and characteristic functions, Joint statistics and statistical independence, Correlation functions and spectra, The central limit theorem
3,260 citations
01 Jan 1972
TL;DR: In this article, it was shown that under the assumption of exponential decrease of dependence is strengthened tormdependence, the error in the normal approximation is of the order of n 1/2(log n)2.
Abstract: This paper has two aims, one fairly concrete and the other more abstract. In Section 3, bounds are obtained under certain conditions for the departure of the distribution of the sum of n terms of a stationary random sequence from a normal distribution. These bounds are derived from a more abstract normal approximation theorem proved in Section 2. I regret that, in order to complete this paper in time for publication, I have been forced to submit it with many defects remaining. In particular the proof of the concrete results of Section 3 is somewhat incomplete. A well known theorem of A. Berry [1] and C-G. Esseen [2] asserts that if X1, X2, . is a sequence of independent identically distributed random variables with EXi = 0, EXV = 1, and ,B = EIXij3 < oo. then the cumulative distribution function of (1//;n) Yi=l Xi differs from the unit normal distribution by at most Kf3/ n where K is a constant, which can be taken to be 2. It seems likely, but has never been proved and will not be proved here, that a similar result holds for stationary sequences in which the dependence falls off sufficiently rapidly and the variance of(1//;n) X1.1 Xi approaches a positive constant. I. Ibragimov and Yu. Linnik ([3], pp. 423-432) prove that, under these conditions, the limiting distribution of (1/ /n) E Xi is normal with mean 0 and a certain variance G2 Perhaps the best published results on bounds for the error are those of Phillip [5]. who shows that if in addition the Xi are bounded, with exponentially decreasing dependence, then the discrepancy is roughly of the order of n-114 In Corollary 3.2 of the present paper it is proved that under these conditions the discrepancy is of the order of n 1/2(log n)2. Actually the assumption of boundedness is weakened to the finiteness of eighth moments. In Corollary 3.1 it is proved that if the assumption of exponential decrease of dependence is strengthened tormdependence, the error in the normal approximation is of the order of n1/2 The abstract normal approximation theorem of Section 2 is elementary in the sense that it uses only the basic properties of conditional expectation and the elements of analysis, including the solution of a first order linear differential equation. It is also direct, in the sense that the expectation of a fairly arbitrary
1,233 citations
Book•
19 Jul 1972
TL;DR: In this article, the analysis of cocycles of first order on some special groups is discussed, including continuous tensor products and positive definite kernels and limit theorems for uniformly infinitesimal families of positive definite kernel families.
Abstract: Positive definite kernels and continuous tensor products.- Limit theorems for uniformly infinitesimal families of positive definite kernels.- The analysis of cocycles of first order on some special groups.
270 citations
TL;DR: In this paper, it was shown how martingale theorems can be used to widen the scope of classical inferential results concerning autocorrelations in time series analysis.
Abstract: In this paper it is shown how martingale theorems can be used to appreciably widen the scope of classical inferential results concerning autocorrelations in time series analysis. The object of study is a process which is basically the second-order stationary purely non-deterministic process and contains, in particular, the mixed autoregressive and moving average process. We obtain a strong law and a central limit theorem for the autocorrelations of this process under very general conditions. These results show in particular that, subject to mild regularity conditions, the classical theory of inference for the process in question goes through if the best linear predictor is the best predictor (both in the least squares sense).
158 citations
TL;DR: In this paper, it was shown that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of φ-mixing random variables.
138 citations
TL;DR: Theoretical foundations for an analytical model of urban radio-noise environments are presented at a level of generality broad enough to include the pertinent physical and statistical elements which critically influence the temporal and statistical character of such interference in radio receivers as discussed by the authors.
