The empirical process of some long-range dependent sequences with an application to U-statistics

Let (Xj)∞ j = 1 be a stationary, mean-zero Gaussian process with covariances r(k) = EXk + 1 X1 satisfying r(0) = 1 and r(k) = k-DL(k) where D is small and L is slowly varying at infinity. Consider the two-parameter empirical process for G(Xj), $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where G is any measurable function. Noncentral limit theorems are obtained for FN(x, t) and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and U-statistics based on the G(Xj)'s. The limiting processes are structurally different from those encountered in the i.i.d. case

Weak convergence of sums of moving averages in the α-stable domain of attraction

Skorohod has shown that the convergence of sums of i.i.d. random variables to an a-stable Levy motion, with 0 < a < 2, holds in the weak-J1 sense. J1 is the commonly used Skorohod topology. We show that for sums of moving averages with at least two nonzero coefficients, weak-J1 conver- gence cannot hold because adjacent jumps of the process can coalesce in the limit; however, if the moving average coefficients are positive, then the adjacent jumps are essentially monotone and one can have weak-M1 con- vergence. M1 is weaker than J1, but it is strong enough for the sup and inf functionals to be continuous

Hermite rank, power rank and the generalized Weierstrass transform

Using the theory of generalized Weierstrass transform, we show that the Hermite rank is identical to the power rank in the Gaussian case, and that an Hermite rank higher than one is unstable with respect to a level shift.Accepted manuscrip

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

(1/α)-Self similar α-stable processes with stationary increments

Originally published as a technical report no. 892, February 1990 for Cornell University Operations Research and Industrial Engineering. Available online: http://hdl.handle.net/1813/8775In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of α-stable processes with stationary increments for 1 < α < 2.https://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubhttps://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubAccepted mansucrip

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 \citet{maejima:tudor:2012:selfsimilar} and \citet{tudor:2013:analysis}. 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

A survey of functional laws of the iterated logarithm for self-similar processes

A process X(t) is self-similar with index H > 0 if the finite-dimensional distributions of X(at) are identical to those of aHX(t) for all a > 0. Consider self-similar processes X(t) that are Gaussian or that can be represented throught Wiener-Itô integrals. The paper surveys functional laws of the iterated logarithm for such processes X(t) and for sequences whose normalized sums coverage weakly to X(t). The goal is to motivate the results by including outline of proofs and by highlighting relationships between the various assumptions. The paper starts with a general discussion fo functional laws of the iterated logarithm, states some of their formulations and sketches the reproducing kernal Hilbert space set-up.ECS-80-15585 - National Science Foundatio

Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes $\{Z(t),t\ge 0\}$ called generalized Hermite process. They have stationary increments, are defined on a Wiener chaos with Hurst index $H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel $g$, called "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 $Z^\beta(t)$ of $Z(t)$, which allows $H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.Comment: Corrected some error

Central limit theorems for double Poisson integrals

Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as examples concerning random hazard rates.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ123 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm