Non-central Limit Theorems Research Articles

Non-central limit theorems for quadratic functionals of Hermite-driven long memory moving average processes

Let [Formula: see text] denote a Hermite process of order [Formula: see text] and self-similarity parameter [Formula: see text]. Consider the Hermite–driven moving average process [Formula: see text] In the special case of [Formula: see text], [Formula: see text] is the non-stationary Hermite Ornstein–Uhlenbeck process of order [Formula: see text]. Under suitable integrability conditions on the kernel [Formula: see text], we prove that as [Formula: see text], the normalized quadratic functional [Formula: see text] where [Formula: see text], converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter [Formula: see text], up to a multiplicative constant, irrespective of self-similarity parameter whenever [Formula: see text]. In the Gaussian case [Formula: see text], our result complements the study started by Nourdin et al. in [10], where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener–Itô integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.

Symmetric Weighted Odd-Power Variations of Fractional Brownian Motion and Applications

We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H = 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.

Non-central limit theorems and convergence rates

Non-central limit theorems and convergence rates

Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter

This paper provides several statistical estimators for the drift and volatility parameters of an Ornstein-Uhlenbeck process driven by fractional Brownian motion, whose observations can be made either continuously or at discrete time instants. First and higher order power variations are used to estimate the volatility parameter. The almost sure convergence of the estimators and the corresponding central limit theorems are obtained for all the Hurst parameter range $H\in (0, 1)$. The least squares estimator is used for the drift parameter. A central limit theorem is proved when the Hurst parameter $H \in (0, 1/2)$ and a noncentral limit theorem is proved for $H\in[3/4, 1)$. Thus, the open problem left in the paper by Hu and Nualart (2010) is completely solved, where a central limit theorem for least squares estimator is proved for $H\in [1/2, 3/4)$.

Non-central limit theorems for random fields subordinated to gamma-correlated random fields

A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in dd-dimensional space. As a particular case, integrals of non-linear functions of chi-squared random fields, with Laguerre rank being equal to one and two, are studied. When the Laguerre rank is equal to one, the characteristic function of the limit random variable, given by a Rosenblatt-type distribution, is obtained. When the Laguerre rank is equal to two, a multiple Wiener–Ito stochastic integral representation of the limit distribution is derived and an infinite series representation, in terms of independent random variables, is obtained for the limit.

Outage probability of SSTS for distributed antenna systems over Rayleigh fading channels in multi-cell environment

In this paper, downlink outage probability for distributed antenna systems (DAS) in multicell environment is proposed, while based on maximum desired signal criterion, a single selection transmission scheme (SSTS) is proposed. Usually, adopting central limit theorem (CLT) method, the component of interference plus noise is considered as a fixed-variance Gaussian random variable in most papers. However, the aforementioned method does not reflect the effect of short-term fading on interference and its usage is in the constraints of restrictive conditions. To relax the constraints, non-central limit theorem (NCLT) is introduced, which treats the variance of interference plus noise as a changeable-variance random variable influenced by short-term fading. It is assumed that channels are independent identical Rayleigh fading with propagation path-loss, and the closed-form expression of outage probability for DAS is derived. Finally, simulation results demonstrate the validity of theoretical analysis.

Block Bootstrap for the Empirical Process of Long‐Range Dependent Data

We consider the bootstrapped empirical process of long‐range dependent data. It is shown that this process converges to a semi‐degenerate limit, where the random part of this limit is always Gaussian. Thus the bootstrap might fail when the original empirical process accomplishes a noncentral limit theorem. However, even in this case our results can be used to estimate a nuisance parameter that appears in the limit of many nonparametric tests under long memory. Moreover, we develop a new resampling procedure for goodness‐of‐fit tests and a test for monotonicity of transformations.

Noncentral limit theorem for the generalized Hermite process

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes $Z_\gamma $ with kernels defined by parameters $\gamma $ taking values in a tetrahedral region $\Delta $ of $\mathbb{R} ^q$. We prove that, as $\gamma $ converges to a face of $\Delta $, the process $Z_\gamma $ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.

Functional limit theorems for generalized variations of the fractional Brownian sheet

We prove functional central and non-central limit theorems for generalized variations of the anisotropic $d$-parameter fractional Brownian sheet (fBs) for any natural number $d$. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes

We study the asymptotic behavior of a suitable normalized stochastic process {QT(t),t∈[0,1]}. This stochastic process is generated by a Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process. We show that under some Lp-type conditions imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard T normalized process QT(t) tend to those of a normalized standard Brownian motion. In contrast, when the covariance function of the model and the kernel of the quadratic functional have a slow power decay, then we have a non-central limit theorem for QT(t), that is, the finite-dimensional distributions of the process QT(t), normalized by Tγ for some γ>1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener–Itô integral on R2.

Intersection and proximity of processes of flats

Weakly stationary random processes of k-dimensional affine subspaces (flats) in Rn are considered. If 2k≥n, then intersection processes are investigated, while in the complementary case 2k<n a proximity process is introduced. The intensity measures of these processes are described in terms of parameters of the underlying k-flat process. By a translation into geometric parameters of associated zonoids and by means of integral transformations, several new uniqueness and stability results for these processes of flats are derived. They rely on a combination of known and novel estimates for area measures of zonoids, which are also developed in the paper. Finally, an asymptotic second-order analysis as well as central and non-central limit theorems for length-power direction functionals of proximity processes derived from stationary Poisson k-flat process complement earlier works for intersection processes.

Convergence of long-memory discrete [formula omitted]th order Volterra processes

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0<i1,…,ik<∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅) is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for X(n) to be well-defined in L2(Ω), and focus on central and non-central limit theorems. We show that normalized partial sums of centered X(n) obey the central limit theorem if a(⋅) decays fast enough so that X(n) has short memory. We prove a non-central limit theorem if, on the other hand, a(⋅) is asymptotically some slowly decaying homogeneous function so that X(n) has long memory. In the non-central case the limit is a linear combination of Hermite-type processes of different orders. This linear combination can be expressed as a centered multiple Wiener–Stratonovich integral.

On the rate of convergence to Rosenblatt-type distribution

The main result of the article is the rate of convergence to the Rosenblatt-type distributions in non-central limit theorems. Specifications of the main theorem are discussed for several scenarios. In particular, special attention is paid to the Cauchy, generalized Linnik's, and local–global distinguisher random processes and fields. Direct analytical methods are used to investigate the rate of convergence in the uniform metric.

Bit Error Rate of SSTS for Downlink Distributed Antenna Systems in Multicell Environment

Based on maximum desired signal criterion, a novel single selection transmission scheme (SSTS) is proposed. In this case, the bit error rate (BER) of downlink distributed antenna systems in multicell environment is investigated. In particular, non-central limit theorem is adopted for SSTS to embody the effect of short-term fading on interference, where the variance of interference plus noise is considered as random variable with changeable variance influenced by the short-term fading. It is assumed that the channels suffer from independent identical Rayleigh fading together with propagation pathloss, and the closed-form expression of BER is derived. Extensive simulations are carried out to validate the theoretical derivation.

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.

Structure of the third moment of the generalized Rosenblatt distribution

The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in Maejima and Tudor (2012) and Tudor (2013). Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.

Inference of weighted $V$-statistics for nonstationary time series and its applications

We investigate the behavior of Fourier transforms for a wide class of nonstationary nonlinear processes. Asymptotic central and noncentral limit theorems are established for a class of nondegenerate and degenerate weighted $V$-statistics through the angle of Fourier analysis. The established theory for $V$-statistics provides a unified treatment for many important time and spectral domain problems in the analysis of nonstationary time series, ranging from nonparametric estimation to the inference of periodograms and spectral densities.

The trace problem for Toeplitz matrices and operators and its impact in probability

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szegö, Toeplitz forms and their applications (University of California Press, Berkeley, 1958). It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, hypotheses testing about the spectrum, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory.

Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes {Z(t),t≥0} called generalized Hermite processes. They have stationary increments, are defined on a Wiener chaos with Hurst index H∈(1/2,1), and include Hermite processes as a special case. They are defined through a homogeneous kernel g, called the “generalized Hermite kernel”, which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels g can also be used to generate long-range dependent stationary sequences forming a discrete chaos process {X(n)}. In addition, we consider a fractionally-filtered version Zβ(t) of Z(t), which allows H∈(0,1/2). Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.

Non-central limit theorem of the weighted power variations of Gaussian processes

By using the techniques of the Malliavin calculus, we investigate the asymptotic behavior of the weighted q-variations of continuous Gaussian process of the form Bt=∫0tK(t,s)dW(s), where W is the standard Brownian motion and K is a square integrable kernel. In particular, in the case of fractional Brownian motion with the Hurst parameter H, the limit can be expressed as the sum of q+1 Skorohod integrals of the Hermite process with self-similarity q(H−1)+1. This result gives the relation between the Skorohod integral and a pathwise Young integral of the Hermite process.