Search

It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper we provide a deeper understanding of these intricate limiting phenomena. We show that intermittent processes may exhibit a multiscale behavior involving growth at different rates. To these rates correspond different scales. In addition to a dominant scale, intermittent processes may exhibit secondary scales. The probability of these scales decreases to zero as a power function of time. For the analysis, we consider large deviations of the rate of growth of the processes. Our approach is quite general and covers different possible scenarios with special focus on the so-called supOU processes.

This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or L\'evy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fej\'er-type singular integrals, and L\'evy-It\^o-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.

We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial asset pricing. The GFBM is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the regions of these two parameter values where the GFBM is a semimartingale. We next study the mixed process made up of an independent BM and a GFBM and identify the range of parameters for it to be a semimartingale, which leads to $H \in (1/2,1)$ for the GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with $H \in \{1/2\}\cup(3/4,1]$ proved by Cheridito (2001) and shows the significance of the additional parameter introduced in the GFBM. We then study the semimartingale asset pricing theory with the mixed GFBM, in presence of long range dependence, and applications in option pricing and portfolio optimization. Finally we discuss the implications of using GFBM on arbitrage theory, in particular, providing an example of semimartingale asset pricing model of long range dependence without arbitrage.

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Holder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.

One of the main problem in prediction theory of discrete-time second-order stationary processes $X(t)$ is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting $X(0)$ given $ X(t),$ $-n\le t\le-1$, as $n$ goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process $X(t)$. In his seminal paper {\it "Some purely deterministic processes" (J. of Math. and Mech.,} 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for discrete-time deterministic processes in the following two cases: (a) the spectral density $f(\lambda)$ of $X(t)$ is continuous and vanishes on an interval, (b) the spectral density $f(\lambda)$ has a very high order contact with zero. He showed that in the case (a) the prediction error variance behaves exponentially, while in the case (b), it behaves hyperbolically as $n\to\infty$. In this paper, using a new approach, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.

SupOU processes are superpositions of Ornstein-Uhlenbeck type processes with a random intensity parameter. They are stationary processes whose marginal distribution and dependence structure can be specified independently. Integrated supOU processes have then stationary increments and satisfy central and non-central limit theorems. Their moments, however, can display an unusual behavior known as "intermittency". We show here that intermittency can also appear when the processes have a heavy tailed marginal distribution and, in particular, an infinite variance.

The so-called "supOU" processes, namely the superpositions of Ornstein-Uhlenbeck type processes are stationary processes for which one can specify separately the marginal distribution and the dependence structure. They can have finite or infinite variance. We study the limit behavior of integrated infinite variance supOU processes adequately normalized. Depending on the specific circumstances, the limit can be fractional Brownian motion but it can also be a process with infinite variance, a L\'evy stable process with independent increments or a stable process with dependent increments. We show that it is even possible to have infinite variance integrated supOU processes converging to processes whose moments are all finite. A number of examples are provided., Comment: arXiv admin note: text overlap with arXiv:1904.00100

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carr\'e du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments., Comment: 24 pages

Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution. To show that supOU processes may exhibit intermittency, we establish the rate of growth of moments for each of the four limiting scenarios. The rate change indicates that there is intermittency, which is expressed here as a change-point in the asymptotic behavior of the absolute moments., Comment: Stochastic Processes and their Applications

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">)

We study the limiting behavior of continuous time trawl processes which are defined using an infinitely divisible random measure of a time dependent set. In this way one is able to define separately the marginal distribution and the dependence structure. One can have long-range dependence or short-range dependence by choosing the time set accordingly. We introduce the scaling function of the integrated process and show that its behavior displays intermittency, a phenomenon associated with an unusual behavior of moments.

Superpositions of Ornstein-Uhlenbeck type (supOU) processes form a rich class of stationary processes with a flexible dependence structure. The asymptotic behavior of the integrated and partial sum supOU processes can be, however, unusual. Their cumulants and moments turn out to have an unexpected rate of growth. We identify the property of fast growth of moments or cumulants as intermittency.

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 $L^2(\Omega)$., Comment: 37 pages. To appear in The Annals of Probability

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., Comment: 31 pages

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., Comment: Proofs greatly simplified

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., Comment: 32 pages, minor revision

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 Statistics

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 pages

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 Probability

\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.

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.

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)

We obtain limit theorems for a class of nonlinear discrete-time processes $X(n)$ called the $k$-th order Volterra processes of order $k$. These are moving average $k$-th order polynomial forms: \[ X(n)=\sum_{0

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 errors

This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the L\'{e}vy-Khintchine formula and derive asymptotic properties of the L\'{e}vy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ421 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". 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, 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.

We consider the multilinear polynomial-form process \[X(n)=\sum_{1\le i_1<\ldots

We investigate the power of the CUSUM test and the Wilcoxon change-point test for a shift in the mean of a process with long-range dependent noise. We derive analytiv formulas for the power of these tests under local alternatives. These results enable us to calculate the asymptotic relative efficiency (ARE) of the CUSUM test and the Wilcoxon change point test. We obtain the surprising result that for Gaussian data, the ARE of these two tests equals 1, in contrast to the case of i.i.d. noise when the ARE is known to be $3/\pi$.

Although fractional Brownian motion was not invented by Benoit Mandelbrot, it was he who recognized the importance of this random process and gave it the name by which it is known today. This is a personal account of the history behind fractional Brownian motion and some subsequent developments., Comment: Published in at http://dx.doi.org/10.1214/12-STS389 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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.

Let $G$ be a non--linear function of a Gaussian process $\{X_t\}_{t\in\mathbb{Z}}$ with long--range dependence. The resulting process $\{G(X_t)\}_{t\in\mathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with $\{G(X_t)\}_{t\in\mathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-It\^o integral of order one or two. We show, however, that there are large classes of functions $G$ which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-It\^o integral of order greater than two.

We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener It\^o integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.

Consider the sum $Z = \sum_{n=1}^\infty \lambda_n (\eta_n - \mathbb{E}\eta_n)$, where $\eta_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $\lambda_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty \lambda_n (\eta_n - \mathbb{E}\eta_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions., Comment: 19 pages, 2 figures

We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non--linear filter with Gaussian input. The wavelet coefficients that appear in the limit are random, typically non--Gaussian and belong to a Wiener chaos. They can be interpreted as wavelet coefficients of a generalized self-similar process.

Max-stable random sketches can be computed efficiently on fast streaming positive data sets by using only sequential access to the data. They can be used to answer point and Lp-norm queries for the signal. There is an intriguing connection between the so-called p-stable (or sum-stable) and the max-stable sketches. Rigorous performance guarantees through error-probability estimates are derived and the algorithmic implementation is discussed.

We present a method for computing the PDF and CDF of a non-negative infinitely divisible random variable $X$. Our method uses the L\'{e}vy-Khintchine representation of the Laplace transform $\mathbb{E} e^{-\lambda X} = e^{-\phi(\lambda)}$, where $\phi$ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples including the stable distribution, mixtures thereof, and integrals with respect to non-negative L\'{e}vy processes. Software to implement this method is available from the authors and we illustrate its use at the end of the paper., Comment: 24 pages, 7 figures, 1 table

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral $I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds)$, where $N$ is a Poisson random measure with control measure $n$ and $\krnl$ is a suitable kernel function. We do so by combining a Kolmogorov-Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper., Comment: 28 pages, 8 figures

A desirable property of an autocovariance estimator is to be robust to the presence of additive outliers. It is well-known that the sample autocovariance, being based on moments, does not have this property. Hence, the use of an autocovariance estimator which is robust to additive outliers can be very useful for time-series modeling. In this paper, the asymptotic properties of the robust scale and autocovariance estimators proposed by Rousseeuw and Croux (1993) and Genton and Ma (2000) are established for Gaussian processes, with either short-range or long-range dependence. It is shown in the short-range dependence setting that this robust estimator is asymptotically normal at the rate $\sqrt{n}$, where $n$ is the number of observations. An explicit expression of the asymptotic variance is also given and compared to the asymptotic variance of the classical autocovariance estimator. In the long-range dependence setting, the limiting distribution displays the same behavior than that of the classical autocovariance estimator, with a Gaussian limit and rate $\sqrt{n}$ when the Hurst parameter $H$ is less 3/4 and with a non-Gaussian limit (belonging to the second Wiener chaos) with rate depending on the Hurst parameter when $H \in (3/4,1)$. Some Monte-Carlo experiments are presented to illustrate our claims and the Nile River data is analyzed as an application. The theoretical results and the empirical evidence strongly suggest using the robust estimators as an alternative to estimate the dependence structure of Gaussian processes.

Let $(X_i)_{i\geq 1}$ be a stationary mean-zero Gaussian process with covariances $\rho(k)=\PE(X_{1}X_{k+1})$ satisfying: $\rho(0)=1$ and $\rho(k)=k^{-D} L(k)$ where $D$ is in $(0,1)$ and $L$ is slowly varying at infinity. Consider the $U$-process $\{U_n(r),\; r\in I\}$ defined as $$ U_n(r)=\frac{1}{n(n-1)}\sum_{1\leq i\neq j\leq n}\1_{\{G(X_i,X_j)\leq r\}}\; , $$ where $I$ is an interval included in $\rset$ and $G$ is a symmetric function. In this paper, we provide central and non-central limit theorems for $U_n$. They are used to derive the asymptotic behavior of the Hodges-Lehmann estimator, the Wilcoxon-signed rank statistic, the sample correlation integral and an associated scale estimator. The limiting distributions are expressed through multiple Wiener-It\^o integrals.

Let $\{D(s), s \geq 0 \}$ be a L\'evy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that $D(0) = 0$. We study the first-hitting time of the process $D$, namely, the process $E(t) = \inf \{s: D(s) > t \}$, $t \geq 0$. The process $E$ is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the $n$-time tail distribution function $P[E(t_1) > s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution given natural boundary conditions. This PDE can be used to derive all $n$-time moments of the process $E$., Comment: 13 pages, one figure