Showing papers on "Central limit theorem published in 1974"

Dependent Central Limit Theorems and Invariance Principles

Abstract: Central limit theorems are proved for martingales and near-martingales without the existence of moments or the full Lindeberg condition. These theorems are extended to invariance principles with a discussion of both random and nonrandom norming.

On the Law of the Iterated Logarithm

Abstract: A triumvirate of sufficient conditions is given for unbounded, independent random variables to obey the Law of the Iterated Logarithm (LIL). As special cases, new results for weighted i.i.d. random variables and the Hartman-Wintner theorem are obtained. Necessity of finite variance for the two-sided LIL is shown to carry over for a large class of weighted i.i.d. random variables and the Marcinkiewicz-Zygmund example is generalized, simplified and clarified.

Summability methods for independent identically distributed random variables

Abstract: In this paper, we present certain theorems concerning the Cesaro (C, a), Abel (A), Euler (E, q) and Borel (B) summability of ZYp where Yr = Xi-X,_ 1 Xo = 0 and XI' X2, e * * are i.i.d. random variables. While the Kolmogorov strong law of large numbers and the Hartman-Wintner law of the iterated logarithm are related to (C, 1) summability and involve the finiteness of, respectively, the first and secord moments of XV their analogues for Euler and Borel summability involve different moment conditions, and the analogues for (C, a) and Abel summability remain essentially the same. Let X1. X2, be independent, identically distributed (i.i.d.) random variables. Let Y n Xn Xn (XO = = 0). Kolmogorov's strong law of large numbers asserts that EXI = ,u iff E Yi is a.e. (C, 1) summable to , i.e., the (C, 1) limit of Xn is 1t a.e. By the well-known inclusion theorems involving Cesaro and Abel summability (cf. [5, Theorems 43 and 551), this implies that E Y. is a.e. (C, a) summable to yt for any a > 1 and that E Yi is a.e. (A) summable to p. In fact, the converse also holds in the present case, and we have the following theorem. Theorem 1. If XI, X2, .. is a sequence of i.i.d, random variables and a > 1 and y are given real numbers, then the following statements are equiva lent: (1) EX, =ph (2) X 1t(C, 1) a.e., i.e., lim (l/n)(X1 + *.. + X )= a.e. (3) Xn i(C a) a.e., ice. . lim Sin(i+ctal)X 1(n+a) = ,i a.e. n ~ ~ ~ ~ ~ ~~il i nn where (i+13) ( + 1) ... (j3 + j)(j!). (4) X -s(A) a.e., i.e., lim 1 (1) f"' ?x =ti a. e. Received by the editors June 5 1973 and, in revised form, November 12, 1973. AMS (MOS) subject classifications (1970). Primary 60F15; Secondary 40D25, 40GI0.

Metric entropy and the central limit theorem in $C(S)$

Abstract: © Annales de l’institut Fourier, 1974, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Tail sums of convergent series of independent random variables

Abstract: Let X 1 , X 2 , … be a sequence of independent random variables such that, for each n ≥ 1, EX n = 0 and and assume that then converges almost surely as N → ∞. Let and let F n ( x ) denote the distribution function of X n . Loynes (2) observed that the sequence { S n } is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence { S n }, and hence the way in which converges to its limit.

Dynamics of Macrovariables for Nonuniform Systems --Scaling Theory--

Abstract: A scale transformation of the nonequilibrium macroscopic system to larger similar sys­ tems is introduced to find kinetic equations for the evolution and fluctuation of the macro­ variables. In the scale transformation, we postulate that the probability distribution for the fluctuation of the macroscopic degrees of freedom and the quantities determined by the micro­ scopic degrees of freedom per unit volume are invariant. The characteristic length of the macroscopic state l, the macroscopic state variables Yand their fluctuation variables z. are transformed by h=Ll, (L):-1), YL=L-"yand z.L=L-Pz., respectively. The probability dis­ . tribution then takes the form P( {z;,lP}, {ql}, SJjl", t/l'), where q, SJ, d and t denote the wave vectors, volume, dimensionality and time, respectively. If a being the frequency. § I. Introduction Macroscopic systems have characteristic properties which do not appear in systems of small numbers of degrees of freedom. The central limit theorem and the phase transitions are outstanding examples. In a previous paper/> we have proposed a general type of kinetic equations from the statistical-mechanical point of view. In this paper we shall explore the most dominant features of macro­ scopic systems by introducing a new method of asymptotic evaluation for large systems and deriving the asymptotic form of the kinetic equations. A similar attempt has been jD-ade by van Kampen2> and by Kubo, Kitahara and Matsuo 3> for uniform systems by the use of the Kramers-Moyal expansion of the master. equation. They extended the central limit theorem in the form of the system-size expansion of the master equation, and showed that, for large values of the system size !2, the master equation is reduced to a linear Fokker-Planck equation and the probability distribution of the macrovariables is normal or Gaus­ sian around their mean evolution. These works, however, are limited to the uni­ form disordered systems which are described by a small number of macrovariables.

Stochastic difference equations with non-integral differences

Abstract: As an alternative to conventional discrete time models for stochastic processes that fluctuate within the sampling interval, we propose difference equations containing non-integral lags. We discuss the problems of stability, identification and estimation, for which an approximate model is needed. Least squaresa pplied to an approximateF ourier-transformedm odel yields estimators of the coefficients that are consistent with respect to the true model under some conditions. The conditions are weak when the model contains predetermined variables that obey an "aliasing condition"; estimators of the lags as well as coefficients can then be found that are consistent, efficient and satisfy a central limit theorem. Optimal estimators for stochastic differencedifferential equations are also available.

