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

Non-stationary self-similar Gaussian processes as scaling limits of power law shot noise processes and generalizations of fractional Brownian motion

We study shot noise processes with Poisson arrivals and non-stationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: 1) the conditional variance functions of the noises have a power law and 2) the conditional noise distributions are piecewise. In both cases, the limit processes are self-similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self-similar Gaussian processes with non-stationary increments, via the time-domain integral representation, which are natural generalizations of fractional Brownian motions.Published versio

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

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

Stable stationary processes related to cyclic flows

We study stationary stable processes related to periodic and cyclic flows in the sense of Rosinski [Ann. Probab. 23 (1995) 1163-1187]. These processes are not ergodic. We provide their canonical representations, consider examples and show how to identify them among general stationary stable processes. We conclude with the unique decomposition in distribution of stationary stable processes into the sum of four major independent components: 1. A mixed moving average component. 2. A harmonizable (or ``trivial'') component. 3. A cyclic component 4. A component which is different from these.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000010