Abstract: Theoretical foundations for an analytical model of urban radio-noise environments are presented at a level of generality broad enough to include the pertinent physical and statistical elements which critically influence the temporal and statistical character of such interference in radio receivers. The central roles of geometry, directionality (beam patterns), waveforms (from the interfering sources), motion (Doppler), source density and distribution, and secular variations of source parameters, as well as the radiation mechanism, are specifically developed. The basic statistical model (BSM) involves independent sources, in space (and in time), leading to Poisson radiation fields and a Poisson process X(t) in a typical receiver. This received process X(t) can be considered the sum of a Gauss process (by the central limit theorem) representing the cumulative effects of a large number of sources, none of which is very large vis-a-vis the rest, and a Poisson process produced by those few strong transients which when present dominate the background. The process X(t) is essentially stationary for periods comparable to the secular period of changes in traffic intensity and flow, which permits the construction of usefully large experimental ensembles from which to estimate the process statistics. A semiempirical, but more analytically tractable model similar to that introduced by Hall [6] for impulsive atmospheric noise but used here for independent sources is also constructed. This model is represented by X(t) = a(t) Z(t), where a,Z are independent processes, both zero mean, and Z is regarded as Gaussian.
125 citations
TL;DR: In this article, analogues of classical limit laws for sums of independent random variables (central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed.
Abstract: Analogues of classical limit laws for sums of independent random variables (central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed. We stress results which go through without moment or smoothness assumptions on the underlying distributions. These include (i) estimates for the spread of the distribution of $S_n = \sum_1^nX_i$ in terms of concentration functions (Levy-Rogozin inequality), (ii) comparison of the distribution of $S_n$ on different intervals (ratio limit theorems and Spitzer's theorem for the existence of the potential kernel for recurrent random walk), (iii) study of the set of accumulation points of $S_n/\Upsilon(n)$ for suitable $\Upsilon (n) \uparrow \infty$. Only the following parallel to the law of the iterated logarithm is new: If $X_1, X_2, \cdots$ are independent random variables all with distribution $F, S_n = \sum_1^nX_1, m_n = \operatorname{med} (S_n)$, then there exists a sequence $\{\Upsilon (n)\}$ such that $\Upsilon (n) \rightarrow \infty$ and $- \infty < \lim \inf (S_n - m_n)/\Upsilon (n) < \lim \sup (S_n - m_n)/\Upsilon (n) < \infty$ w.p. 1, if and only if $F$ belongs to the domain of partial attraction of the normal law.
64 citations
48 citations
TL;DR: In this paper, a class of processes with stationary independent increments whose basic parameters are allowed to vary randomly over time is studied and several limiting theorems are presented including weak and strong laws of large numbers and a functional central limit theorem.
Abstract: We study a class of processes which are essentially processes with stationary independent increments whose basic parameters are allowed to vary randomly over time. These processes are equivalent to random time transformations of processes with stationary independent increments where the time process is independent of the original process. Several limiting theorems are presented including weak and strong laws of large numbers and a functional central limit theorem.
46 citations
TL;DR: For a wide class of (dependent) random variables, a limit law was proved for the maximum, with suitable normalization, of $X 1, X 2, X 3, X 4, X 5, X n as discussed by the authors.
Abstract: For a wide class of (dependent) random variables $X_1, X_2, \cdots, X_n$, a limit law is proved for the maximum, with suitable normalization, of $X_1, X_2, \cdots, X_n$. The results are more general in two aspects than the ones obtained earlier by several authors, namely, the stationary of the $X$'s is not assumed and secondly, the assumptions on the dependence of the $X$'s are weaker than those occurring in previous papers. A generalization of the method of inclusion and exclusion is one of the main tools.
TL;DR: In this article, a necessary condition is given for the convergence of distributions of the sums of a random number of independent random variables, which is made on the basis of a theorem which gives sufficient conditions for convergence of distribution of randomly stopped stochastic processes.
Abstract: A necessary condition is given for the convergence of distributions of the sums of a random number of independent random variables. This is made on the basis of a theorem which gives sufficient conditions for the convergence of distributions of randomly stopped stochastic processes. The random indices are supposed to be independent of the sequence of summands.
01 Jan 1972
TL;DR: In this paper, the authors considered the problem of determining the distribution of the length of the time interval from the first registration of a particle to the first loss in a Geiger-Muller counter, where a particle having struck the counter and having been counted by it causes a breakdown in it lasting some time.
Abstract: The work of A. Wald [11] and H. Robbins [10] played an important role in stimulating interest in the investigation of sums of a random number of independent random variables. Of late a large number of papers related to this topic have been published and their importance for numerous applied questions as well as theoretical mathematics has been shown. Results of research, conducted over the last three years by myself and my students in connection with problems in the theory of reliability, theory of queuing, physics, and production organization, are presented in this paper. We shall begin the presentation with the consideration of one of these problems. Some of the formulations of problems and theorems which will be included here have been presented before when I delivered lectures at the Universities of London, Sheffield, Rome, Budapest and Warsaw where I was a guest in recent years. Several statements of problems and their solutions originated there. Geiger-Muller counters are used in nuclear physics and also in the study of cosmic radiation. A particle having struck the counter and having been counted by it causes a breakdown in it lasting some time z. Any particle striking the counter in the period of the breakdown is not registered by it. Usually one assumes that r is a constant; however, a more realistic assumption is that T is a random variable with some distribution G(x). Our problem is to determine the distribution of the length of the time interval from the first registration of a particle to the first loss. Here we shall assume that the time intervals between successive entries of particles are independent and identically distributed with distribution function F(x). It is obvious that the counter does not lose a particle until the duration of the breakdown is as large as the interval between successive entries of particles. (See Figure 1.) This permits us to write the following equality:
01 Jan 1972
TL;DR: In this article, the rate of convergence of the central limit theorem in Rk has been studied and two recent results which are most closely related to the estimate obtained in the present paper are discussed.
Abstract: : In recent years many papers concerned with estimation of the rate of convergencein the central limit theorem in Rk have appeared (see [1], [2], [6]-[8],[10], [13]-[16]). They have significantly extended our knowledge in this area. We shall mention here two recent results which are most closely related to the estimate obtained in the present paper.
TL;DR: In this article, a large class of functions and inequalities of a somewhat different form that are appropriate for considering exponential convergence rates are given for exponential convergence rate theorems, as well as examples of how the inequalities can be used to prove rate theorem.
Abstract: Let $X_1, X_2, X_3, \cdots$ be independent random variables and $a_1, a_2, a_3, \cdots$ positive real numbers. Define $F(t) = \sup_k P\{|X_k| > t\}$ and $S_m = \sum^m_{k=1} a_k X_k.$ Inequalities of the form $P\{\sup_m|S_m| > \delta\} \leqq C \sum_k \int^1_0 \varphi'(u)F(u/a_k) du$ are given for a large class of functions $\varphi$, as well as inequalities of a somewhat different form that are appropriate for considering exponential convergence rates. Examples of how the inequalities can be used to prove rate theorems are also given.
TL;DR: A frequently used technique for generating non-uniform random numbers is to combine uniform randomNumbers, where the sum of two uniform random numbers yields a triangular distribution and the difference yields another which will be statistically independent.
Abstract: A frequently used technique for generating non-uniform random numbers is to combine uniform random numbers. For example, the sum of two uniform random numbers yields a triangular distribution. The difference yields another which will be statistically independent. A normally distributed random number is obtained from the sum of n uniformly distributed random numbers if n is large according to the central limit theorem. Actually, each of the n linearly independent combinations of n random numbers will be a statistically independent normally distributed random number if n is sufficiently large. Exponential random numbers can be obtained approximately with a trigometric transformation of a normal random number. However, unless one wants a large array of random numbers {of length n or more}, the relatively long computation to produce multiple uniform random numbers is likely to determine the entire program execution time.
TL;DR: In this article, the convergence rate of functional central limit theorems for GI/G/1 queue and general multiple channel system is estimated for the case where the traffic intensity is fixed > 1 and the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.
