SPECTRAL MEASURES OF PARMA SEQUENCES By

Transkrypt

doi:10.1111/j.1467-9892.2007.00541.x
SPECTRAL MEASURES OF PARMA SEQUENCES
By Agnieszka WyŁomańska
Wrocław University of Technology
First Version received February 2006
Abstract. The aim of this article is to give a complete description of the spectral
measure of periodic autoregressive moving-average (PARMA) system in terms of its
coefficients. In the analysis we use the spectral theory of strongly harmonizable sequences
presented in Hurd [Journal of Multivariate Analysis (1989) Vol. 29, pp. 53–67] and the
form of the unique bounded solution of ARMA model with periodic coefficients. As an
application of the theoretical results, we present some examples of the spectral measures
for PARMA models.
Keywords. PARMA models; periodically correlated process; spectral measure;
strongly harmonizable sequence.
1.
INTRODUCTION
Let (x(n)), n 2 Z, where Z is the group of integers, be a sequence in a Hilbert
space H ¼ L2(X, F, P), and let Rx(n, m) ¼ (x(n), x(m)) be its correlation
function. The dual group of Z is identified here with T ¼ [0, 2p) with addition
modulo 2p.
Definition 1 (Hurd, 1989). The sequence (x(n)), n 2 Z is called strongly
harmonizable if there is a measure F on T2 (called spectral measure of (x(n))) such
that
Rx ðn; mÞ ¼
Z
2p
Z
0
2p
eiðnsmtÞ F ðds; dtÞ:
0
Definition 2 (Dehay, 1994). A sequence (x(n)), n 2 Z is called periodically
correlated (PC) with period T, if for every n, k 2 Z the following hold
mx ðnÞ ¼ EðxðnÞÞ ¼ mðn þ T Þ and
Rx ðn þ k; kÞ ¼ Rx ðn þ k þ T ; k þ T Þ:
In the sequel we will be assuming that mx(n) 0.
Any PC sequence is strongly harmonizable and its spectrum is supported on the
sets
0143-9782/08/01 1–13
JOURNAL OF TIME SERIES ANALYSIS Vol. 29, No. 1
2007 The Author
Journal compilation 2007 Blackwell Publishing Ltd., 9600 Garsington Road, Oxford OX4 2DQ, UK
and 350 Main Street, Malden, MA 02148, USA.
2
A. WYŁOMAŃSKA
Ln ¼
ðs; tÞ 2 T2 : t ¼ s þ
2np
;
T
where n ¼ 0; 1; . . . ; T 1:
The spectral measure F of a PC sequence with period T has the following form
(see Gladyshev, 1961; Dehay and Hurd, 1994; Makagon, 2001)
F ðds; dtÞ ¼
T 1
X
Cn ðds; dtÞ;
n¼0
where Cn are supported on the sets Ln, n ¼ 0, 1, 2, . . . , T 1.
In this article, we study the spectral measures of certain harmonizable
sequences which admit a moving-average representation of the form
xðnÞ ¼
1
X
ak ðnÞnnk ;
ð1Þ
k¼1
where (nn) is an orthonormal basis in Mx ¼ spfx(n) : n 2 Zg. Of course, each
sequence can be written in this form in infinite ways, depending on the choice of
(nn). However, very often this representation is somewhat unique, for example
when x(n) is required to be a bounded solution to ARMA systems with varying
coefficients (Makagon et al., 2004; Weron and Wyłomańska, 2004). In the latter
case, (nn) is its forward or backward innovation sequence and the representation
(1) is one-sided. This article is therefore motivated by the needs to describe the
spectra of harmonizable solutions to such systems.
We extend the results obtained in Hurd et al. (2002) for AR(1) models with
periodic coefficients. Moreover, we use the form of the unique bounded solution
of PARMA sequences (Makagon et al., 2004; Weron & Wyłomańska, 2004) to
calculate the formulas of the spectral measures for such systems.
