Next, we will summarize different moment assumptions of the two papers Gi-raitis et al. (2000b) and (2003c). Note that, under (A5.2), orthogonality in (5.3) directly implies finiteness of the second moment E[σt2] respectively E[Xt2]. How-ever, for higher order moments, the complex structure of the Volterra-expansion (5.3) makes the evaluation of E[σtp] for p≥3 much more involved and combina-torial approaches comparable to section 4.1.4 are necessary. Since the analysis in the next chapter is based on the subsequent results, we will describe one of these conditions in detail. Further, we will argue that finiteness of moments of random variables related to σt is implied by the conditions as well. Throughout this section, the sequence Xt = εtσt is assumed to be a LARCH process with coefficients (bj)j=0,1,... satisfying assumptions (A5.1) and (A5.2). For an integer p≥2, define the following notations:
µp =E[εpt], |µ|p =E[|εt|p] and
kbkp = X∞
j=1
|bj|p
!1/p
.
Assumption (M′p)
The first condition, due to Giraitis et al. (2000b), can be formulated as follows:
(M′p) p∈N, p≥2, |µ|p <∞ and (2p−p−1)1/2|µ|1/pp kbk2 <1.
To explain the preceding condition we will show how finiteness of thep-th moment ofσtcan be deduced from (M′p). The proof is adopted from Giraitis et al. (2000b, lemma 3.1) and will be modified for our purposes later.
We start with some notations: Let Pk,t
S denote the sum over all subsets S = {sk, sk−1, . . . , s1} ⊂Z with sk< sk−1 <· · ·s1 < s0 =t, k = 0,1, . . .. For such an S we define
bS :=bt−s1bs1−s2· · ·bsk−1−sk
and
εS :=εskεsk−1· · ·εs1.
CHAPTER 5. LARCH 84 For S =∅, set bS =εS = 1.
The stationary solution of σt can then be written as σt = b0 +b0 Moreover, the p-th power of σt is given by
σtp = bp0
The essential point is that almost all terms in the sum in (5.4) are zero due to the fact that µ(S)p = 0, if there is a Sj ∈ (S)p containing a s ∈ Sj that is not
For this reason, Giraitis et al. (2000b) introduce the following diagram formalism which results in a representation of E[σtp] where most of the vanishing terms in (5.4) are omitted. Therefore, note that a necessary condition for a non-zero term in (5.4) is, that all εs in the product ε(S)p appear at least twice. These products of identical εs will be collected in the so-called edges of the following diagrams:
For a given collection (k)p, let I := I((k)p) be the table consisting of p rows Ij := Ij((k)p) = {(kj, j), . . . ,(1, j)} of length kj ≥ 0, j = 1, . . . , p. Then, an ordered partition γ = (V1, . . . , Vr) of the table I by nonempty subsets Vq, q = 1, . . . , r,r= 1,2, . . ., containing at most one element of any row|Vq∩Ij| ≤1, q = 1, . . . , r, j = 1, . . . , p, is called a diagram. Vq are called edges.
CHAPTER 5. LARCH 85 One can then simplify the sum in (5.4) as follows: Denote the elements of Sj ∈ (S)p by si,j (i = 1, . . . , kj, j = 1, . . . , p) and notice that for each non-zero summand in (5.4)
06=b(S)pE[ε(S)p] = bt−s1,1 · · ·bsk1−1,1−sk1,1· · ·bt−s1,p· · ·bskp−1,p−skp,p (5.6)
×E[εs1,1· · ·εsk
1,1· · ·εs1,p· · ·εskp,p],
there must be for every (i, j) an index (i′, j′) withsi,j =si′,j′. Thus each summand (5.6) can be uniquely characterized either by
(i) a collection (k)p = (k1, . . . , kp),
(ii) the set (S)p of subsetsSj = (si,j, i= 1, . . . , kj) satisfying skj,j < skj−1,j <· · ·< s1,j < t, j = 1, . . . , p
and where each si,j ∈Sj appears in at least one different Sj′,j′ 6=j. or by
(iii) a collection (k)p = (k1, . . . , kp),
(iv) a diagram γ = (V1, . . . , Vr) over I((k)p), where all edges consist of at least two elements.
(v) numbers ˜s1 < · · · < s˜r < t corresponding to the edges Vq via ˜sq = si,j for (i, j)∈Vq, q= 1, . . . , q.
Indeed, assuming (i) and (ii), the diagram needed in (iv) is given by the partition of I((k)p) that pools elements (i, j) and (i′, j′) in an edge Vq iff si,j and si′,j′ coin-cide (note that due to (5.5) these edges always consist of at least two elements).