Abstract: Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in the GI/G/1 queue and a general multiple channel system. The traffic intensity is fixed > 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem. RATES OF CONVERGENCE; ERROR TERMS; WEAK CONVERGENCE; FUNCTIONAL LIMIT THEOREMS; HEAVY TRAFFIC; MULTIPLE CHANNEL QUEUES; QUEUES
TL;DR: In this paper, it was shown that for discrete martingales with increments in $L_2$ the resulting trajectory is distributed approximately like Brownian motion, subject to a natural side condition.
Abstract: Many discrete martingales with increments in $L_2$ can be normalized so that the resulting trajectory is distributed approximately like Brownian motion. This paper will find all such martingales, subject to a natural side condition. Two techniques of normalization are possible: The usual one involving the partial sums of conditional variances of the increments given the past, and the analogous method using the partial sums of squares of the increments. This result is applied to obtain a central limit theorem and an arc sin law for dependent random variables.
TL;DR: In this paper, the weak convergence of weighted empirical cumulatives based on the ranks of independent, not necessarily identically distributed, observations to a continuous Gaussian process is proved.
Abstract: The weak convergence of weighted empirical cumulatives based on the ranks of independent, not necessarily identically distributed, observations to a continuous Gaussian process is proved. The results contain a shorter proof of a central limit theorem by Dupac and Hajek (1969) Ann. Math. Statist. Analogous results are proved for signed rank processes.
TL;DR: In this paper, the authors prove central limit theorems for sums of mutually dependent random vector variables for a Markov process with stationary transition mechanism, given that a sequence of random vector variable satisfies a certain type of decoupling condition and two milder restrictions.
Abstract: We prove the following central limit theorems for sums of mutually dependent random vector variables: Given that a sequence of random vector variables satisfies a certain type of decoupling condition (and two milder restrictions), we present a Lindeberg-Feller condition which we show to be both necessary and sufficient for central limit behavior. The decoupling condition and one of the two milder conditions is then applied to a Markov process with stationary transition mechanism.
01 Jan 1972
TL;DR: In this paper, it was shown that the distribution Qn of the normalized sum n-'12(X1 + *-+ X") converges weakly to the k dimensional standard normal distribution (D.i.d.).
Abstract: Let {X"} be a sequence of independent and identically distributed (i.i.d.) random vectors in Rk with zero mean vector and identity covariance matrix. The distribution Qn of the normalized sum n-'12(X1 + *-+ X") converges weakly to the k dimensional standard normal distribution (D. Although many important results on rates of convergence have been obtained in the past, most of them refer to approximations of the distribution function F. of Qn by the normal distribution function. An exception to this is the case where Qn is assumed to have a density with respect to Lebesgue measure or to have a lattice distribution. In this situation, one obtained local limit theorems as well (see [14], Chapter 16, and [19]). The first notable exception was a result of Esseen [13] which states that iffourth moments are finite, then, uniformly over all spheres S (open or closed) with center at the origin, one has
01 Jan 1972
TL;DR: In this paper, it is shown that the existence of all integral scales is necessary and sufficient for integrals over adjacent segments of a stationary process to become asymptotically independent, and sufficient to ensure that existing moments of integrals will become Gaussian.
Abstract: It is shown that (to the extent that the moments involved exist) the existence (≠0) of all (generalized) integral scales is necessary (and sufficient if all moments exist) for integrals over adjacent segments of a stationary process to become asymptotically independent, and sufficient to ensure that existing moments of integrals will become Gaussian. The conditions under which several recent central limit and related theorems for dependent variables have been proven, are shown to be closely related to this requirement. As a consequence of this examination, a slight weakening is suggested of the common condition that the spectrum be non-zero. Several physical problems are described, which may be resolved by the application of such a central limit theorem: longitudinal dispersion in a channel flow (previously treated semi-empirically); the spreading of hot spots, or the expansion of macromolecules; the weak interaction hypothesis (of Kraichnan) for Fourier components. Finally, it is shown that dispersion in homogeneous turbulence is unlikely to be explicable on the basis of a central limit theorem.