The article is organized as follows. In Section 2 the conditions of existence of
the unique, bounded, periodically correlated solution of general PARMA model
are obtained. The complete description of periodically correlated PARMA
spectral measure expressed in terms of the coefficients is formulated in Section 3
and the main results are obtained in Theorems 2 and 3. Some examples of the
spectral measures of PARMA models are shown in Section 4. In the Appendix we
present the proofs of Theorems 1, 2 and 3.
2.
Definition 3 (Sakai 1991).
linear equations
xðnÞ p
X
k¼1
PARMA(p,q) MODELS
A PARMA(p,q) system with period T is a system of
bk ðnÞxðn kÞ ¼
q1
X
aj ðnÞnnj ;
j¼0
2007 The Author
Journal compilation 2007 Blackwell Publishing Ltd.
n 2 Z;
ð2Þ
3
where (bk(n)), k ¼ 1, . . . , p and (aj(n)), j ¼ 0, . . . , q 1, are periodic (with period
T) sequences of complex numbers, bp(n) 6¼ 0 for all n 2 Z and (nn) is a sequence of
uncorrelated complex random variables with mean zero and unit variances.
Throughout the article, the L2-closed linear space generated by a sequence of
random variables nn, n 2 Z, is denoted by Mn. The space Mn with norm kÆk is a
Hilbert space. Any sequence (x(n)) in Mn that satisfies eqn (2) is called a solution.
The solution (x(n)) is bounded if supn2Zkx(n)k2 < 1. The special case of system
(2) was considered in Makagon et al. (2004) where a complete description of the
unique bounded PC solutions to PARMA(1, q) models is obtained.
Let us denote
2
b1 ðnÞ
6 1
6
0
Bn ¼ 6
6 .
4 ..
0
b2 ðnÞ
0
1
..
.
...
...
...
..
.
0
...
3
bp1 ðnÞ bp ðnÞ
0
0 7
7
0
0 7;
.. 7
..
. 5
.
1
0
Bnk ¼ Bn Bn1 Bk ;
n k:
Let
Yn ¼ ½Yn ; 0; . . . ; 00
and
xðnÞ ¼ ½ xðnÞ; xðn 1Þ; . . . ; xðn p þ 1Þ0 ;
where
Yn ¼
q1
X
ap ðnÞnnp
p¼0
and [. . .]0 denotes the column vector. Then (see Boshnakov (1997))
xðnÞ ¼ Bn xðn 1Þ þ Yn :
Consequently
xðnÞ ¼ Bnnkþ1 xðn kÞ þ
k1
X
Bnnþ1s Yns ;
k > 0;
s¼0
where Bnj ¼ I if j>n. Since all matrices Bnj are invertible
1
xðnÞ ¼ ðBnþk
nþ1 Þ xðn þ kÞ k
X
1
ðBnþs
nþ1 Þ Ynþs ;
k > 0:
s¼1
Therefore we obtain
xðnÞ ¼
p
X
j¼1
Uj ðn k þ 1; nÞxðn k j þ 1Þ þ
k1
X
U1 ðn s þ 1; nÞYns ;
ð3Þ
s¼0
2007 The Author
4
A. WYŁOMAŃSKA
where
Uj ðl; kÞ ¼ Bkl ð1; jÞ
for l < k. On the other hand we have:
xðnÞ ¼
p
X
Wj ðn þ 1; n þ kÞxðn þ k j þ 1Þ k
X
W1 ðn þ 1; n þ sÞYnþs ;
ð4Þ
s¼1
j¼1
where
Wj ðl; kÞ ¼ ðBkl Þ1 ð1; jÞ
for l < k.
Theorem 1 shows that the bounded solution of the PARMA system (2) can be
written in causal (forward) or backward form under conditions I and II,
respectively. The conditions are defined as follows:
Condition I.
2
1 q1þs
X
X
W1 ðn þ 1; n þ jÞajs ðn þ jÞ < 1
sup
n2Z s¼2qj¼maxð1;sÞ
and
inf
k1
p
X
jWj ðn þ 1; n þ kÞj2 ¼ 0;
n 2 Z:
j¼1
Condition II.