The other way around, assuming (iii)-(v), the set (S)p in (ii) is determined by
|Sj| =kj, j = 1. . . , p and si,j = ˜sq for all (i, j) ∈ Vq, q = 1, . . . , r. Further, the expected value
E[εs1,1· · ·εsk
1,1· · ·εs1,p· · ·εskp,p]
only depends on γ and will thus be denoted by µγ. Let ΓI((k)p) denote all dia-grams where all edges contain at least two elements. This leads to the following alternative representation of (5.4):
E[σtp] = bp0X
(k)p
X
γ∈ΓI((k)p)
µγ
X
γ,tb(S)p, (5.7)
CHAPTER 5. LARCH 86 where the sum P
γ,t is taken over all integers ˜sq, q = 1, . . . , r satisfying (v) and for which the resulting numbers of the set (S)p, which is determined as described above, fulfill (ii). One can then proof the following lemma:
Lemma 5.1 Let (A5.1), (A5.2) and (M′p) hold. Then the p-th unconditional moment of σt is finite and given by
E[σtp] =bp0X
(k)p
X(k)p,t
(S)p
b(S)pµ(S)p.
Consequently, E[Xtp] =µpE[σtp].
Proof: One has to prove that (5.7) converges which is implied by the following estimates:
(a) |µγ| ≤ |µ|(p|S1|+···|Sp|)/p, (b) P
γ,t|b(S)p| ≤ kbk|2S1|+···|Sp|,
(c) |ΓI((k)p)| ≤(2p−p−1)(k1+···+kp)/2.
H¨older’s inequality immediately leads to (a). For (c), recall that, according to (v), the edges of a diagram are ordered, and any edge can be chosen inPp
i=2 p i
= 2p−p−1 ways. Moreover, the number r of edges of a diagram can not be grater than (k1+· · ·+kp)/2 implying (c).
To prove (b), one proceeds iteratively by first showing for the left-most edge V1
X
γ,t|b(S)p| ≤ kbk|2V1|
X
γ′,t|b(S′)p|, (5.8) where γ′ := (V2, . . . , Vr), (S′)p := (S1 \V1, . . . , Sp \V1), (k′)p = (k′1, . . . , kp′), k′j :=|Sj\V1| and γ′ ∈Γ(I((k′)p)), and then repeating (5.8) resulting in
X
γ,t|b(S)p| ≤ kbk|2S1|+···+|Sp|. (5.9) Thus the proof of (b) is finished by showing (5.8). Therefore, assume that V1
has length |V1| =m ≥2 and that V1 connects the first m rowsIj, j = 1, . . . , m.
Then V1 corresponds to the numbers sk1,1 = · · · = skm,m =: ˜s and one can split P
γ,t|b(S)p| into X
γ,t|b(S)p|=X
γ′,t|b(S′)p| ·X
˜ s
Ym j=1
|bskj−1,j−˜s|.
CHAPTER 5. LARCH 87 The later sum is bounded by
X
Now, (5.7) can be dominated by bp0X
The preceding proof can directly by generalized to more general sums similar toσt, which will be useful in section 6.2. There, a process is analyzed which is formally given by (5.2), however the coefficients need not be the true coefficients defining the LARCH process Xt. Hence, for arbitrary square summable coefficients β :=
(βj)j=0,1,..., i.e. P∞
Note, that the sum of the squared βj’s need not be less then one. Also recall, that Xt=εtσt is a LARCH process corresponding to coefficients (bj)j=0,1,... satisfying (A5.1) and (A5.2). We can then prove the following proposition concerning higher moments of σt(β):
Proposition 5.2 Let (A5.1), (A5.2) and (M′p) hold. Then thep-th unconditional moment of σt(β) is finite and further
(E[|σt(β)|p]))1/p≤ |β0|+|β0| b
p−1 p
0 kbk−21
1− |µ|1/pp kbk2(2p−p−1)1/2kβk2.
Proof: By Minkowski’s inequality, we can neglectβ0. Analogously toσt, represen-tation (5.7) still holds forσt(β) where for each set S={sk, . . . , s1}the coefficient bt−s1 has to be replaced byβt−s1. This means that in each row Ij respectively Sj
exactly one coefficient comes from β and the others arebj’s. Thus (5.6) changes
CHAPTER 5. LARCH 88 to
βt−s1,1bs1,1−s2,1· · ·bsk1−1,1−sk1,1· · ·βt−s1,pbs1,p−s2,p· · ·bskp−1,p−skp,p
×E[εs1,1· · ·εsk1,1· · ·εs1,p· · ·εskp,p]
and the bound (b) in the proof of lemma 5.1 to kbk|2S1|−1+···+|Sp|−1kβkp2. For the latter note that (5.11) is replaced by
bp0−1kβkp2kbk−2p
X
(k)p6=(0,...,0)
|µ|1/pp kbk2(2p−p−1)1/2(k1+···+kp)
.
