Behavior of the generalized Rosenblatt process at extreme critical exponent values

The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0,1]. These limits cannot be strengthened to convergence in L2(Ω).Supported in part by NSF Grants DMS-10-07616 and DMS-13-09009 at Boston University. (DMS-10-07616 - NSF at Boston University; DMS-13-09009 - NSF at Boston University)Accepted manuscrip

The impact of the diagonals of polynomial forms on limit theorems with long memory

We start with an i.i.d. sequence and consider the product of two polynomial-forms moving averages based on that sequence. The coefficients of the polynomial forms are asymptotically slowly decaying homogeneous functions so that these processes have long memory. The product of these two polynomial forms is a stationary nonlinear process. Our goal is to obtain limit theorems for the normalized sums of this product process in three cases: exclusion of the diagonal terms of the polynomial form, inclusion, or the mixed case (one polynomial form excludes the diagonals while the other one includes them). In any one of these cases, if the product has long memory, then the limits are given by Wiener chaos. But the limits in each of the cases are quite different. If the diagonals are excluded, then the limit is expressed as in the product formula of two Wiener-It\^{o} integrals. When the diagonals are included, the limit stochastic integrals are typically due to a single factor of the product, namely the one with the strongest memory. In the mixed case, the limit stochastic integral is due to the polynomial form without the diagonals irrespective of the strength of the memory.Comment: Published at http://dx.doi.org/10.3150/15-BEJ697 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Sensivity of the Hermite rank

The Hermite rank appears in limit theorems involving long memory. We show that an Hermite rank higher than one is unstable when the data is slightly perturbed by transformations such as shift and scaling. We carry out a "near higher order rank analysis" to illustrate how the limit theorems are affected by a shift perturbation that is decreasing in size. As a byproduct of our analysis, we also prove the coincidence of the Hermite rank and the power rank in the Gaussian context. The paper is a technical companion of \citet{bai:taqqu:2017:instability} which discusses the instability of the Hermite rank in the statistical context. (Older title "Some properties of the Hermite rank">

Short-range dependent processes subordinated to the Gaussian may not be strong mixing

There are all kinds of weak dependence. For example, strong mixing. Short-range dependence (SRD) is also a form of weak dependence. It occurs in the context of processes that are subordinated to the Gaussian. Is a SRD process strong mixing if the underlying Gaussian process is long-range dependent? We show that this is not necessarily the case.Comment: 3 page

Behavior of the generalized Rosenblatt process at extreme critical exponent values

The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0,1]. These limits cannot be strengthened to convergence in L2(Ω).Supported in part by NSF Grants DMS-10-07616 and DMS-13-09009 at Boston University. (DMS-10-07616 - NSF at Boston University; DMS-13-09009 - NSF at Boston University)Accepted manuscrip

How the instability of ranks under long memory affects large-sample inference

Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue that a rank other than one is unstable, whereas, a rank equal to one is stable. We provide empirical evidence supporting this argument. This has important consequences. Assuming a higher-order rank when it is not really there usually results in underestimating the order of the fluctuations of the statistic of interest. We illustrate this through various examples involving the sample variance, the empirical processes and the Whittle estimator.Accepted manuscrip

On the validity of resampling methods under long memory

For long-memory time series, inference based on resampling is of crucial importance, since the asymptotic distribution can often be non-Gaussian and is difficult to determine statistically. However due to the strong dependence, establishing the asymptotic validity of resampling methods is nontrivial. In this paper, we derive an efficient bound for the canonical correlation between two finite blocks of a long-memory time series. We show how this bound can be applied to establish the asymptotic consistency of subsampling procedures for general statistics under long memory. It allows the subsample size $b$ to be $o(n)$, where $n$ is the sample size, irrespective of the strength of the memory. We are then able to improve many results found in the literature. We also consider applications of subsampling procedures under long memory to the sample covariance, M-estimation and empirical processes.Comment: 36 pages. To appear in The Annals of Statistic

The universality of homogeneous polynomial forms and critical limits

Nourdin et al. [9] established the following universality result: if a sequence of off-diagonal homogeneous polynomial forms in i.i.d. standard normal random variables converges in distribution to a normal, then the convergence also holds if one replaces these i.i.d. standard normal random variables in the polynomial forms by any independent standardized random variables with uniformly bounded third absolute moment. The result, which was stated for polynomial forms with a finite number of terms, can be extended to allow an infinite number of terms in the polynomial forms. Based on a contraction criterion derived from this extended universality result, we prove a central limit theorem for a strongly dependent nonlinear processes, whose memory parameter lies at the boundary between short and long memory.Comment: 13 pages; to appear in Journal of Theoretical Probabilit

Multivariate limit theorems in the context of long-range dependence

We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multivariate Gaussian process involving dependent Brownian motion marginals, or (b) a multivariate process involving dependent Hermite processes as marginals, or (c) a combination. We treat cases (a), (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary

Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes

\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained