The asymptotic behavior of partial sumsPn
t=1Ytfor stationary processes Ytwith linear long-range dependence is well known (see e.g. Rosenblatt 1961, Taqqu 1975, 1979, Dobrushin and Major 1979, Surgailis 1981, 1982, Giraitis 1983, 1985, Giraitis and Surgailis 1985, 1986, 1999, Avram and Taqqu 1987, Dehling and Taqqu 1989, Arcones and Yu 1994, Ho and Hsing 1996, 1997). One should already mention that asymptotic theory for processes that exhibit long-range dependence in volatility, which will be defined later, is much less developed. Two recent references are, for instance, Berkes and Horv´ath (2003) and Giraitis et al. (2000b), where certain limit theorems for LARCH processes are derived. The results of the latter two papers are summarized in section 5.4.
Here, we will describe limit theorems for Yt = f(Xt), where (i) Xt is gaussian and f admits a Hermite expansion and (ii) Xt is linear and f admits an Appell expansion. Start with case (i) and let (Xt)t∈Z be a stationary gaussian process with E[Xt] = 0, E[Xt2] = 1 and covariances such that
γX(t) =E[X0Xt]∼t−qΛ(t), t→ ∞ (2.1) for some 0 < q <1 and a slowly varying function Λ(t) (in fact Λ may take negative values as well). For a function H with E[H(X0)] = 0 and E[H(X0)2] <∞, we
CHAPTER 2. LONG MEMORY AND LIMIT THEOREMS 12 derive the asymptotic behavior of
Sn,H(u) :=
X[nu]
t=1
H(Xt).
(We will also use the notation Sn,H :=Sn,H(1).) Therefore, define Hermite polynomials as follows:
Definition 2.2 For m≥0, the Hermite polynomials Hm are defined by Hm(x) =ex2/2 dm
dxme−x2/2.
Since (Hm)m≥0 constitutes a complete orthogonal system in the space{H :R→ R : E[H(X0)] = 0, E[H2(X0)] < ∞}, see e.g. Abramowitz and Stegun (1972), the following expansion holds:
H(Xt) = X∞ m=0
cmHm(Xt), where P∞
m=0c2mm!<∞, that means the series converges in L2 since the variance of Hm(Xt) is given by var(Hm(Xt)) =m!(γX(0))m, see (2.2). Moreover, Hermite polynomials are uncorrelated and thus the coefficients can be calculated by cm =
1
m!E[H(X0)Hm(X0)]. The lowest integer m∗ ∈N with cm∗ 6= 0 is then called the Hermite rank ofH. DenoteγH(t) the covariance function ofH(Xt) and note that the (cross-)covariances of Hermite polynomials are given by
E[Hk(X0)Hj(Xt)] =δk,jk!(γX(t))k. (2.2) (The latter can be shown, e.g., by the diagram formula in theorem 2.5 below.) Thus
γH(t) =
X∞ n,m=m∗
cncmE[Hn(Xt)Hm(X0)]
= γX(t)m∗ X∞ m=m∗
c2mm!γX(t)m−m∗. (2.3) For m∗ >1/q, the covariances of Hm∗(Xt) are absolutely summable, since
X∞ t=1
|γX(t)|m∗ = X∞
t=1
t−qm∗Λm∗(t)<∞.
CHAPTER 2. LONG MEMORY AND LIMIT THEOREMS 13 (therefore note that γX(t) →0), i.e. one gets summability of the covariances of the process H(Xt).
The other way around, (2.3) and γX(t)→0 imply
|γH(t)| ≥ |γX(t)|m∗c2m∗m∗!/2, for t large. Consequently, we have the equivalence
X∞
meaning that a function H with Hermite rank m∗ >1/q leads to short memory in the process H(Xt). Indeed, Giraitis and Surgailis (1985) proved the following theorem:
Theorem 2.1 Let(Xt)t∈Z be a zero mean gaussian process for which (2.1) holds.
If P∞
t=0|γH(t)|<∞ (i.e. m∗ >1/q) and σ2 :=P∞
t=0γH(t)>0, then n−1/2Sn,H(u)→D σB(u),
where B(u) is a standard Brownian motion.
Next, we consider the case m∗ <1/q. First, we derive the order of growth of the variance ofSn,m:=Sn,Hmin the simple caseγ(t) =ct−q for a constant 0< c <∞:
CHAPTER 2. LONG MEMORY AND LIMIT THEOREMS 14 since the sum is a Riemann approximation of the latter integral. Analogously, we get for the latter term in (2.4) that
Xn−1 the usual central limit theorem with standard √
n-scaling can not hold for sums of Hm∗(Xt). Indeed, even a non-normal limiting distribution can arise and the same is true for Sn,H(u), as the next theorem, which goes back to Taqqu (1979) and Dobrushin and Major (1979), states.
Theorem 2.2 Let m∗ be the Hermite rank of H und (Xt)t∈Z a zero-mean where cm∗ is the m∗-th coefficient in the Hermite expansion of H.
