Convergence of longmemory discrete $k$th order Volterra processes
Abstract
We obtain limit theorems for a class of nonlinear discretetime processes called the th order Volterra processes of order . These are moving average th order polynomial forms:
where is i.i.d. with , , where is a nonrandom coefficient, and where the diagonals are included in the summation. We specify conditions for to be welldefined in , and focus on central and noncentral limit theorems. We show that normalized partial sums of centered obey the central limit theorem if decays fast enough so that has short memory. We prove a noncentral limit theorem if, on the other hand, is asymptotically some slowly decaying homogeneous function so that has long memory. In the noncentral case the limit is a linear combination of Hermitetype processes of different orders. This linear combination can be expressed as a centered multiple WienerStratonovich integral.
Key words Long memory; Longrange dependence; Volterra process; Wiener chaos; Wiener; Stratonovich; Limit theorems
2010 AMS Classification: 60G18, 60F05
1 Introduction
A common assumption when analyzing a stationary time series , is that is a causal linear process, that is,
(1) 
where is a sequence of i.i.d. random variables with mean and variance . This assumption is based on the Wold’s decomposition, which states that if is stationary with mean and finite second moment, and is also purely nondeterministic, then the representation (1) always holds with a sequence of uncorrelated random variables (Brockwell and Davis [5] §5.7). The independence assumption of in (1) obliterates the higherorder dependence structure. In some applications, linear processes provide good approximations, while in others, not, as in the case of the ARCH model for volatility data.
The Volterra process extends linear process by incorporating nonlinearity. A (causal) Volterra process with highest order is of the form
(2) 
To understand the importance of (2), suppose that the stationary process is for some regular function . Then (2) can be heuristically regarded as its th order Taylor series approximation. The homogeneous polynomialform expansion in (2) and its continuoustime counterpart where the sums are replaced with integrals, was originally proposed by Vito Volterra (see Volterra [22]) for modeling deterministic nonlinear systems, and later extended by Norbert Wiener (see Wiener [23]) to random systems, which eventually lead to the welldeveloped theory of Wiener chaos (see, e.g., Cameron and Martin [8], Itô [14], and the recent survey Peccati and Taqqu [19]). In the context of approximation of stationary processes, Nisio [18] shows that any stationary process can be approximated in the sense of finitedimensional distributions by a Volterra process with ’s Gaussian. Some nonlinear time series models admit Volterra expansions (2) with . For example, the LARCH() model
under suitable conditions admits the following Volterra expansion (see, e.g., Theorem 2.1 of Giraitis et al. [12]):
We are interested here in stationary processes that have long memory, or longrange dependence. A common choice is a linear process in (1) with as , where is the memory parameter, and is some constant. This is the case, for instance, when is the stationary solution of the fractional difference equation
where is the difference operator with being identity operator and being the backward shift operator, and is understood as a binomial series (see, e.g., Giraitis et al. [11] Chapter 7.2). We note that such longmemory linear processes have an autocovariance decaying like as , and a spectral density exploding at the origin as as .
If one wants to consider a nonlinear long memory model, a natural choice is to have a Volterra process (2) with coefficients decaying slowly as tends to infinity, so that the autocovariance has a slow hyperbolic decay. The major goal in this paper is to study the limit of normalized partial sum of some longmemory Volterra processes. When is a longmemory linear process, that is, a longmemory Volterra process with , then the limit, as is wellknown, is fractional Brownian motion (Davydov [9]). When is polynomial of a long memory linear processes, that is, when in (2) for some constant , and is large enough, then the limit is a Hermite process of a fixed order (Surgailis [20], Avram and Taqqu [1]). Such limit theorems involving nonBrownian motion limits are often called noncentral limit theorems.
In this paper, we focus on Volterra processes of a single order :
(3) 
which avoids possible cancellations between terms of different orders. Note that the multiple sum (3) includes diagonals, that is, it allows to be equal to each other. In the literature, one often considers multiple sums of the type (3) where summation over the diagonals is excluded, which greatly simplifies the theory. Although the exclusion of the diagonals is a typical theoretical assumption, it is, from a practical perspective, an artificial one. Expression (3) is the natural one since it includes all the terms.
