SHIFT OPERATOR FOR PERIODICALLY
CORRELATED PROCESSES
A.G. Miamee
1,*and G.H. Shahkar
21 Department of Mathematics, Hampton University, Hampton, VA, 23668, USA
2 Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran
Abstract
The existence of shift for periodically correlated processes and its boundedness
are investigated. Spectral criteria for these non-stationary processes to have such
shifts are obtained.
Keywords: Periodically correlated; Shift; Stationary; Linearly stationary; Bounded linearly stationary
AMS (1991) Subject Classification: 60G10, 60G12, 60G25
* E-mail: [email protected]
1. Introduction
Prediction theory of stationary stochastic processes has been extensively developed and is now considered to be complete. The existence of bounded shift for stationary processes has played a major role in this development. The existence and boundedness of shift for non-stationary processes is important [1]. An interesting class of non-stationary stochastic processes is that of periodically correlated processes. This class of processes has been studied by several authors [2-14]. However, questions concerning their shift have not yet been considered. In this note, we study these questions and obtain spectral criteria for the existence of bounded shift for PC processes.
) PC (
2. Preliminaries
Let (Ω,β,Ρ) be a probability space and 2( , , )
0 Ωβ Ρ
L
denote the space of all complex-valued random variables on Ω with zero mean and finite variance. The
inner product and norm here are given by
∫
Ω Ρ=
= ( ) ( ) ( ) ( ) )
,
(X Y E XY X ωY ω d ω
and X = (X,X).
Any sequence
{
Xn,n∈Z}
of random variables in) , , (
2 0 Ω β Ρ
L will be called a stochastic process and its correlation function R(m,n) is defined by
) ( ) ,
(mn E XmXn
R = .
Given a stochastic process , its shift operator V is a linear transformation which sends to , for each
n
X
n
X Xn+1
Z
to impose certain restrictions on . Before we proceed further, let’s now state the formal definition of shift operator and consider an example where this operator is not well-defined.
n
X
Definition 2.1. (a) A stochastic process is said to have a shift if the linear transformation V on
which sends each to is well defined. (b) A process is said to have a bounded shift if it has a shift which can be extended to
n
X
{
X n Z SΡX
L( )= n: ∈
}
Xn Xn+1 nX
{
X n Z}
SP X
H( )= n: ∈
as a bounded operator.
Example. Let be any nonzero stochastic process and define a new stochastic process by
n
Y
n
X
⎪⎩ ⎪ ⎨ ⎧
+ =
= =
1 2 0
2
k n if
k n if Y
Xn k
The shift operator for is clearly ill-defined because it sends a zero vector, say to a nonzero vector, say . If we take the original stochastic process to be a nondeterministic, then we get an example of a stochastic process which has no shift.
n
X
1
X
2
X
n
Y
n
X
Definition 2.2. A stochastic process is called stationary if
n
X
) , ( ) ,
(m n =R m+1n+1
R
For all m,n∈Z.
It is well-known that any stationary stochastic pro-cess has a bounded shift and that it is a unitary operator. However, for a non-stationary process as we saw above the shift may not even exist and in order for a stochastic process to have a shift we must impose some restrictions on the process or its correlation function
. For the following lemma one can see [1].
n
X
n
X
) , (mn
R
Lemma 2.3. Let be a stochastic process with correlation function as defined above. Then
n
X
) , (mn
R
(a) In order for to have a shift it is necessary and
sufficient that for any finite sequence
{
of complex numbersn
X
}
n
a
0 ) 1 , 1 ( 0
) ,
( = ⇒
∑
+ + =∑
amanR mn amanR m n .(b) In order for to have a bounded shift it is necessary and sufficient to have a positive number
n
X
M
such that for any finite sequence
{
of complex numbers}
n
a
∑
∑
amanR(m+1,n+1)≤M amanR(m,n).We close this section with a brief introduction to periodically correlated processes.
Definition 2.4. A stochastic process is called periodically correlated with period p if for all
n
X
Z n m, ∈ , we have
) , ( ) ,
(mn R m p n p
R = + + .
Such a process will be briefly called a process. Let be a process with period . Then for each integer
PC
n
X PC p
τ , the function R(n,n+τ) is periodic in with period . Therefore it has Fourier expansion
n p
∑
− == +
1
0
) 2 exp( ) ( )
, (
p
k k
p ikn R
n n
R τ τ π ,
where Rk(τ) are given by
∑
− =− +
= 1
0
) 2 exp( ) , ( 1 ) (
p
n
k p R nn in
R τ τ
π
τ
.For convenience, we extend the definition of these
1 1
, 0 , )
( k= , ,p−
Rk τ K to all integers by k Rk(τ)
) (τ
p k R +
= .
It is shown in [3] that each Rk(τ) has a spectral representation of the form
∫
−= π θ
τ τθ
π
2
0 ( )
2 1 )
( i k
k e dF
where each dFk is a complex-valued measure on
) 2 , 0
[ π . One can then see that
∫ ∫
− −= π π θ λ θ λ
π
2
0 2
0
) (
2 ( , )
4 1 ) ,
(m n e dF
R im n
where the spectral measure dF of Xn is given by
∑
− − =− =
1
1
)) 2 ( ( )
, (
p
p k
k p
k B A F B
A
F I π .