Here, the Hermite process Hk(u) of orderk ≥1 is defined by
where B(·) is a standard Brownian motion and Cq is a positive constant (see e.g.
Taqqu 1979). Fork = 1,Hk(u) is fractional Brownian motion, and thus gaussian, while Hk(u) has non-normal marginal distributions for k ≥2.
The preceding theorem can be understood as a reduction principle, in the sense that the asymptotic properties of Sn,H(u) only depend on the Hermite rank m∗ of H. In particular, the sum ofH(Xt) and the sum ofcm∗Hm∗(Xt) have the same limiting distribution.
Giraitis (1985), Surgailis (1982) and Avram and Taqqu (1987) showed that this reduction principle, together with the central limit theorem2.1, can be generalized for linear processes (our case (ii)), which will be described in the following. Let Xt be given by
Xt=X
s∈Z
b(t+s)ξs, t ∈Z (2.5)
CHAPTER 2. LONG MEMORY AND LIMIT THEOREMS 15 with coefficients
b(t) = Λ(|t|)|t|d−1, t∈Z, (2.6) where Λ is a slowly varying function and d ∈ (0,12). (Here, the reason for the two-sided moving average representation of Xt is a more convenient notation in section 2.3.2.) Moreover, ξs, s ∈ Z are i.i.d. with E[ξs] = 0, E[ξs2] = 1 and E[|ξs|k] < ∞ for all k ≥ 0. The covariance function is then given by γX(t) = E[X0Xt] = Λ1(t)t2d−1. Due to the more general marginal distributions of Xt, the Hermite polynomials have to be replaced by Appell polynomials defined as follows:
Definition 2.3 Let X be a random variable with finite moments up to order M, i.e. E[|X|M] < ∞. Then, the corresponding Appell polynomials Am are defined by A0(X) = 1 and for m= 1, . . . , M recursively by
d
dxAm(x) =mAm−1(x), E[Am(X)] = δ0,m.
In the next section, we will give a more detailed description of the polynomials Am. However, one immediately sees that eachAm is of orderm form≥0. Thus, every polynomial Gof order M can be uniquely expanded as
G(Xt) = XM m=0
cmAm(Xt).
Correspondingly to the gaussian case, the lowest m with cm 6= 0 is called Appell rank and is denoted by m∗. Write Sn,G(u) := P[nu]
t=1 G(Xt) and let γG(t) be the covariance function ofG(Xt). Then, the following theorem, due to Giraitis (1985), is the (partial) analogon to theorem 2.1.
Theorem 2.3 Let (2.5) and (2.6) hold and denotem∗ the Appell rank of G. For m∗ ≥2, the conditions
X∞ t=0
|γG(t)|<∞ and X∞
t=0
|γ(t)|m∗ <∞ (2.7) are equivalent. Moreover, if one of them holds and σ2 :=P∞
t=0γG(t)>0, then n−1/2Sn,G(u)→D σB(u). (2.8)
CHAPTER 2. LONG MEMORY AND LIMIT THEOREMS 16 Hence, if m∗ > (1−12d), (2.7) is fulfilled and the central limit theorem holds. The next theorem considers the case m∗ < (1−12d). The result was primarily proven by Surgailis (1982), whereas the connection to Appell polynomials was carried out in Avram and Taqqu (1987), see also Surgailis (2003).
Theorem 2.4 For1≤m <1/(1−2d), letAmdenote them-th Appell polynomial corresponding to the linear process (2.5) and define
Sn,Am(u) :=
X[nu]
t=1
Am(Xt).
Then, Bn,m2 :=E[Sn,A2 m(1)]∼bmΛm(n)n2−m(1−2d) with a constant bm >0 and Bn,m−1 Sn,Am(u)→ HD m(u),
as n tends to infinity.
Thus, given a polynomial function G with Appell rank m∗ < (1−2d)−1, one gets var(Sn,Am(u)) = o(Bn,m2 ∗) for m > m∗ and the leading term in the Appell expansion of G(Xt) is cm∗Am∗(Xt). Therefore, the preceding non-central limit theorem also holds for G(Xt).
Finally, note that we only considered polynomials G until now since the Appell expansion of more general functions is rather complicated, see the next section.
However, one should mention that Giraitis (1985) also proved that an expansion G(Xt) =
X∞ m=m∗
cmAm(Xt)
still holds for entire functionsGsatisfying some growth condition (see theorem2.7 below) by which theorem 2.3 can be proven forG(Xt) ifm∗ >(1−2d)−1. Thus, if m∗ < (1−2d)−1, theorem 2.4 can also be applied to G(Xt) by decomposing G=G1+G2, where G1 is a polynomial with Appell rankm(1)∗ <(1−2d)−1 and G2 is an entire function with rank m(2)∗ >(1−2d)−1. Then, theorem 2.4 can be applied to G1(Xt), while the variance of Sn,G2(u) is of order O(n), and hence of smaller order than var(Sn,G1(u)), leading to asymptotic negligibility of Sn,G2(u).