2
1 min ð0;q1sÞ
X
X
sup
U1 ðn j þ 1; nÞasþj ðn þ jÞ < 1
n2Z s¼0
j¼s
and
inf
k1
p
X
jUj ðn k þ 1; nÞj2 ¼ 0;
n 2 Z:
j¼1
Theorem 1. If the sequence (x(n)) is the unique bounded PC solution of system
(2) then it has the following form:
xðnÞ ¼
8
1
P
>
>
>
<
>
>
>
:
q1þs
P
W1 ðn þ 1; n þ jÞajs ðn þ jÞnsþn ; if I holds,
s¼2q j¼maxð1;sÞ
q1sÞ
1
P minð0;P
U1 ðn þ j þ 1; nÞasþj ðn þ jÞnns ;
s¼0
j¼s
2007 The Author
ð5Þ
if II holds.
5
3.
SPECTRA OF PERIODICALLY CORRELATED SOLUTIONS OF PARMA(p,q) MODELS
Because of eqn (5) we can suppose MX ¼ Mn. Let us denote V as a forward shift
operator V(nn) ¼ nnþ1. The operator V is unitary and the sequence x(n) ¼ VnQn,
where
8
q1þs
1
P
P
>
>
W1 ðn þ 1; n þ jÞajs ðn þ jÞns if I holds,
>
<
s¼2q j¼maxf1;sg
Hn ¼
1 minf0;q1sg
>
P
P
>
>
U1 ðn þ j þ 1; nÞajþs ðn þ jÞns
if II holds.
:
s¼0
j¼s
The operator V is an isometry, therefore the correlation function of x(n) and
x(n þ k) is given by:
Rx ðn; n þ kÞ ¼ ðHn ; V k Hnþk Þ:
When Condition I holds, we obtain
1
X
Rx ðn; n þ kÞ ¼
q1þs
X
q1þsk
X
s¼2q j¼maxf1;sg l¼maxf1; skg
W1 ðn þ 1; n þ jÞajs ðn þ jÞW1 ðn þ k þ 1; n þ k þ lÞalsþk ðn þ k þ lÞ:
On the other hand, when condition II is fulfilled, we have
Rx ðn; n þ kÞ ¼
minf0;q1kþsg
1 minf0;q1sg
X
X
X
s¼0
j¼s
l¼kþs
U1 ðn þ j þ 1; nÞajþs ðn þ jÞU1 ðn þ l þ k þ 1; n þ kÞalþks ðn þ k þ lÞ:
Let D be an unitary mapping D : nn ! einÆ. D maps MX into L2([0, 2p), dt),
where dt is the normalized Lebesgue measure on [0, 2p), that is the Lebesgue
measure divided by 2p. If we denote fn ¼ D(x(n)), then system (3) takes the
following form
fn ðÞ ¼
p
X
bk ðnÞfnk ðÞ þ
k¼1
q1
X
aj ðnÞeiðnjÞ ;
n 2 Z:
j¼0
Thus we have
8
q1þs
1
P
P
>
in
>
W1 ðn þ 1; n þ jÞajs ðn þ jÞeis
>
< e
s¼2q j¼maxf1;sg
DðxðnÞÞðÞ ¼
q1sg
1 minf0;P
>
P
>
>
U1 ðn þ j þ 1; nÞajþs ðn þ jÞeis
: ein
s¼0
if I holds,
if II holds.
j¼s
If we denote
2007 The Author
6
A. WYŁOMAŃSKA
gn ðkÞ ¼
8
1
P
>
>
>
<
>
>
>
:
q1þs
P
W1 ðn þ 1; n þ jÞajs ðn þ jÞeisk
s¼2q j¼maxf1;sg
1 minf0;q1sg
P
P
if I holds,
ð6Þ
U1 ðn þ j þ 1; nÞajþs ðn þ jÞe
isk
if II holds,
j¼s
s¼0
we can see that x(n) is unitary equivalent to
fn ðkÞ ¼ eink gn ðkÞ
in
L2 ð½0; 2pÞ; dtÞ:
Lemma 1. The correlation function of PC sequence with period T has the
following form
Z 2p
T 1
X
ik 2pl
T
Bn ðkÞ ¼ Rx ðn þ k; kÞ ¼
e
eins cl ðdsÞ;
0
l¼0
where
2pl
:
cl ðdsÞ ¼ Cl ds; ds þ
T
Proof.
