Central limit theorem for stochastically continuous processes. Convergence to stable limit

LetX={X(t), t[0, 1]} be a stochastically continuous cadlag process. Assume that thek dimensional finite joint distributions ofX are in the domain of normal attraction of strictlyp-stable, 0<p<2, measure onR ^k for all 1k<. For functionsf, g such that _p (|X(x–X(u)|) >g(u–s) and _p (|X(s–X(t|)|X(t)–X(u|)>f(u–s), 0 s t u 1, conditions are found which imply that the distributions –(n ^–1/p (X ₁+···+X _n )),n1, converge weakly inD[0, 1] to the distribution of ap-stable process. HereX ₁,X ₂, ... are independent copies ofX and _p (Z)=sup _t<0 t ^pP{|Z|<t} denotes the weakpth moment of a random variable Z.     

Extremes of deterministic sub‐sampled moving averages with heavy‐tailed innovations

Let {X_k}_k?1 be a strictly stationary time series. For a strictly increasing sampling function g:?→? define Y_k=X_g(k) as the deterministic sub‐sampled time series. In this paper, the extreme value theory of {Y_k} is studied when X_k has representation as a moving average driven by heavy‐tailed innovations. Under mild conditions, convergence results for a sequence of point processes based on {Y_k} are proved and extremal properties of the deterministic sub‐sampled time series are derived. In particular, we obtain the limiting distribution of the maximum and the corresponding extremal index. Copyright © 2003 John Wiley & Sons, Ltd.     

A functional central limit theorem for a class of urn models

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.     

The central limit theorem for stationary associated sequences

We study the problem of convergence in distribution of a suitably normalized sum of stationary associated random variables. We focus on the infinite variance case. New results are announced. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rate of convergence in the central limit theorem for empirical processes

Let _n and be an empirical process and a generalized Brownian bridge, respectively, indexed by a class of real measurable functions. From the central limit theorem for empirical processes it follows that for allr0. In this paper, assuming the class to be countably determined, under certain conditions we obtain an estimate for some constantC. Vapnik-ervonenkis class and the indicators of lower left orthants provide examples of classes considered here.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.     

Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Central limit theorems for the large-spin asymptotics of quantum spins

We use a generalized form of Dysons spin wave formalism to prove several central limit theorems for the large-spin asymptotics of quantum spins in a coherent state.T. Michoel is a Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.–Vlaanderen)This material is based on work supported by the National Science Foundation under Grant No. DMS0303316.This article may be reproduced in its entirety for non-commercial purposes.Mathematics Subject Classification (2000): 60F05, 82B10, 82B24, 82D40     

Central limit theorems for the radial spanning tree

Consider a homogeneous Poisson point process in a compact convex set in d‐dimensional Euclidean space which has interior points and contains the origin. The radial spanning tree is constructed by connecting each point of the Poisson point process with its nearest neighbour that is closer to the origin. For increasing intensity of the underlying Poisson point process the paper provides expectation and variance asymptotics as well as central limit theorems with rates of convergence for a class of edge functionals including the total edge length. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 262–286, 2017     

Almost sure central limit theory for products of sums of partial sums

Consider a sequence of i.i.d. positive random variables. An universal result in almost sure limit theorem for products of sums of partial sums is established.We will show that the almost sure limit the...     

A note on the central limit theorem for bipower variation of general functions

In this paper we present a central limit theorem for general functions of the increments of Brownian semimartingales. This provides a natural extension of the results derived in [O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij, N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, in: From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, Springer, 2006], where the central limit theorem was shown for even functions. We prove an infeasible central limit theorem for general functions and state some assumptions under which a feasible version of our results can be obtained. Finally, we present some examples from the literature to which our theory can be applied.     

Central limit theorems for power variation of Gaussian integral processes with jumps

This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with stationary increments,ξis a purely non-Gaussian L′evy process independent from G,andφis a stochastic process ensuring that the integral is well defined as a pathwise Riemann-Stieltjes integral.We obtain the central limit theorems for the case that both the continuous term and the jump term are presented simultaneously in the law of large numbers.     

Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

Refinement of the almost sure central limit theorem for associated processes

In this paper, for the partial sumsS _n of a stationary associated random process it is proved that the logarithmic averages converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their limiting value is established. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000.     

Two-sided bounds and leading term for rates of convergence in the multivariate central limit theorem

For a sequence of independent and identically distributed random vectors, upper and lower bounds are obtained for the discrepancy between the probability measure P_n, induced by their normalized sum, and the Normal measure Φ. The upper and lower bounds are of the same order of magnitude. These results may be derived by a “leading term” approach, in which a signed measure Q_n is introduced as a first order approximation to P_n − Φ. The purpose of this paper is to investigate properties of the leading term.     

The hydrodynamic limit for the reaction diffusion equation — An approach in terms of the GPV method

We study the hydrodynamic limit of the reaction diffusion process by means of the GPV technique (Guoet al. ⁽⁴⁾). To this end, we first derivea priori bounds on the moments of the occupation numbers using the local central limit theorem and results of stochastic analysis. The result of De Masi and Presutti⁽²⁾ for the hydrodynamic limit of the reaction diffusion process is generalized here.     

Density estimates and central limit theorem for the functional of fractional SDEs

In this paper, we consider a general class of functionals of stochastic differential equations driven by fractional Brownian motion. For this class, we obtain Gaussian estimates for the density and a quantitative central limit theorem. The main tools of the paper are the techniques of Malliavin calculus.