TL;DR: In this paper, the authors deal with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity, and the rate of approximation is given for the case of the M/G/1 system.
Abstract: The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/MIm system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity p > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums. QUEUEING THEORY; SERVICE SYSTEM WITH FINITE CAPACITY; CENTRAL LIMIT THEOREM; RANDOMLY STOPPED SUMS; RATE OF CONVERGENCE
01 Jan 1972
TL;DR: In this article, a proof of theorem for the distribution of the sum of independent, absolutely continuous random variables is presented. And the Chi-square distribution is shown to be a function of the mean and variance of independent and normally distributed random variables.
Abstract: This chapter presents some properties of normally distributed random variables and also derives certain distribution functions that are important in statistics. It presents a proof of theorem for the distribution of the sum of independent, absolutely continuous random variables. In addition of independent normal random variables, one considers the two independently and normally distributed random variables X 1 and X 2 , and admits the possibility that their means and variances are different. The Chi-Square distribution discusses the determination of the distribution function of the sum of squares of independent normal random variables. This chapter also presents a proof for Student's distribution.
TL;DR: In this paper, the rate at which the distribution of a random walk with independent and identically distributed steps converges to the stable distribution with index (1/2) is given.
Abstract: Let $T_r$ be the time of first passage to the level $r > 0$ by a random walk with independent and identically distributed steps and mean $
u \geqq 0$. Estimates are given for the rate at which the distribution of $T_r$, suitably scaled and normalized, converges to the stable distribution with index $\frac{1}{2}$ when $
u = 0$ and to the normal distribution when $
u > 0$ as $r \rightarrow \infty$.
01 Jan 1972
TL;DR: The concepts of uniform ergodicity and strong mixing are introduced in this paper, and the derivation of a new result indicating that strong mixing is strong mixing when there is a trivial tail field is briefly sketched.
Abstract: A number of problems in fluid mechanics which have been dealt with by making use of a central limit theorem (and asymptotic normality) are mentioned. A discussion of central limit theorems for stationary processes and the need for some form of asymptotic independence is given.The concepts of uniform ergodicity and strong mixing are introduced.An example of asymptotic nonnormality is given when the form of asymptotic independence is not sufficiently strong.The derivation of a new result indicating that uniform ergodicity is strong mixing when there is trivial tail field is briefly sketched.
TL;DR: In this article, conditions are given which imply that the partial sums of a sequence of independent integer-valued variables which satisfy the classical Lindeberg conditions for the central limit theorem also obey a strong version of the local limit theorem.
Abstract: Conditions are given which imply that the partial sums of a sequence of independent integer-valued variables which satisfy the classical Lindeberg conditions for the central limit theorem also obey a strong version of the local limit theorem. Application is made to the problem of establishing the interval recurrence of the partial sums.
TL;DR: In this paper, a characterization of the asymptotic distribution of a general capitalized random variable as the time-invariant rate of discount approaches zero is presented, and the main result is a central limit theorem demonstrating that the standardized distribution of the discounted random variable converges to the unit normal as the discount rate approaches zero.
Abstract: A renewal process is defined to represent the present value of a stream of money payments occurring at random times.The objective is to partially close a gap in the state of knowledge regarding the properties (asymptotic behavior) of certain cashflow models, scheduling models, and inventory models, all of which can be embedded in the context of replacement-reliability theory in general. The contribution of this paper is a characterization of the asymptotic distribution of a general capitalized random variable as the time-invariant rate of discount approaches zero. The main result is a central limit theorem demonstrating that the standardized distribution of the discounted random variable converges to the unit normal as the discount rate approaches zero.
TL;DR: In this article, a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0.
Abstract: Let X 1, X 2,... be a sequence of independent, identically distributed random variables with P(X⩽0)=0, and such that pκ = ƒ0 ∞ x κ dP(x) u) for u⩾0.