Also notice that the collection (k)p = (0, . . . ,0) is excluded, since the lowest coef-ficientβ0 is considered separately. However, for a more convenient representation, the sum is taken over all collections, since then we get geometric sums leading to the bound for E[|σt(β)|p] as claimed in the proposition.
Assumptions (M3) and (M′′p)
Though one could base the analysis of LARCH processes only on assumption (M′p) for suitable p≥ 2, we consider further moment assumptions introduced in Giraitis et al. (2003c). These assumptions seem to be weaker than (M′p) and thus allow for a greater variety of coefficients. Since the problem of finiteness of higher moments is more difficult for odd powers, see Giraitis et al. (2003c), only additional conditions for the third moment and even moments of arbitrary order have been considered yet. The conditions are given as follows:
(M3) |µ|3 < ∞ and |µ|1/33 kbk3 + 3ζkbk2 < 1, where ζ ≈ 1.27 is the positive solution of the equation 3ζ2−3ζ−1 = 0.
(M′′p) p∈N,p≥4 and peven. |µ|p <∞,Pp j=2
p j
kbkjj|µj|<1.
Giraitis et al. (2003c) use a diagram formalism related to the preceding subsection to prove the following lemma:
Lemma 5.2 Let (A5.1) and (A5.2) hold.
(a) Under (M3), E[|σt|3]<∞ and E[|Xt|3]<∞. (b) Under (M′′p),E[σtp]<∞ and E[Xtp]<∞.
CHAPTER 5. LARCH 89
Proof: See Giraitis et al. (2003c).
By adaption of the results in Giraitis et al. (2003c) (in particular their proposi-tions 2.1, 2.2 and the corresponding lemmas B1-B3), it is also possible to gener-alize the preceding lemma for moments of general sums σt(β), which will be used in lemma 6.2 in section 6.2.
Comparison of the moment assumptions
All assumptions discussed in the present section are sufficient but not necessary for finiteness of the corresponding moments, since crude bounds are used in the proofs. There are a few simplifications leading to weaker or more convenient conditions:
• First consider (M′p) for p = 2. Then (M′p) can obviously be replaced by µ2b2 <1, since only the second order properties ofσthave to be considered.
• Also (M′p) for p = 4 and gaussian (or symmetric) εt can be simplified to 7µ412b2 <1. This can be seen by
|γ ∈ΓI((k)4) :µγ = 0| ≤ 4
2
+ 4
4
(k1+···+k4)/2
= 7(k1+···+k4)/2, since E[εt] = E[ε3t] = 0 and µγ disappears if γ is a diagram containing an edge of length 1 or 3.
• For even p ≥ 4, (M′′p) is weaker than (M′p): Note that kbkj ≤ kbk2 and
|µj|1/j ≤ |µp|1/p for 2 ≤ j ≤ p. Also (M′p) implies kbk22µ2/pp < 1. Thus kbkjj|µj| ≤(kbk22µ2/pp )j/2 ≤ kbk22µ2/pp and hence
Xp j=2
p j
kbkjj|µj| ≤ kbk22µ2/pp Xp
j=2
p j
= (2p−p−1)kbk22µ2/pp .
• For standard gaussian εt, it holds |µ|3 = (8/π)12 and for even p ≥ 2: µp = (p−1)(p−3)· · ·1. Thus one can calculate the bounds implied by (M′3) and (M′4):
kbk2 <(23−3−1)−1/2|µ|−31/3 ≈0.4279 and
kbk2 <(24−4−1)−1/2µ−41/4 ≈0.2291.
CHAPTER 5. LARCH 90 (Note, that the results also illustrate the fact that (M′p) implies (M′q) for q ≤ p. The latter holds since the functions (2p −p−1)1/2 and µ1/pp are increasing.)
• Again, assume εt to be standard gaussian. Using kbkj ≤ kbk2 for j ≥ 2, a simplified version of (M3) can be derived by
|µ|1/33 kbk3+ 3ζkbk2 ≤ kbk2(|µ|1/33 + 3ζ) (5.13) leading to
kbk2 <((8/π)1/6 + 3ζ)−1 ≈0.2008.
Analogously, one gets a simplified version for (M′′4):
kbk2 <√
0.1547≈0.3933.
(Note that the very low bound 0.2008 corresponding to (M3) is partly due to the estimate (5.13). Hence, in general it is not clear whether (M′3) or (M3) is weaker.)