Discrete time age-dependent branching processes in relation to stable population theory in demography

Abstract: A discrete time version of a generalized one-type age-dependent branching process is considered in relation to stable population theory in demography. The motivation underlying the discrete time version of the theory is to make it amenable to computations involving demographic data. After giving a brief discussion of the foundations underlying the process, discrete type renewal equations for the mean and covariance functions of the process are derived. It is then shown how those renewal type equations may be used for making population projections with respect to age-specific birth rates, rates of population growth, and the number of live individuals in each age group, given an initial population with an arbitrary age distribution. A novel feature of the method of population projection introduced in this paper is that it is possible to derive confidence bounds for the projected number of individuals in any age group by utilizing the covariance functions of the process and the central limit theorem.

A New General Estimate of the Rate of Convergence in the Central Limit Theorem in Rk

Abstract: A general theorem is proved which gives an estimate of the rate of convergence on convex sets in the multidimensional central limit theorem for identically distributed summands. The estimate depends on the distance of the boundary of the convex set from the origin (the larger the distance, the better the estimate). The estimate makes sense under minimal requirements on the moments. Furthermore, the dependence on the distribution of a summand in it is in terms of pseudo-moment type quantities which may be small even if the moments are large.

Some continuous probability distributions

Abstract: This chapter describes the laws of variation appropriate to continuous random variables, that is, for application to measured data. By far, the most important of such a model is the so-called normal curve. It is of wide use, has many convenient properties, some of which are unique, and it can be called a first approximation to the law of variation in a great many cases. But there are many situations in which the normal curve is clearly not applicable, and one needs to use other frequency or probability curves to obtain a better approximation to the law of variation in question. Thus, the chapter first explains the normal curve, and then discusses a number of other probability distributions. It also discusses briefly the central limit theorem, and for completeness an inequality of rather limited practical application.

Strong Laws of Large Numbers for Weakly Orthogonal Sequences of Banach Space-Valued Random Variables

Abstract: : A generalization of the Strong Law of Large Numbers to weakly orthogonal sequences of Banach-Space valued random variables is given. (Author)

On the remained term for the central limit theorem

Abstract: An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution.

Limit theorems for random evolutions with explicit error estimates

Abstract: We think of x (t, y) as the position of a particle at time t when its velocity is v (t). The process x (t, y) is the simplest example of a random evolution: one-dimensional motion at a constant but random velocity determined by the state of the Markov chain associated with v(t). We denote by P(y,~i){" }, Y real, v~sA, the probability laws of the joint process (x (t, y), v (t)), where v (0)= v/. E(y, v,) will denote integration with respect to P(y, v~). The purpose of this paper is to prove the following two theorems, which correspond respectively to the weak law of large numbers and the central limit theorem for x (t).

Functionals of an infinite particle system with independent motions, creation, and annihilation

Abstract: Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained. PARTICLE SYSTEMS; INDEPENDENT MOTIONS; FUNCTIONALS; COMPOUND STOCHASTIC PROCESSES 0. Summary Consider a countable collection of particles. The nth particle enters the state space (E, &) at time Tn; it then travels in E according to a stochastic process Xn = {Xn(s); s > O} until its death. Associated with its motion there arises a secondary process An such that (a) for fixed t > 0, B -+ An(B, t) is a a-finite measure on (E, 8); and (b) for each B e d, t -> An(B, t) is right-continuous and nondecreasing. The processes (Xn,A,), (n = 1,2,...), are assumed to be independent of each other and to have the same probability law independent of {T1}. For B E , define Dt(B) = z An(B, t-Tn). Tn S t For fixed t > 0, the mapping B -4 Dt(B) is a random measure on (E, od). The joint probability law of Dt(B), "', Dt(Bn) is completely determined for any finite sequence of sets B1, -,Bn in g. Strong laws and central limit theorems are obtained for Dt as t -+ oo.

Convexity and Complete Monotonicity in Queueing Distributions and Associated Limit Behavior

Abstract: Congestion theory is often called upon for insight into the behavior of a random system for which governing distribution forms and parameters are unavailable, and the fidelity of any simple mathematical model subject to question Indeed, most applied problems have this character To be helpful, the guidance offered should be simple and substantially insensitive to missing structural detail, ie, should be “robust” Such robustness may be hoped for when the system behavior of interest is in the limit-theoretic domain eg, is associated with the law of large numbers, or the central limit theorem, or the limit theorem for rare events When limit theorems are of interest, one must know how valid the limit theorem is and how close one is to normality or exponentiality Inequalities and bounds obtained from underlying structural features of the system may be needed For such objectives, an awareness of convexity present and exploitation of this convexity is often useful

Ergodlc Properties of the Equilibrium Process Associated with Infinitely Many Markovlan Particles

Abstract: Consider a system of independent identically distributed Markov processes which have an invariant measure A. It is known that if each process starts from each point of a ^-Poisson point process at time zero,, these particles are /l-Poisson distributed at every later time £>0 Q]. In the present paper we are concerned with the ergodic properties of the stationary processes obtained from such a system of particles, which is called the equilibrium process. Sinai's ideal gas model is a special example of the equilibrium processes Q4]. In §1 we will give some preliminaries and the definition of the equilibrium process, and §2 is devoted to the study of the ergodic properties (metrical transitivity, mixing properties and pure nondeterminism) of the equilibrium processes. In §3 we will discuss the Bernoulli property in the strong sense of the shift flow {® t}-