Limit theorems for partial sums

dB(y1) Z y1

−∞

dB(y2)· · ·

Z ym−1

−∞

dB(ym) Z t

0 m

Y

i=0

(s−yi)^H0−32ds+

Z t 0

dB(y₁) Z y1

−∞

dB(y₂)· · ·

Z ym−1

−∞

dB(y_m) Z t

y1

m

Y

i=0

(s−y_i)^H0−32ds )

is well-defined in the sense of Ito [Ito51]. Here, c_H,m denotes a normalising coefficient ensuring that Var (H^m_H(1)) = 1. Recall that by symmetry of the multiple Wiener integral

H^m_H(t) = ˜cH,m

Z

R^m

Z t 0

m

Y

j=1

(s−yj)⁻(¹2+^1−H_m )

+

!

dsdB(y1). . . dB(ym),

see property (I.2) in [Ito51]. The process {H^m_H(t), t∈R+}, usually referred to asHermite process of order m, is H-sssi with mean zero and finite variance, see [Taq77, Taq78]. In particular, its second order structure is given by (1.17). Note, that H^m_H(·) is in fact a fractional Brownian motion if m= 1 and non-Gaussian if m >1.

1.3 Limit theorems for partial sums

1.3.1. The sole assumption that {ξ(t), t∈Z} is second order stationarity is usually not strong enough to draw any conclusions about the asymptotic distribution of the bro-ken line process Zn,ξ(·). Therefore, we need to impose more restrictive assumptions on {ξ(t), t∈Z}.

We discuss linear process with independent innovations to start with. More precisely, we assume that

ξ(t) =

∞

X

j=0

bjε(t−j)

with (b_j)j∈Z ∈ l²(Z) and {ε(t), t ∈Z} denoting an iid sequence such that E(ε_t) = 0 and Var (ε_t)<∞.

Note that a linear process is always strictly stationary. By [Han70, theorem 3, p. 204], a linear process is mixing, in particular ergodic. Since {ξ(t), t∈Z} has a finite mean, it follows from the ergodic theorem of Birkhoff that the strong law of large numbers holds, see e.g. [Han70, theorem 2, p. 203].

Theorem 1.3.2. Letξ(t) =P∞

j=0b_jε(t−j) be a linear process as in 1.3.1. Assume that Var [Pn

t=1ξ(t)]∼n^2HL²(n) asn → ∞, with 0< H <1 andLis slowly varying at infinity.

Then

Z_n,ξ(t)−→^d B_H(t) (1.19)

where −→^d denotes convergence in the sense of the finite dimensional distributions.

Proof. The original result is due to Davydov [Dav70]. For additional comments, see [Taq75, Taq03].

Remark 1.3.3. By [Taq75, lemma 2.1], the sequence{Z_n,ξ(·), n= 1,2,3, . . .} is tight in D([0,1]) for ¹₂ < H < 1. In particular, we have Z_n,ξ(t) −→^D B_H(t) where −→^D denotes weak convergence in the sense of Skorokhod, see e.g [Bil68]. If E(|ε(t)|^2r) < ∞ for some r ≥ 2 and _r+2¹ < H < 1, then it follows from the arguments of Davydov, that Zn,ξ(·), n= 1,2,3, . . . is tight in D([0,1]), cf. [Taq75, page 228].

As an immediate consequence of theorem 1.3.2, we obtain the following corollary:

Corollary 1.3.4. Let{ξ(t), t∈Z} be as in theorem 1.3.2.

(i) If {ξ(t), t∈Z} is a linear short-memory process, then Z_n,ξ(t)−→^d B(t).

(ii) If {ξ(t), t∈Z} is either a long-memory or antipersistent process, then Z_n,ξ(t) −→^d B_H(t) withH = ^2−α₂ and α as in definition 1.1.5.

Theorem 1.3.5. Let ξ(t) = P∞

j=0b_jε(t−j) as in theorem (1.3.2). Suppose that for all γ >0 exists δ <1 such that

lim sup

n

δn≤i≤nmax D²(i)/D²(n)<1 +γ, (1.20) E(|ε(t)|^r)<∞ for some r >2, and (1.21)

lim inf

n D²(Kn)/D²(n)> K^2/r for some integer K ≥2. (1.22) Then

lim sup

n

|Z_n,ξ(1)|

p2 log log (D²(n)) ≤1 almost surely. (1.23) Proof. This result is due to Lai and Wei [LW82, theorem 3].

Remark 1.3.6. Assume D²(n) = n^2−αL(n) for some α ∈ (0,2) and a slowly varying L, then

lim sup

n

δn≤i≤nmax D²(i)

D²(n) = lim sup

n

δn≤i≤nmax

i^2−αL(i)

n^2−αL(n) ≤lim sup

n

δn≤i≤nmax L(i) L(n) →1, by (1.4). In particular, lim sup_nmaxδn≤i≤n D²(i)

D²(n) < 1 +γ for all γ > 0, in other words condition (1.20) always holds if {ξ(t), t∈Z} has short memory, long memory or is an-tipersistent.