Bn ðkÞ ¼
Z
2p
2p
e
0
¼
Z
Z
0
2p
Z
iðnþkÞsikt
F ðds; dtÞ ¼
0
2p
0
2p
eikðstÞ eins
0
Z
T 1
X
Cl ðds; dtÞ ¼
l¼0
Z
2p
0
T 1
X
l¼0
eikðstÞ eins F ðds; dtÞ
2pl
eik T
Z
2p
eink cl ðdkÞ:
u
0
Corollary 1. The measures (ck) (k ¼ 0,1, . . . , T 1) given in Lemma 1 satisfy
the following condition
Z 2p
T 1
1X
2ikpl
eink cl ðdkÞ ¼
e T Bn ðkÞ:
ð7Þ
T
0
k¼0
The measures (ck), k ¼ 0,1, . . . , T 1 are called spectrum of the PC sequence
(x(n)).
Theorem 2. Let (bk(n)), k ¼ 1, 2, . . . , p and (aj(n)), j ¼ 0, 1, . . . , q 1 be
periodic (in n) sequences with period T and condition I or II holds. Let (ck), k ¼
0, 1, . . . , T 1 be spectrum of the unique solution of PARMA(p,q) system defined
in eqn (2). Then the measures (ck) are absolutely continuous with respect to the
normalized Lebesgue measure dk on [0, 2p) and
2007 The Author
7
T 1 dck ðkÞ X
2pl
2pl
^
^glk k þ
¼
;
gl k þ
dk
T
T
l¼0
where
^
gj ðkÞ ¼
T 1
2pinj
1X
gn ðkÞe T
T n¼0
and gn(k) is given in eqn (6).
Now we consider a PARMA(1,q) system given by:
xðnÞ bn xðn 1Þ ¼
q1
X
ap ðnÞnnp ;
n 2 Z;
ð8Þ
p¼0
where the coefficients and innovations have the same properties like in the general
model. If the sequence (x(n)) is the unique bounded PC solution of system (3) then
it has the following form (Theorem 2.2 in Makagon et al., 2004):
8
q1þs
1
P
P
>
1
>
ðBnþj
>
nþ1 Þ ajs ðn þ jÞnnþs ; if jP j > 1
<
s¼2q j¼maxf1;sg
xðnÞ ¼
1 minf0;q1sg
>
P
P
>
>
ðBnnþjþ1 Þajþs ðn þ jÞnns ;
if jP j < 1,
:
s¼0
j¼s
where P ¼ b1b2 bT and Blk ¼ bk bkþ1 bl (with the convention Blk ¼ 1 for
k > l). PARMA(1,q) models satisfy Theorem 3:
Theorem 3. Let fn(k) be unitary equivalent to the unique bounded solution (x(n))
of PARMA(1,q) sequence through the mapping D [that means fn(k) ¼ D(x(n))(k)],
then the following formula holds:
fn ðkÞ ¼ eink ½1 P eiT k 1 Gn ðkÞ þ eink Rn ðkÞ;
where
Gn ðkÞ ¼
and
Rn ðkÞ ¼
8 T þq2 q1
P P n
>
>
Bnkþjþ1 aj ðn þ j kÞeikk
>
<
k¼q1 j¼0
1 q1
P n
> TP
>
>
Bnkþjþ1 aj ðn
:
k¼0 j¼0
if jP j < 1,
þ j kÞeikk
if jP j > 1,
8 q2
s
PP
>
>
Bnnsþjþ1 aj ðn þ j sÞeiks
>
<
if jP j < 1,
>
>
:
if jP j > 1.
s¼0 j¼0
P q1þs
P
> q2
s¼0 j¼1
1
iks
ðBnþj
nþ1 Þ ajþs ðn þ jÞe
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A. WYŁOMAŃSKA
4.
Example 1.
