Shuyang Bai

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.

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 multi‐variate Gaussian process involving dependent Brownian motion marginals, (b) a multi‐variate process involving dependent Hermite processes as marginals or (c) a combination. We treat cases (a) and (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, although the conjecture can be resolved in some special cases.

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]C[0,1]. These limits cannot be strengthened to convergence in L2(Ω)L2(Ω).

A unified approach to self-normalized block sampling

The inference procedure for the mean of a stationary time series is usually quite different under various model assumptions because the partial sum process behaves differently depending on whether the time series is short or long-range dependent, or whether it has a light or heavy-tailed marginal distribution. In the current paper, we develop an asymptotic theory for the self-normalized block sampling, and prove that the corresponding block sampling method can provide a unified inference approach for the aforementioned different situations in the sense that it does not require the {\em a priori} estimation of auxiliary parameters. Monte Carlo simulations are presented to illustrate its finite-sample performance. The R function implementing the method is available from the authors.

Convergence of long-memory discrete kth 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 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

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

b

Structure of the third moment of the generalized Rosenblatt distribution

to be

The Universality of Homogeneous Polynomial Forms and Critical Limits

o(n)

Canonical correlation between blocks of long-memory time series and consistency of subsampling

, where