Likewise, under the same assumptions lim inf

n

D²(Kn)

D²(n) = lim inf

n K^2−αL(Kn)

L(n) =K^2−α.

Thus, condition (1.22) holds if r(2−α)>2. Sincer >2 by assumption (1.21) and (1.22) does not impose any additional restriction if{ξ(t), t∈Z}is either a long- or short-memory process. One should note that this is not the case for α ∈ (1,2), i.e. antipersistence. In fact, since r → ∞ for α → 2, the conditions on the moments imposed by (1.21) can be rather strict.

Remark 1.3.7. One should also note that we can replace p

2 log log(D²(n)) in (1.23) by the more familiar term p

2 log log(n). Since log log n^β

log log(n) = log [βlog (n)]

log log(n) = log (β) + log log (n) log log(n) →1 for all β∈(0,2), simply choose β₁, β₂ such that ^D_n²β⁽ⁿ⁾1 → ∞and ^D_n²β⁽ⁿ⁾2 →0.

1.3.8. In a more general setup, we now assume that {ξ(t), t∈Z} is not necessarily lin-ear itself but only subordinated, see [Maj81, page 3], to another linear process. More specifically, we assume that ξ(t) is given by

ξ(t) = G(η(t)) (1.24)

whereη(t) denotes a linear process yet to be specified andG:R→Ra sufficiently regular function.

1.3.9. We start with the case of a stationary Gaussian process η(t). More precisely, let {η(t), t∈Z}be a standardised¹, stationary Gaussian process such that

k→∞lim Cov [η(t), η(t+k)] = 0 and denote by G a measurable function satisfying

E[G(ε(t))] = 0, E

G²(ε(t))

<∞.

Under these assumptions, ξ(t) as given in (1.24) defines a strictly stationary process with mean zero and finite variance. It is ergodic by [Sto74, theorem 3.5.8].

Denoting

G =L²

R,^√1_2πexp

−^x₂²

, (1.25)

we immediately find that G ∈ G. It is well-known, see e.g. [Roz67] or [Abr72], that the Hermite polynomials

H_k(x) = (−1)^kexp

x² 2

d^k

dx^k exp

−^x₂²

(k = 0,1,2, . . .) (1.26) form a complete orthogonal system in G. Thus

G(·) =

∞

X

k=0

C_k k!H_k(·)

withCk=hHk, Gi_G and convergence understood with respect to the scalar producth·,·i_G. Note that kH_kk²_G =k! and thus P∞

k=0 C_k²

k! <∞. It follows that ξ(t) =

∞

X

k=0

C_k

k!H_k η(t)

where convergence is understood in the sense of the L²(P)-norm.

As in [Taq75], we define theHermite rank of Gby m_G ={k ∈N, C_k 6= 0}. Note that m_g >0, since E[G(η_t)] = 0. The covariance structure ofξ(t) is given by

Cov (ξ(t), ξ(s)) =

∞

X

k=mg

C_k²

k!γ_η^k(t−s), compare e.g. [Roz67, page 183]. Consequently, P

n∈Z|γ_ξ(n)| <∞ if P

n∈Z|γ_η(n)| < ∞.

On the other hand, if η(t) is a long-memory process, we find γ_ξ(n)∼ C_m²_g

mg!γ_η^mg(n). (1.27)

1Throughout this section we refer to a finite variance process as standardised, if E[η(t)] = 0 and Var [η(t)] = 1.

Hence, ξ(t) has long memory with parameter αmg, if αmg <1 and P

k∈Z|γξ(k)| <∞ if αm_g >1.

Finally, note that antipersistence of η(t) does in general not entail antipersistence of ξ(t): if C_m = 0 for all uneven m then γ_ξ(k)≥0 and thusP

k∈Zγ_ξ(k)>0.

Theorem 1.3.10. Let{η(t), t∈Z}andGbe as in 1.3.9. Defineξ(t) =G(η(t)). Assume that {ξ(t), t∈Z} is a short-memory process in the sense of definition 1.1.5. Then

Z_n,ξ(·)−→^d B(·).

Proof. This particular result is due to Giraitis and Surgailis [GS85, theorem 5]. Compare also [BM83].

Remark 1.3.11. Note that the tightness criteria given by Taqqu [Taq75] and Davy-dov [Dav70] require the existence of higher moments than 2, if applied to short-memory processes.

