Limit Theorem For Partial Sums Research Articles

On the local limit theorems for linear sequences of lower psi-mixing Markov chains

In this paper we investigate the local limit theorem for partial sums of linear sequences of the form Xj=∑i∈Zaiξj−i. Here (ai)i∈Z is a sequence of constants satisfying ∑i∈Zai2<∞ and (ξi)i∈Z are functions of a stationary Markov chain with mean zero and finite second moment. The Markov chain is assumed to satisfy one-sided lower psi-mixing condition.

Central limit theorems for nearly long range dependent subordinated linear processes

AbstractThe limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

Asymptotic modelling of corn plant data using moving sums (MOSUM) technique

Empirical model building using linear regression has been widely used in agricultural study. The validity of an assumed model is usually check using either F of likelihood ratio test under normally distributed observations. In this paper an asymptotic method based on the moving sums (MOSUM) of triangular array of ordinary least squares (OLS) residuals is proposed. Reasonable tests statistics for detecting valid model are defined as the Kolmogorov-Smirnov (KS) and Cramér-von Mises (MvM) functionals of the residual MOSUM process. The limit process is obtained for the condition under H0 and under H1 by applying the continuous mapping theorem to the existing limit theorem for partial sums of the residuals. The quantiles of the KS and CvM statistics under H0 are approximated by simulation. The application of the method in obtaining a valid model for the speed of growth of corn plants results in the conclusion that a first-order polynomial regression model is shown to be plausible.

Stable limits for Markov chains via the Principle of Conditioning

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions.We show that if the transition operator of the chain is operator uniformly integrable and the chain is ρ-mixing, then the limit is the same as if the summands were independent.We provide an example of a Markov chain that is operator uniformly integrable (and admits a spectral gap) while it is not hyperbounded.What makes our assumptions working is a new, efficient version of the Principle of Conditioning.

Optimal rates for parameter estimation of stationary Gaussian processes

We study rates of convergence in central limit theorems for partial sums of polynomial functionals of general stationary and asymptotically stationary Gaussian sequences, using tools from analysis on Wiener space. In the quadratic case, thanks to newly developed optimal tools, we derive sharp results, i.e. upper and lower bounds of the same order, where the convergence rates are given explicitly in the Wasserstein distance via an analysis of the functionals’ absolute third moments. These results are tailored to the question of parameter estimation, which introduces a need to control variance convergence rates. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.

Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows

We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag–Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240–285] dealt with positive dependence in the increment process, whereas this paper derives the functional limit theorems under negative dependence. The negative dependence is due to cancellations arising from Gaussian-type fluctuations of functionals of the associated Harris chain. The new types of limiting processes involve stable random measures, due to heavy tails, Mittag–Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations. Along the way, we prove a function central limit theorem for fluctuations of functionals of Harris chains which is of independent interest as it extends a result of Chen [Probab. Theory Related Fields 116 (2000) 89–123].

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

In the paper, we discuss how to define long, short, and negative memories for stationary processes on ℤ and fields on ℤ d with infinite variances. We propose to distinguish the properties of dependence and memory, and to attribute memory properties not only to a stationary random process (or a field) but also to a process and the operation that we apply to this process. We deal exclusively with the summation operation, that is, we consider the limit behavior of partial sums of random processes or fields. In order to have a unified approach to processes and fields with finite and infinite variances, we propose to define memory properties via the growth of normalizing sequences in limit theorems for partial sums. Also, we propose to change a little bit the terminology: instead of terms “long and short memories,” to use positive and zero memories, respectively, leaving the term “negative memory” and introducing “strongly negative memory.” For random fields, we introduce the notions of isotropic and directional memories.

Limit theorems for weighted Bernoulli random fields under Hannan’s condition

Recently, invariance principles for partial sums of Bernoulli random fields over rectangular index sets have been proved under Hannan’s condition. In this note we complement previous results by establishing limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu’s condition, which is stronger than Hannan’s condition.

An Invariance Principle for Fractional Brownian Sheets

We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (Bernoulli 17:88–113, 2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an \(m\)-approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (Stoch. Process. Appl. 123:1–14, 2013).

On the random functional central limit theorems with almost sure convergence for subsequences

Abstract In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems for subsequences.

Approximating class approach for empirical processes of dependent sequences indexed by functions

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class ${\mathcal{F}}$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.

Almost sure central limit theorem for partial sums of Markov chain

The authors prove an almost sure central limit theorem for partial sums based on an irreducible and positive recurrent Markov chain using logarithmic means, which realizes the extension of the almost sure central limit theorem for partial sums from an i.i.d. sequence of random variables to a Markov chain.

Almost sure central limit theorem for partial sums and maxima

AbstractLet X, X1, X2, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX2 = 1. Let Xi and Mn = max{Xi, 1 ≤ i ≤ n }. Suppose there exists constants an &gt; 0, bn ∈ R and a nondegenrate distribution G (y) such that equation image Then, we have equation image almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the tvGARCH(1,1) model: Existence, CLT, and tail index

We study the properties of the tvGARCH(1, 1) model (1.8) with logistic coefficients. We obtain conditions for the existence of an L p -bounded solution (p ⩾ 0) of Eq. (1.8). For p ⩾ 4, we prove an exponential decay of the lagged correlation function and the central limit theorem for partial sums of the squared tvGARCH(1, 1) process under similar conditions as in the stationary case. In the second part of the paper, we study the (weak) tail index of the tvGARCH(1, 1) model.

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Renyi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L p -norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Renyi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.

Nonlinear system theory: Another look at dependence

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.

Central limit theorems for partial sums of bounded functionals of infinite-variance\\moving averages

For j= 1, ..., J, let Kj : R -+ R be measurable bounded functions and Xj,,, = -fR aj(n cjx)M(dx), n > 1, be a-stable moving averages where a E (0, 2), cj > 0 for j= 1, ..., J, and M(dx) is an astable random measure on R with the Lebesgue control measure and skewness intensity f# G [-1, 1]. We provide conditions on the functions aj and Kj, j =1,..., J, for the normalized partial sums vector Nj 1/2 L Ni.-EjXjAi-L..iJ n=vector N1/2 (Kj(Xj, n) EK(Xn)), j = 1, ..., J, to be asymptotically normal as Nj 00. This extends a result established by Tailen Hsing in the context of causal moving averages with discrete-time stable innovations. We also consider the case of moving averages with innovations that are in the stable domain of attraction.

A limit theorem for partial sums of random variables and its applications

In this paper, we establish a new limit theorem for partial sums of random variables. As corollaries, we generalize the extended Borel–Cantelli lemma, and obtain some strong laws of large numbers for Markov chains as well as a generalized strong ergodic theorem for irreducible and positive recurrent Markov chains.

LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS

We introduce Gibbs–Markov maps T as maps with a (possibly countable) Markov partition and a certain type of bounded distortion property, and investigate its Frobenius–Perron operator P acting on (locally) Lipschitz continuous functions ϕ. If such a function ϕ belongs to the domain of attraction of a stable law of order in (0,2), we derive the expansion of the eigenvalue function t↦λ(t) of the characteristic function operators Ptf=Pf exp [i&lt; t,ϕ&gt; (perturbations of P) around 0. From this representation local and distributional limit theorems for partial sums ϕ+…+ϕ◦ Tn are easily obtained, provided ϕ is aperiodic. Applications to recurrence properties of group extensions are also given.

On partial sums of a gaussian sequence with stationary increments

We establish large increment properties for a Gaussian sequence with stationary increments under global conditions. Limit theorems for partial sums of the sequence with nonpositive correlation functions or the correlation functions on their speed of convergence to zero are proved via estimating a probability inequality on the supremum of Gaussian processes