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Uniform integrability and the central limit theorem for strongly mixing processes
... Let us now focus on term B j in (C-6) and let us define S T = T t=1 ψ(W j,t , ν 2 j ), with ψ(W j,t , ν 2 j ) being a stationary ergodic process with E[S T ] = 0 by definition and σ 2 T = V ar[S T ] → ∞. Then, following Theorem 3 of Denker [1986], we have ...
... By using the same argument as for Z t we have that ( T t=1 Rt) 2 /σ 2 R is a uniformly integrable sequence. Based on these results and again following Theorem 3 of Denker [1986] ...
We present a new framework for the robust estimation of latent time series
models which is fairly general and, for example, covers models going from ARMA
to state-space models. This approach provides estimators which are (i)
consistent and asymptotically normally distributed, (ii) applicable to various
classes of time series models, (iii) straightforward to implement and (iv)
computationally efficient. The framework is based on the recently developed
Generalized Method of Wavelet Moments (GMWM) and a new robust estimator of the
wavelet variance. Compared to existing methods, the latter directly estimates
the quantity of interest while performing better in finite samples and using
milder conditions for its asymptotic properties to hold. Moreover, results are
given showing the identifiability of the GMWM for various classes of time
series models thereby allowing this method to consistently estimate many models
(and combinations thereof) under mild conditions. Hence, not only does this
paper provide an alternative estimator which allows to perform wavelet variance
analysis when data are contaminated but also a general approach to robustly
estimate the parameters of a variety of (latent) time series models. The
simulation studies carried out confirm the better performance of the proposed
estimators and the usefulness and broadness of the proposed methodology is
shown using practical examples from the domains of economics and engineering
with sample sizes up to 900,000.
... Moreover, in the same paper it was also established that the weak invariance principle for +-mixing sequences is equivalent (in one direction assuming 4, < 1) to the Lindeberg's condition. By a result of Denker (1986) the CLT for the class described in the conjectures is equivalent to the uniform integrability of {$/a~}, and by Peligrad (1985) this is equivalent to uniform integrability of {max,,;,, X3/o',},. (For one of the implications (+), we assume 4, < 1.) ...
... Dehling, Denker and Philipp (1986) obtained a general CLT for strong mixing sequences using as a normalizing constant v'$ b,,. Hahn, Kuelbs and Samur (1987) obtained a CLT for +-mixing sequences with the maximal terms deleted. ...
The aim of this paper is to give new central limit theorems and invariance principles for φ-mixing sequences of random variables that support the Ibragimov–Iosifescu conjecture. A related conjecture is formulated and a positive answer is given for the distributions that have tails regularly varying with the exponent –2.
... Most of the research in this direction addresses the problem of a particular sequence of b n 's and little is known about (1.3) in its full generality. For the particular case of b 2 n = Var S n we would like to mention the characterization due to Denker (1986). ...
... It is possible that a central limit theorem might hold in this setting with a more general normalization. To see that more general normalizations are possible, examine the example given by Ibragimov and Rozanov [(1978), pages 179-180, Example 1], or the central limit theorems derived by Dehling, Denker and Philipp (1986), Denker (1986), Mori and Yoshihara (1986), Peligrad (1992) or Rosenblatt (1956). In all these papers b 2 n = nh n where h n is slowly varying as n → ∞. ...
We prove a central limit theorem for strongly mixing sequences under a sharp sufficient condition which combines the rate of the strong mixing coefficient with the quantile function. The result improves on all earlier central limit theorems for this type of dependence and answers a conjecture raised by Bradley in 1997. ¶ Moreover, we derive the corresponding functional central limit theorem.
... Using (6), the CLT (2) is proved in [10, Proof of Theorem 8 a)] via blocking type arguments; we refer to Denker [16] for a classical reference. Furthermore, as shown in [10, Proof of Theorem 8], the limit law (2) together with a tightness argument for a truncated version of κ ρ provides another refinement of the CLT, namely, the Weak Invariance Principle (WIP): ...