To obtain a noncentral limit theorem for (3), we assume that the coefficient behaves asymptotically as a homogeneous function on which is bounded excluding a neighborhood of the origin. We shall show that in this case, the limit of a normalized sum of centered is a linear combination of Hermitetype processes of different orders. These Hermitetype processes that appear in the limit were first introduced in Mori and Oodaira [17], and were called in Bai and Taqqu [2] generalized Hermite processes. They live in Wiener chaos, and extend in a natural way the usual Hermite processes considered in the literature, e.g., Dobrushin and Major [10] and Taqqu [21].
The limit, which is a linear combination involving different orders of multiple WienerItô integrals, can be reexpressed as a single centered multiple WienerStratonovich integral with the zerothorder term excluded. These integrals were introduced by Hu and Meyer [13]. Loosely speaking, in contrast to the usual WienerItô integrals, the multiple WienerStratonovich integrals include diagonals, and intuitively they are the continuous counterpart of the multiple sums in (3) which, as was noted, do include diagonals.
The paper is organized as follows. In Section 2, we introduce the generalized Hermite processes which appear in the formulation of the noncentral limit theorem. In Section 3, we provide conditions for the polynomial form (3) to be welldefined in . In Section 4, we introduce the class of longmemory Volterra processes of interest in the noncentral limit theorem. In Section 5, we establish central limit theorems when in (3) decays fast enough so that has short memory. In Section 6, we state a noncentral limit theorem for processes in (3). Before launching into the article, the reader may want to have a look at this result, formulated as Theorem 6.2, and also at the illustrative Example 6.4. The connection between the limit and multiple WienerStratonovich integrals is indicated in Section 7. Section 8 contains an extended hypercontractivity formula.
2 Generalized Hermite processes and kernels
We introduce here the kernels which will be used to define both the coefficient in (3), and the processes that will appear in the noncentral limit.
First, some notation which will be used throughout the paper. Let , , , , and let denote the vector made of ’s. If , then , and . We write if , and use the following standard notations: denotes a norm in some suitable space, is the indicator function of a set , denotes the cardinality of set , and if and are two functions on and respectively, then defines a scalar function on as .
The following class of functions was introduced in Bai and Taqqu [2]:
Definition 2.1.
A generalized Hermite kernel (GHK) is a nonzero measurable function defined on satisfying:

, , ;

.
Remark 2.2.
As shown in Theorem 3.5 and Remark 3.6 in Bai and Taqqu [2], if is a GHK on , then for every ,
for a.e. . Furthermore,
is a.e. defined, and . In addition, if is nonzero, then .
These functions were used in Bai and Taqqu [2] as defining kernels for a class of stochastic processes called generalized Hermite processes.
Definition 2.3.
The generalized Hermite processes are defined through the following multiple WienerItô integrals:
(4) 
where the prime indicates that one does not integrate on the diagonals , , is a Brownian random measure, and is a GHK defined in Definition 2.1.
The generalized Hermite processes are selfsimilar with Hurst exponent
(5) 
that is, has the same finitedimensional distributions as , and they have also stationary increments.
Example 2.4.
In Bai and Taqqu [2] the following subclass of functions , called generalized Hermite kernel of Class (B) was considered.
Definition 2.5.
We say that a nonzero homogeneous function on having homogeneity exponent is of Class (B) (abbreviated as “GHK(B)”, “B” stands for “boundedness”), if

is a.e. continuous on ;

for some constant , where is as in Definition 2.1.
Remark 2.6.
The norm in Definition 2.5 can be any norm in the finitedimensional space since all the norms are equivalent. For convenience, we choose throughout this paper . The GHK(B) class is a subset of the GHK class, because if is a GHK(B), then it is homogeneous and hence satisfies Condition 1 of Definition 2.1. It also satisfies Condition 2 of Definition 2.1. Indeed, we have for some that
(6) 
where the last inequality follows from the arithmeticgeometric mean inequality
In view of Condition 1 of Definition 2.1, since , we hence have
Example 2.7.
As an example of a GHK(B), we can simply set equal to
since .
Example 2.8.
As another example, consider
and
is continuous and homogeneous with exponent . It is a GHK(B) because the functions and are bounded on the dimensional unit sphere restricted to . For instance,
by the equivalence of norms on . Thus .
Example 2.9.
It is easy to see that the set of GHK(B) functions on with fixed homogeneity exponent (with the zero function added) is closed under linear combinations and taking maximum or minimum. Thus one can consider , and using the and in the foregoing examples.