EXAMPLES
Let us consider AR(1) system given by the equation:
xðnÞ bxðn 1Þ ¼ nn ;
for |b| < 1. It is a PAR(1) system with period T ¼ 1. Using Corollary 1 and
formula (7) we obtain the following:
Z 2p
Bn ð0Þ ¼
eink c0 ðdkÞ:
0
Because the measure c0 is absolutely continuous with respect to the normalized
Lebesque measure on [0, 2p), then from Theorem 2 we have:
Z 2p
Z 2p
Bn ð0Þ ¼
eink ^
eink g0 ðkÞg0 ðkÞdk:
g0 ðkÞ^
g0 ðkÞdk ¼
0
0
Using Theorem 3 we can see that
R0 ðkÞ 0;
f0 ðkÞ ¼ ð1 beik Þ1 :
G0 ðkÞ 1;
From the relation between f0(k) and g0(k) we conclude that g0(k) ¼
(1 beik)1. Thus we have:
Z 2p
Bn ð0Þ ¼
eink j1 beik j2 dk:
0
If we take n ¼ 1 we can write
Z 2p
Z
ik
ik 2
e j1 be j dk ¼ 2
B1 ð0Þ ¼
0
p
0
cosðkÞ
pð1 þ b2 Þ p
dk
¼
:
1 þ b2 2b cosðkÞ
bj1 b2 j b
If an estimate of B1(0) is available, then the above equation can be solved to get
an estimate of the parameter b. In this case, the spectral measure
c0 ðkÞ ¼
1
j1 beik j2
:
.
Example 2.
Let us consider PAR(1) system with period T ¼ 2 given by:
xðnÞ bn xðn 1Þ ¼ nn :
Let suppose |P| ¼ |b0b1| < 1. Using results presented in Theorem 3 we obtain
for n ¼ 0,1:
Rn ðkÞ 0;
Gn ðkÞ ¼ 1 þ bn eik ;
fn ðkÞ ¼ eink ½1 P e2ik 1 Gn ðkÞ;
which gives gn(k) ¼ (1þbneik)/(1Pe2ik).
2007 The Author
9
To obtain the spectral measures of the sequence (x(n)) we use the relation
between c0 and c1 (defined in Lemma 1) and the normalized Lebesgue measure on
[0, 2p) presented in Theorem 2:
c0 ðkÞ ¼
c1 ðkÞ ¼
¼
jb0 b1 j2 þ j2 þ b0 eik þ b1 eik j2
4j1 b0 b1
e2ik j2
¼
2 þ 2ðb0 þ b1 Þ cosðkÞ þ b20 þ b21
;
2 4b0 b1 cosð2kÞ þ 2b20 b21
eik ðb1 b0 Þð2 þ b0 eik þ b1 eik Þ þ eik ðb1 b0 Þð2 b0 eik b1 eik Þ
4j1 b0 b1 e2ik j2
cosðkÞðb0 b1 Þ
:
1 2b0 b1 cosð2kÞ þ b20 b21
To obtain the coefficients b0 and b1 we use formula (7) from Corollary 1 and
substitute theoretical correlations with estimated ones. For the sample of length
NT from PARMA models with period T the estimators of Bn(k) have the
following form
T 1
X
^ n ðkÞ ¼ 1
B
xðjT þ k þ nÞxðjT þ kÞ:
N j¼0
We then obtain
2pðb0 þ b1 Þb0 b1
1 ðb0 b1 Þ
2
^ 1 ð0Þ þ B
^ 1 ð1Þ;
¼B
2pðb0 b1 Þb0 b1
1 ðb0 b1 Þ2
^ 1 ð0Þ B
^ 1 ð1Þ:
¼B
We conclude that
b0 ¼
.
^ 1 ð0Þ
B
b
^ 1 ð1Þ 1
B
The coefficient b1 can be obtained from the following equation:
0
!
^
B1 ð0Þ
2p
þ 1 b31 ¼ @1 ^ 1 ð1Þ
B
^ 1 ð0Þ
B
^ 1 ð1Þ
B
!2 1
^ 1 ð0Þ þ B
^ 1 ð1ÞÞ:
b4 A ð B
1
APPENDIX: PROOFS
Proof of Theorem 1. Suppose that I holds and (x(n)) is bounded solution of eqn (2) then
for every n 2 Z there is a sequence kr such that
lim
r
p
X
jWj ðn þ 1; n þ kr Þj2 ¼ 0:
j¼1
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A. WYŁOMAŃSKA
Hence, from eqn (4), we conclude that for every n
xðnÞ ¼ lim
r
kr
X
W1 ðn þ 1; n þ sÞYnþs :
s¼1
From formula (9) in Lemma 2.1 in Makagon et al. (2004) we have that
kr
X
W1 ðn þ 1; n þ jÞðYnþj ; nk Þ ¼
j¼1
minðkr ;q1þknÞ
X
W1 ðn þ 1; n þ jÞanþjk ðn þ jÞ:
j¼maxð1;knÞ
Letting r ! 1, we obtain that
ðxðnÞ; nk Þ ¼ lim
r
kr
X
W1 ðn þ 1; n þ jÞðYnþj ; nk Þ
j¼1
q1þkn
X
¼
W1 ðn þ 1; n þ jÞanþjk ðn þ jÞ;
j¼maxð1; knÞ
if k > 1 q þ n, and (x(n),nk) ¼ 0 if k 1 q þ n. Therefore from formula (6) in
Lemma 2.1 in Makagon et al. (2004) we obtain
xðnÞ ¼ 1
X
2
4
q1þkn
X
3
W1 ðn þ 1; n þ jÞanþjk ðn þ jÞ5nk ;
k¼2þnq j¼maxð1;knÞ
which after substituting s ¼ k n gives the thesis. This also shows that the solution is
unique. The variance of x(n) is given by
2
1 q1þs
X
X
:
W
ðn
þ
1;
n
þ
jÞa
ðn
þ
jÞ
kxðnÞk2 ¼
1
js
s¼2qj¼maxð1;sÞ
Suppose now Condition II holds. If
inf
r
p
X
jUj ðn r þ 1; nÞj2 ¼ 0;
j¼1
then for every n there is a sequence kr ! 1, such that
lim
r
p
X
jUj ðn kr þ 1; nÞj2 ¼ 0:
j¼1
If the sequence (x(n)) is a bounded solution of eqn (2), then from eqn (3) it follows that
xðnÞ ¼ lim
r
0
X
U1 ðn þ j þ 1; nÞYnþj :
j¼1kr
Using formula (6) Lemma 2.1 in Makagon et al. (2004) we conclude that
2007 The Author
xðnÞ ¼
"
1 minð0;q1sÞ
X
X
11
#
U1 ðn þ j þ 1; n þ 1Þasþj ðn þ jÞ nns :
j¼s
s¼0
Moreover
2
1 minð0;q1sÞ
X
X
U1 ðn j þ 1; nÞasþj ðn þ jÞ :
kxðnÞk ¼
j¼s
s¼0
2
u
Proof of Theorem 2.
T 1
T 1
T 1
T 1
X
X
1X
1X
^l ðkÞe2iplðnþpÞ=T
^j ðkÞe2ipjn=T
g
g
ei2pnk=T gnþp ðkÞgn ðkÞ ¼
ei2pnk=T
T n¼0
T n¼0
l¼0
j¼0
¼
¼
T 1 X
T 1
T 1
X
1X
e2pilp=T ^
e2ipn=T ðkþjlÞ
gl ðkÞ^
gj ðkÞ
T j¼0 l¼0
n¼0
T 1
X
e2pilp=T ^gl ðkÞ
l¼0
¼
T 1
X
e2pilp=T
l¼0
¼
T 1
X
T 1 X
T 1
1X
^
gj ðkÞe2ipn=T ðjðlkÞÞ
T j¼0 n¼0
T 1
1X
gl ðkÞ
gn ðkÞe2ipn=T ðlkÞ ^
T n¼0
e2pilp=T ^glk ðkÞ^
gl ðkÞ:
l¼0
If we use the equivalence between x(n) and fn(Æ) ¼ einÆgn(Æ) we obtain
Z
2p
eipk ck ðdkÞ ¼
0
¼
¼
T 1
1X
e2ipnk=T Bp ðnÞ
T n¼0
T 1
1X
e2ipnk=T
T n¼0
Z
2p
eipk
0
¼
¼
Proof of Theorem 3.