We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time $n$ tends to infinity, the scatterer size $\rho$ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive $\sqrt{n\log n}$ scaling (i) for fixed infinite horizon configurations -- letting first $n \to \infty$ and then $\rho \to 0$ -- studied e.g.~by Sz\'asz & Varj\'u (2007) and (ii) Boltzmann-Grad type situations -- letting first $\rho \to 0$ and then $n \to \infty$ -- studied by Marklof & T\'oth (2016).
... (b) In central limit theorems for strictly stationary, strongly mixing sequences (X k , k ∈ Z) of (not necessarily bounded) random variables with mean 0 and finite second moments, an assumption that ES 2 n → ∞ as n → ∞ is often used (in conjunction with other assumptions) in order to insure that S n "grows" (in probability) and examples such as the one in (a) above are avoided. See for example the central limit theorems in [4], [5], [7], [8]. ...
For a given strictly stationary, strongly mixing random sequence for which the distributions of the partial sums are tight, certain “tightness bounds" exist which depend only on the marginal distribution and the mixing rate.
... (Note that σ 2 ≤ Var P i,j .) Since the S (i) n are partial sums of a strictly stationary sequence of random variables and have variance σ 2 n , we know from [6] that ...
Owing to their capability of summarising interactions between elements of a system, networks have become a common type of data in many fields. As networks can be inhomogeneous, in that different regions of the network may exhibit different topologies, an important topic concerns their local properties. This paper focuses on the estimation of the local degree distribution of a vertex in an inhomogeneous network. The contributions are twofold: we propose an estimator based on local weighted averaging, and we set up a Monte Carlo cross-validation procedure to pick the parameters of this estimator. Under a specific modelling assumption we derive an oracle inequality that shows how the model parameters affect the precision of the estimator. We illustrate our method by several numerical experiments, on both real and synthetic data, showing in particular that the approach considerably improves upon the natural, empirical estimator.
... By using the same argument as for Z k in Appendix C.3, we have that ( k∈K * R k ) 2 /σ 2 R is a uniformly integrable sequence. Based on these results and again following Theorem 3 of Denker (1986) we have that ...
We present a general M-estimation framework for inference on the wavelet variance. This framework generalizes the results on the scale-wise properties of the standard estimator and extends them to deliver the joint asymptotic properties of the estimated wavelet variance vector. Moreover, this is achieved by extending the estimation of the wavelet variance to multidimensional random fields and by stating the necessary conditions for these properties to hold when the size of the wavelet variance vector goes to infinity with the sample size. Finally, these results generally hold when using bounded estimating functions thereby delivering a robust framework for the estimation of this quantity which improves over existing methods both in terms of asymptotic properties and in terms of its finite sample performance. The proposed estimator is investigated in simulation studies and different applications highlighting its good properties.
... converges in distribution to a Gaussian law. It was established for d = 1 by Denker[Den86], Mori and Yoshihara[MY86] using a blocking argument. Volný[Vol88] gave a proof for d arbitrary based on approximation by an array of independent random variables.A natural question would be: what if we replace R d by another normed space? ...
We show that in a separable infinite dimensional Hilbert space, uniform
integrability of the square of the norm of normalized partial sums of a
strictly stationary sequence, together with a strong mixing condition, does not
guarantee the central limit theorem.
... is the standard normal distribution, if and only if {σ −2 n S 2 n } n∈N is uniformly integrable (cf. [12], Theorem 3). While there are many examples where {σ −2 n S 2 n } n∈N is not uniformly integrable (cf. ...
Relative stability results for weakly dependent and strongly mixing strictly stationary sequences are established. As a consequence, some infinite memory models, including ARCH(1) processes, are relatively stable.
... (b) In central limit theorems for strictly stationary, strongly mixing sequences (X k , k ∈ Z) of (not necessarily bounded) random variables with mean 0 and finite second moments, an assumption that ES 2 n → ∞ as n → ∞ is often used (in conjunction with other assumptions) in order to insure that S n "grows" (in probability) and examples such as the one in (a) above are avoided. See for example the central limit theorems in [4], [5], [7], [8]. ...