In Bai and Taqqu [2], noncentral limit theorems involving GHK(B) are established
(7) 
where , is a GHK(B), is some asymptotically negligible function (see (25) and the lines below), and the prime means that we do not sum on the diagonals , , i.e., the summation in (7) is only over unequal . We note that when is symmetric, the autocovariance of in (7) is
3 definiteness
In this section, we derive conditions under which a th order polynomial form with diagonals is welldefined.
The th order Volterra process in (3) is a polynomial form in i.i.d. random variables . To allow for long memory and obtain noncentral limit theorems, the coefficient in (3) must be nonzero at an infinite number of . Otherwise is an dependent sequence and thus subject to the central limit theorem (Billingsley [4]). So the first problem is to ensure that such a polynomial form with an infinite number of terms is welldefined, that is, to determine when the following random variable is welldefined:
(8) 
where is an i.i.d. sequence such that
(9) 
One can restrict to be a symmetric function in , since a permutation of the variables does not affect , but we shall not do so unless indicated, because it is easier to write down nonsymmetric ’s.
First, we have the following straightforward criterion for the welldefinedness of :
Proposition 3.1.
If , then in (8) is welldefined in the sense.
Proof.
Let
It suffices to check that is a Cauchy sequence in . This is true since for any ,
where is bounded above by a constant because of the assumption in (9). ∎
The absolute summability assumption in Proposition 3.1 is easy to work with, but it is unfortunately too restrictive for incorporating long memory. We will introduce instead a condition on so that is welldefined in the sense. Beside the obvious assumption , some delicate assumptions on need to be imposed, which are stated in Proposition 3.3 below. We first give an outline of the idea. If in (8) is instead defined as an offdiagonal polynomial form:
(10) 
then due to the offdiagonality, it is easy to see that the welldefinedness of is guaranteed by the simple squaresummability condition:
which equals if is symmetric. In fact, this defineness criterion still holds if one has more generally
(11) 
where forms an i.i.d. sequence of dimensional vector with mean and finite variance in each component. We will need this fact below.
In order to check that the polynomialform in (8), which includes diagonals, is welldefined, we shall decompose it into a finite number of offdiagonal polynomial forms, and check the welldefinedness of each using the simple squaresummability condition. In order to do this, we introduce some further notation, which will also be useful in the sequel.
We let denote all the partitions of . If , then denotes the number of sets in the partition. If we have a variable , then denotes a new variable where its components are identified according to . For example, if , and , then . In this case we write where and . If is a function on , then , where . In the preceding example, with .
Suppose that , where , . We suppose throughout that the ’s are ordered according to their smallest element. In the preceding example, and . We define the following summation operation on a function on .
Definition 3.2.
For any , the summation is obtained by summing over its variables indicated by offdiagonally, yielding a function with variables.
For instance, if , then and if , then
(12) 
provided that it is welldefined. Note that in this offdiagonal sum, we require, in addition to , that neither nor equals to . If , is understood to be the identity operator, where no summation is performed.
We need also Appell polynomials which we briefly introduce here. For more details, see, e.g. Avram and Taqqu [1] or Chapter 3.3 of Beran et al. [3]. Given a random variable with , the th order Appell polynomial with respect to the law of , is defined through the following recursive relation:
For example, if , then , , etc. If in addition , then , and . For consistency, one sets . We will use an important property of Appell polynomials, namely, for any integer ,
(13) 
Proposition 3.3.
Remark 3.4.
An example of satisfying (14) but not (15) is given by:
Note that is summable because is finite by the integral test, while is not summable.
Proof of Proposition 3.3.
By collecting various diagonal cases, we express as
(16) 
where , , , , . Since is finite, one can focus on the definedness of each term
Let be the th order Appell polynomial with respect to the law of . Let
Then by (13),
Thus to ensure , it suffices to show that
(17) 
is welldefined in for any .
Note now the following crucial fact. Since by assumption, we do not need to consider in (17). Thus:
If , then we need to consider only .  (18) 
Suppose first that . Since by assumption and for , then in view of the discussion concerning (11), it is sufficient to require (14). Now suppose that some , and observe that is then the constant . Thus if is the set of ’s such that , then
(19) 
where , and
(20) 
So one can bound by a constant times the sum in (15) since (19) has the form (11).
∎
Remark 3.5.
We now state here a practical sufficient condition for Proposition 3.3:
Proposition 3.6.
Let be a function on such that
where is some constant and , . Then
is a welldefined random variable in , where is i.i.d. with mean and variance and .
4 Volterra processes with long memory
We introduce in this section the th order Volterra processes for which we establish noncentral limit theorems in Section 6.