Theorem 1.3.12. Let {η(t), t ∈Z} be a stationary, standardised Gaussian process and let G be as in 1.3.9. Define ξ(t) =G(η(t)). Assumeη(t) is a long-memory process such that αm_G <1. Then

Z_n,ξ(·)−→^D C_mG

m_G!H^m_H^G(·)

where H^m_H^G denotes a Hermite process of order m_G with self-similarity parameter H =

2−αm_G

2 .

Proof. This pioneering result is due to Taqqu [Taq75] [Taq79] and Dobrushin and Major [DM79].

Remark 1.3.13. For the case γ_ξ(k)∼ ^L(|k|)_k , see [GS85, theorem 6].

Theorem 1.3.14. Let {ε(t), t∈Z} be a standardised Gaussian iid sequence and let (b_j)_j∈

N as in 1.1.8 and such thatP

b²_j = 1. Define η(t) =

∞

X

j=0

b_jε(t−j) (t∈Z).

Then,η(t) is long-range dependent withα = 1−2d. LetGas in 1.3.9 such thatαm_G<1 and define

ξ(t) =G(η(t)) (t∈Z).

Furthermore, define the polygonal line process Z˜n,ξ(·) by

Z˜_n,ξ(s) =











0 s = 0

1 D(n)

Pnt

t=1ξ(t) s= _n^t, t= 1, . . . , n Z_n,ξ ^t−1_n

+ s−^t−1_{n t}

n

_t−1

n < s < _n^t, t= 1, . . . , n .

Then, with probability one the sequence

(2 log log (n))^−mG2 Z˜_n,ξ(·)

n∈N

is relatively compact in C([0,1]). The set of limit points is given by ^C_m^mG

G!K where K denotes a compact subset of C([0,1]) consisting functions which have the representation

f(t) = Z t

0

Z v

−∞

(v−u)^−αg(u) du m

dv for some g ∈L²(R) such that R

g²(u) du= 1.

Proof. For a proof of this particular result, see [MO87, theorem 5]. For a similar result, cf.

[Taq77]. Lai and Stout [LS80] obtained the upper half of the law of the iterated logarithm in a more general setting.

Remark 1.3.15. In the case m= 1, K can be interpreted as the unit ball of the repro-ducing kernel Hilbert space of a fractional Brownian motion, see e.g. [Taq77, PT00b].

1.3.16. We saw in (1.27) and theorem 1.3.12 that asymptotic behaviour of γ_ξ and the limit distribution of Z_n,ξ(·) essentially depend on G only via the Hermite rank m_G if the underlying Gaussian process has long memory. This so-called reduction principle still applies if we replace the underlying Gaussian process by a general linear one. For such processes, the role of the Hermite expansion is played by an expansion in Appell polynomials which we will describe in the following. The pioneering result in this field was obtained by Surgailis [Sur82, theorem 2], who derived an analogous result to theorem 1.3.12 for a certain class of linear processes and an entire function G(x) = P∞

k=0c_kx^k satisfying some regularity assumptions. It is worthwhile noting that Surgailis established his result without the use of Appell polynomials. The link between these two topics was discovered later, see [Gir85, GS86, AT87].

This topic is technically more involved, so we restrict ourselves to the very basic definitions and results. For a detailed introduction, we refer to [Sch09, section 2.2 and 2.3]. See also [Sur03b, Sur03a].

Definition 1.3.17. Let X be a random variable such that E X^k

<∞. As in [AT87], we define the Appell polynomials recursively by

A₀(x) = 1, d

dxA_n(x) = nAn−1(x), E[A_n(X)] = δ_0,n

for n = 1, . . . , k.

Remark 1.3.18. Equivalently, the Appell polynomials may be defined in terms of their generation function, i.e. by the relation

∞

X

n=0

zⁿ

n!A_n(x) = exp(zx)

E(exp(zX)). (1.28)

see e.g. [Gir85, GS86, AT87]. For detailed arguments, compare also [Sch09] and references therein. If X is a standardised Gaussian variable, the generating function of the Appell polynomials and the generating function of the Hermite polynomials coincide, see [Abr72].

Hence, the Appell polynomials of a standardised Gaussian distribution and the Hermite polynomials are the same.

If the moment generating function does not exist as an analytical object, the generating function can be defined as a formal power series, see e.g. [AT87, Sch09].

Likewise, Appell polynomials can be defined in terms of the characteristic function of X. In this case, we have the relation

A_k(x) = (−i)^k d^k d^kz

exp(izx) E(exp(izX))

z=0

,

see [Sur03b]. Note that _d^dk^kz

_exp(izx)

E(exp(izX))

exists ifE |X|^k

<∞.

Definition 1.3.19. Let X be a random variable and let G be a function which admits an expansion in Appell polynomials, i.e.

G(x) =

∞

X

k=0

c_kA_k(x)

where Ak(x) denotes the k-th Appell polynomial associated with X or rather the distri-bution of X. We define the Appell rank of G by

m_G = min{k= 0, . . . , n;c_k 6= 0}.