For a given strictly stationary, strongly mixing random sequence for which the distributions of the partial sums are tight, certain \tightness bounds" exist which depend only on the marginal distribution and the mixing rate.
... g . Ibragimov ( 1975 , Theorem 2 . 1 ) or Denker ( 1986 ) . However , Herrndorf ( 1983 ) constructed a strictly stationary strongly mixing sequence X : = ( X , ) with finite second moments ( and arbitrarily fast mixing rate ) , with Var S , - + w at an exactly linear rate ( the X , ' s are uncorrelated ) , such that the family of distributions ( on R ) of the partial sums Si , S2 , S3 , . . . is tight . ...
If a strictly stationary sequence satisfies the strong mixing condition, then either the distributions of the partial sums are "tight" with respect to their own medians, or else they become completely "dissipated". The latter case holds if in addition the first maximal correlation coefficient is less than 1.
... Central limit theory for arrays of dependent r.v.'s {X,,j} saw strong activity in the late 1960's and early 1970's (see e.g. Philipp, 1969; Bergstr6m, 1970 Bergstr6m, , 1973 Ibragimov and Linnik, 1971) and some more recent revisitations (Samur, 1984; Denker, 1986; Yori and Yoshihara, 1986). A good deal of this work was directed towards finding sufficient conditions on the distribution of X,,.j under which ~j X,,j has the same limiting distribution as if the X..j were independent. ...
This paper discusses a general framework common to some varied known and new results involving measures of threshold exceedance by high values of stationary stochastic sequences. In particular these concern the following.(a) Probabilistic modeling of infrequent but potentially damaging physical events such as storms, high stresses, high pollution episodes, describing both repeated occurrences and associated ‘damage’ magnitudes.(b) Statistical estimation of ‘tail parameters’ of a stationary stochastic sequence {Xj}. This includes a variety of estimation problems and in particular cases such as estimation of expected lengths of clusters of high values (e.g. storm durations), of interest in (a).‘Very high’ values (leading to Poisson-based limits for exceedance statistics) and ‘high’ values (giving normal limits) are considered and exhibited as special cases within the general framework of central limit results for ‘random additive interval functions’. The case of array sums of dependent random variables is revisited within this framework, clarifying the role of dependence conditions and providing minimal conditions for characterization of possible limit types. The methods are illustrated by the construction of confidence limits for the mean of an ‘exceedance statistic’ measuring high ozone levels, based on Philadelphia monitoring data.
... Define S n = n i=1 Y i and σ 2 n = ES 2 n . Theorem 3. (Cogburn, 1960; Denker, 1986; Mori and Yoshihara, 1986) Let Y be a centered strictly stationary strongly mixing sequence such that EY 2 0 < ∞. If σ 2 n → ∞ as n → ∞ then the following are equivalent: Remark 6. ...
The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain Monte Carlo settings and hence the focus is on the connections between drift and mixing conditions and their implications. In particular, we consider three commonly cited central limit theorems and discuss their relationship to classical results for mixing processes. Several motivating examples are given which range from toy one-dimensional settings to complicated settings encountered in Markov chain Monte Carlo.
... Ibragimov (1962) and Ibragimov and Linnik (1971) have investigated weak laws of large numbers relating to sums of observations on stationary SM and UM processes. Other works in this area have appeared in Bradley (1983 Bradley ( , 1986 Bradley ( , 2001), Davydov (1968), Denker (1986), Doukhan (1994, Oodaira and Yoshihara (1972), Yoshihara (1978), and Withers (1981) among others. Statistical analyses of time series data, on the other hand, have, traditionally, followed a different path. ...
We consider the sampling properties of U-statistics based on a sample of realization from a class of stationary nonlinear processes which include, in particular,
linear, bilinear and finite order volterra processes. It is shown that if the size n of the realization tends to infinity then certain normalized versions of the U-statistics tend to be distributed normally with zero means and finite variances.
... There exist in the literature very nice results associated to strong mixing conditions in the ergodic-theoretic sense, for gaussian sequences, for stationary Markov chains, etc. We also have interesting results for central limit theorems and invariance principles see for example [4,12,13], we also have many applications of these results in Time Series Analysis, Extreme Values, Markov Chain Theory, and so forth. However from the definitions it is clear that these measures only compare two vectors of the sequence separated by n units, but they do not measure the dependency among the coordinates of any of these vectors. ...
The problem of dependency between two random variables has been studied throughly in the literature. Many dependency measures have been proposed according to concepts such as concordance, quadrant dependency, etc. More recently, the development of the Theory of Copulas has had a great impact in the study of dependence of random variables specially in the case of continuous random variables. In the case of the multivariate setting, the study of the strong mixing conditions has lead to interesting results that extend some results like the central limit theorem to the case of dependent random variables. In this paper, we study the behavior of a multidimensional extension of two well-known dependency measures, finding their basic properties as well as several examples. The main difference between these measures and others previously proposed is that these ones are based on the definition of independence among n random elements or variables, therefore they provide a nice way to measure dependency. The main purpose of this paper is to present a sample version of one of these measures, find its properties, and based on this sample version to propose a test of independence of multivariate observations. We include several references of applications in Statistics.
... A CLT for a stationary strong mixing sequence (X n ) was proved by Denker [8] in the following form. Let EX 1 = 0, EX 2 1 = σ 2 , 0 < σ < +∞, and σ 2 n = ES 2 n = nh(n), where h(n) is a slowly varying function. ...
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the cost function is continuous, non-decreasing and depends on the distance. As an application, the CLT in the transportation distance is proved for independent and some dependent stationary sequences.
... + Y n of a centered stationary strongly mixing sequence {Y i } with finite second moment, the well-known sufficient conditions for the central limit theorem are that Var(S n )/n is slowly varying as n → ∞ and the sequence {S 2 n /σ 2 n } is uniformly integrable (σ 2 n = Var(S n )). The conditions are checkable under various mixing conditions and they lead to the central limit theorem under the normalization σ n (see, Denker (1986) and Peligrad (1986) for a survey). ...
Braverman, Mallows and Shepp (1995), showed that if the absolute moments of partial sums of i.i.d. symmetric variables are equal to those of normal variables, then the marginals have normal distribution. This fact suggested the conjecture that probably the absolute moments alone characterize the homogeneous process with independent increments. In this paper we prove a more general result that gives a positive answer to this conjecture, and then apply it in order to obtain the CLT for a class of dependent random variables under a normalization involving the absolute moments of partial sums.
In the context of stochastic processes, ergodic theory relates the long-run “time-averages” such as the sample mean of an evolving strictly stationary process X
0, X
1, … to a “phase-average” computed as an expected value with respect to a probability distribution on the state space. This is the perspective developed in this chapter.
We prove a central limit theorem for strongly mixing sequences under a sharp sufficient condition which combines the rate of the strong mixing coefficient with the quantile function. The result improves on all earlier central limit theorems for this type of dependence and answers a conjecture raised by Bradley in 1997. Moreover, we derive the corresponding functional central limit theorem.
A minimal (in a certain sense) condition of weak dependency for stationary sequences is suggested which provides the validity of the central limit theorem.
The paper considers distributional limits for families of additive random interval functions ζ
t
= ζt(I), under an array form of strong mixing. This provides a natural general setting for discussing the central limit problem in a variety of situations, including sums of strongly mixing arrays, and certain random measures of interest in continuous parameter extremal theory. In particular previous results on array sums are extended, providing also insights into the role of various mixing conditions used in earlier works.
In this article we will introduce a real valued random field on a restricted indexed set and construct a classical asymptotic limit theorems on them. We will survey the basic properties of weakly dependent random processes and investigate two major mixing conditions for sequences of random variables. The concepts of weakly dependent sequence of random variables will be generalized to the case of random fields. Finally we will construct a central limit theorem and prove it.
We study the asymptotic distribution of
$S_N (A,X) = \sqrt {(2N + 1)} ^{ - d} (\sum {_{n \in A_N } } X_n )$
where A is a subset of
$\mathbb{Z}^d $
, A
N= A∩[−N, N]d
, v(A) = limN
card(A
N) (2N+1)−d
∈(0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S
N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if
$F_N (n;A) = {\text{card\{ }}A_N^c \cap {\text{(}}n + A_N {\text{)\} (}}2N + 1{\text{)}}^{ - d} $
has a limit F(n; A) as N → ∞ for each
$n \in \mathbb{Z}^d $
. We also study the class of sets A that satisfy this condition.
In a largely overlooked paper of Murray Rosenblatt from more than thirty years ago, it was shown that if a strictly stationary Markov chain satisfies "uniform ergodicity" as well as mixing (in the ergodic-theoretic sense), then it satisfies the (Rosenblatt) strong mixing condition. In this note, a strengthened version of that theorem will be presented. As a by-product, a couple of other, somewhat hidden, insights in that paper of Rosenblatt will be brought into sharper focus.
This is a brief history of the Rosenblatt process, how it came about, the role it played, its properties and a detailed description
of its various representations.
This article is a survey of the main results on the central limit theorem (CLT) and its invariance principle (IP) for mixing sequences that have been obtained in the probabilistic literature in the last fifty years or so with a view towards econometric applications. Each of these theorems specifies a set of moment, dependence, and heterogeneity conditions on the underlying sequence that ensures the validity of CLT and IP. Special emphasis is paid to the case in which the underlying sequence has just barely infinite variance, since this case is relevant to econometrics applications that involve high-frequency financial data. Moreover, two new results on IPs that apply to heteroscedastic sequences are obtained. The first IP applies to sequences whose variances evolve over time in a polynomial-like fashion, whereas the second IP concerns sequences that experience a single variance break at some point within the sample.
The aim of this paper is to give new central limit theorems and invariance principles for [phi]-mixing sequences of random variables that support the Ibragimov-Iosifescu conjecture. A related conjecture is formulated and a positive answer is given for the distributions that have tails regularly varying with the exponent -2.
Three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the borderline of the central limit theorem for strictly stationary, strongly mixing random sequences. In these constructions, a key role is played by quantiles, as in a related construction of Doukhan et al.(11)
We prove that under appropriate conditions a sequence of associated, integer-valued random variables is also strongly mixing. The almost sure invariance principle and the central limit theorem for integer-valued associated processes are also discussed.
Resampling methods for dependent variables typically involve selecting blocks of observations. In this paper, we investigate
the effects of different block lengths on the consistency of approximations generated by block bootstrap and subsampling methods.
A complete description is provided for the case of sample mean. It is shown that both methods are consistent if the block
length grows at a rate slower than the sample size. When the growth rate of block lengths is comparable to the sample size,
the resulting approximations are no longer consistent. In the latter case, we present a detailed account of the limit behavior
of these resampling distribution approximations, considered as random elements in the space of all probability measures on
the real line.
In this paper, we study the central limit theorem and its weak invariance principle for sums of a stationary sequence of random
variables, via a martingale decomposition. Our conditions involve the conditional expectation of sums of random variables
with respect to the distant past. The results contribute to the clarification of the central limit question for stationary
sequences.
Weak invariance principles are established for strictly stationary weakly dependent sequences, having a decomposed strong mixing coefficient into two parts, one based on the strong mixing condition with a polynomial mixing rate and other based on the ρ-mixing condition. The result is applied to the output of the Tukey ‘3R smoother’.
An earlier paper gave a construction of a strictly stationary, finite-state, nondegenerate random sequence which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem. Here it will be shown that random sequence is in fact triple-wise independent (though not quadruple-wise independent).
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