eiðnþpÞkink gnþp ðkÞgn ðkÞdk
0
e2pilp=T ^
gl ðkÞdk
glk ðkÞ^
2p
eipðk2pl=T Þ ^
gl ðkÞ^
glk ðkÞdk
0
T 1 Z
X
l¼0
2p
l¼0
T 1 Z
X
l¼0
T 1
X
Z
2p
eipk ^gl ðk þ 2pl=T Þ^
glk ðk þ 2pl=T Þdk:
u
0
If |P| < 1, then we have
2007 The Author
12
A. WYŁOMAŃSKA
gn ðkÞ ¼
1 minf0;q1sg
X
X
s¼0
¼
Bnnþjþ1 ajþs ðn þ jÞeiks
j¼s
q2 X
s
X
Bnnþjsþ1 aj ðn þ j sÞeiks þ
q2 X
s
X
Bnnþjsþ1 aj ðn þ j sÞeiks þ
s¼0 j¼0
¼
q2 X
s
X
Bnnþjþ1 ajþs ðn þ jÞeiks
s¼q1 j¼s
s¼0 j¼0
¼
1 q1s
X
X
q1
1 X
X
Bnnþjsþ1 aj ðn þ j sÞeiks
s¼q1 j¼0
Bnnþjsþ1 aj ðn þ j sÞeiks
s¼0 j¼0
þ
þq2 X
q1
1 TX
X
BnnþjNT kþ1 aj ðn þ j NT kÞeikðNT þkÞ
N ¼0 k¼q1 j¼0
¼ ½1 P eik 1 Gn ðkÞ þ
q2 X
s
X
Bnnþjsþ1 aj ðn þ j sÞei ks:
s¼0 j¼0
For |P| > 1 we obtain
gn ðkÞ ¼ 1
X
q1þs
X
nþj 1
ðBnþ1
Þ ajs ðn þ jÞeiks
s¼2q j¼maxf1;sg
¼
q2 q1s
X
X
s¼0
¼
nþj 1
ðBnþ1
Þ ajþs ðn þ jÞeiks j¼1
1 q1þs
X
X
s¼1
nþj 1
ðBnþ1
Þ ajs ðn þ jÞeiks
j¼s
q1
1 X
T X
X
nþjþNT þk 1
ðBnþ1
Þ aj ðn þ j þ NT þ kÞeikðNT þkÞ
N ¼0 k¼1 j¼0
q2 q1s
X nþj 1
X
ðBnþ1 Þ ajþs ðn þ jÞeiks
s¼0
¼
1
X
j¼1
P N eikTN
N ¼0
q1
T X
X
1
ikk
ðBnþjþk
nþ1 Þ aj ðn þ j þ kÞe
k¼1 j¼0
q2 q1s
X nþj 1
X
s¼0
j¼1
¼ ½1 P eikT 1
q1
T X
X
ðBnþjþkT
Þ1 aj ðn þ j þ kÞeikðkT Þ
nþ1
k¼1 j¼0
q2 q1s
X nþj 1
X
s¼0
j¼1
¼ ½1 P eikT 1
q1
T 1 X
X
1
ikk
ðBnþjþk
nþ1 Þ aj ðn þ j kÞe
k¼0 j¼0
2007 The Author
q2 q1s
X
X
s¼0
13
nþj 1
ðBnþ1
Þ ajþs ðn þ jÞeiks
j¼1
¼ ½1 P eikT 1 Gn ðkÞ q2 q1s
X
X
s¼0
1
iks
ðBnþj
:
nþ1 Þ ajþs ðn þ jÞe
j¼1
ink
Because fn(k) ¼ e gn(k), then we obtain the thesis.
u
NOTE
Corresponding author: Agnieszka Wyłomańska, Institute of Mathematics and
Computer Science, Wrocław University of Technology, Janiszewskiego 14a, 50370 Wrocław, Poland. E-mail: [email protected]
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