Remark 1.3.20. Note that Surgailis [Sur82] [Sur00] [Sur03a] defines the Appell rank by mG = min

k∈N E G^(k)(X) 6= 0

where G^(k)(x) = _d^dk^kxG(x). Both definitions coincide if G is a polynomial. If G is not a polynomial but admits an expansion G(x) = P∞

k=0ckAk(x), both definitions are still equivalent under some regularity assumptions, see [Gir85, proposition 2].

Theorem 1.3.21. Let {ε(t), t∈Z} be a standardised sequence of iid random variables.

Assume thatE ξ^k(t)

<∞for allk ∈N. Let (b_j)_j∈

Zbe real sequence such thatP

j∈Zb²_j = 1 and let G be a polynomial such that E[G(η(t))] = 0 and E[G²(η(t))]<∞. Define

η(t) = X

j∈Z

b_jε(t−j), ξ(t) =G(η(t)). Assume that P

k∈Z|γξ(k)|<∞ and P

k∈Zγξ(k)>0. Then, Zn,ξ(·)−→^d B(·).

Proof. This result can be found in [Sur03a, theorem 1.1]. The original result is due to Giraitis [Gir85, proposition 6 and 8]

Theorem 1.3.22. Let {ε(t), t∈Z} be a standardised sequence of iid random variables.

Assume that ∆ =E[exp (r₀|ξ(t)|)]<∞for somer₀ >0. Let (b_j)_j∈Zbe real sequence such that P

j∈Zb²_j = 1. Let G be a function such that E[G(η(t))] = 0 and E[G²(η(t))]<∞.

Define

η(t) = X

j∈Z

b_jε(t−j), ξ(t) =G(η(t)). Suppose that G(x) = P∞

n=mGc_kA_k(x) where the A_k denote the k-th Appell associated with η(t) and m_G the corresponding Appell rank of G. Furthermore, suppose that the coefficients c_k satisfy P∞

k=mG|c_k|k!d^k <∞ for some d >3(1 + 2∆)/r₀. (i) Assume thatmG>1. Then, the conditions

X

k∈Z

|γ_ξ(k)|<∞ and X

k∈Z

γ_η^mG(k) <∞ are equivalent.

(ii) Assume that P

k∈Z|γξ(k)|<∞ and P

k∈Zγξ(k)>0. Then, Z_n,ξ(·)−→^d B(·).

Proof. The first statement follows from [Gir85, proposition 8 and lemma 5]. The second statement is identical to [Gir85, theorem 2].

Remark 1.3.23. Giraitis also proves a central limit theorem for the case γξ(k)∼ ^L(|k|)_k , see [Gir85, theorem 3].

Theorem 1.3.24. Let {ε(t), t∈Z} be a sequence of iid random variables such that E[ε] = 0 andE[ε^2m(t)]<∞for m∈N\0. Letb_j ∼L(j)j^d−1,j ∈Nas in 1.1.8 such that α = (1−2d)< _m¹ and define

η(t) =

∞

X

j=0

b_jε(j)

Denote byA_m(x) them-th Appell polynomial associated with the distribution ofη(t) and define

ξ(t) =A_m(η(t)). Then,

D²_ξ(n)∼c_mn^2−mαL^2m(n) for some constant c_m >0 and

Z_n,ξ(·)−→ H^D _H^mG(·) with H = 1−^αm₂ .

Proof. This result is taken from [AT87, theorem 2]

1.3.25. Clearly, theorem 1.3.21 and 1.3.22 can be regarded as the counterpart to theorem 1.3.10 whereas theorem 1.3.24 generalises theorem 1.3.12 for polynomials. In order to obtain theorem 1.3.24 for a general function G, let {η(t), t∈Z} be as in theorem 1.3.24 and assume thatGadmits an expansion as in theorem 1.3.22. Suppose there existsm^∗ ∈N such that αm∗<1 and α(m^∗ + 1)>1. Then, we have

G(x) =

m^∗

X

k=mG

c_kA_k(x) +

∞

X

k=m^∗+1

c_kA_k(x) = G₁(x) +G₂(x).

Theorem 1.3.24 implies

Z_n,G1(η)(·)−→ H^{D m}_H^G(·).

Moreover, by by theorem 1.3.22, Var [Pn

t=1G₂(η(t))] =O(n). Thus, we deduce Z_n,G(η)(·)−→ H^{d m}_H^G(·).

If αm∗< 1 and α(m^∗ + 1) = 1, we may apply [GS85, theorem 3] instead. Note also the result of Surgailis [Sur82, theorem 2].

2 Fractional calculus

2.1 Fractional integrals on a compact interval

Im Dokument Piecewise polynomial regression with fractional residuals for the analysis of calcium imaging data (Seite 31-40)