Here stands for the set of all numbers of the form with . This shows that spectral measure of any process is concentrated on
line segments
a B−
a
b− b∈B dF PC 1
2p − θ − λ = 2πk/ p, k =
, contained in the square 1
, ,
1−pK p− [0,2π)×[0,2π), and
∫
− −= −
= π θ
π θ
2
0
)
( ( )
2 1 ) ( ) ,
(m n r m n e dF
R im n
3. Shift for PC Processes
In this section we study the questions of existence and boundedness of shift for processes. We first obtain some basic results and then prove our main result which gives several criteria for existence and boundedness of shift including a spectral criterion.
PC
Lemma 3.1. Let Xn be a PC process with period p. (a) If Xn has shift V then V is invertible.
(b) If has bounded shift V then V is boundedly invertible.
n
X
Proof. (a) Suppose has shift V and assume for some finite sequence
n
X
0
=
∑
anXn{ }
an ofcomplex numbers. This means that
0 ) , ( =
∑
amanR m n . Applying Lemma 2.3(a) we get0 ) , ( +1 +1 =
∑
amanR m n . Applying Lemma 2.3(a)more times to the latter equation, we get 2
−
p
0 ) 1 ,
1
( + − + − =
∑
amanR m p n p .Considering that Xn is PC with period p, we get
0 ) 1 , 1 ( − − =
∑
amanR m n ,which means
∑
anXn−1=0. So we showed that forany sequence of complex numbers an
0 0 ⇒ 1=
=
∑
∑
anXn anXn− .But this is clearly equivalent to the existence of the inverse U of V which sends each Xn to Xn−1.
(b) Suppose has a bounded shift V . and let be a finite sequence of complex numbers. Applying Lemma 2.3(b)
n
X an
1
−
p consecutive times we arrive at
, ) , (
) 1 ,
1 (
1
∑
∑
− ≤
− + − +
n m R a a M
p n p m R a a
n m p n m
which in conjunction with the fact that is periodically correlated with period implies
n
X p
∑
∑
a a R(m−1,n−1)≤M −1 a a R(m,n) n m p nm .
Therefore
2 1
2
1
∑
∑
−− ≤ p n n n
nX M a X
a ,
which implies that backward shift U sending each to has a bounded extension to . Since
clearly , we conclude that V is boundedly invertible.
n
X
1
−
n
X H(X)
1 −
=V U
The following remarks follows from the proof of Lemma 3.1.
Remarks 3.2. Let be any process with period .
n
X PC
p
(a) If has a shift (bounded shift) V , then it has shifts (bounded shifts) , of any order , sending
each to . In fact, it is clear that .
n
X
k
V k∈Z
n
(b) The shift Vp always exists and it is unitary.
(c) If has a bounded shift V , then for any positive integer ,
n
X k
k
k V
V ≤ and V−k ≤V k(p−1).
Before we proceed further, we need to introduce some terminologies. Let be Hilbert
spaces. Their direct sum , equipped
with the Euclidean inner product
q
H H
H1, 2,K, q
q
H H
H1⊕ 2⊕K⊕
∑
==
q
i
i i Y X 1
) , ( )) , ((XY
becomes a Hilbert space. Here X=X1⊕X2⊕K⊕Xq
and , with . Direct
sum of q copies of H will be denoted by .
q
Y Y
Y ⊕ ⊕ ⊕
= 1 2 K
Y Xi,Yi∈Hi
q
H
Lemma 3.3. A stochastic process Xn in H =
is periodically correlated with period if
and only ifits associated process in defined by
) , , (
2
0 Ω β Ρ
L p
n
Z Hq
1
1⊕ ⊕ + −
⊕
=Xn Xn+ Xn p
n K
Z
is stationary.
Proof. “only if” part: If is a process with period , then for any and n in
n
X PC
p m Z, we can write
∑
∑
∑
∑
=
+ +
−
=
+ + +
+
−
=
+ + −
=
+ +
=
+ =
+ =
=
p
i
i n i m
p
i
i n i m p
n p m
p
i
i n i m n
m p
i
i n i m n
m
X X
X X X
X
X X X
X
X X
1
1
1 1
1 1
0
) , (
) , ( ) , (
) , ( ) , (
) , ( )) , ((Z Z
. )) , ((
) , (
1 1 1
0
1 1
+ + −
=
+ + + +
= =
∑
n m p
i
i n i
m X
X
Z Z
This means that is stationary. Proof of “if” part is similar.
m
Z
Lemma 3.4. If is a process with period then the spectral measure of its associated stationary process introduced in last lemma is , where is the part of the spectral measure of supported on the main diagonal of the square mentioned in section 2.
n
X PC p
n
Z pdFo dFo dF Xn
Proof. For the proof, we refer the reader to [7].
Definition 3.5. A stochastic process in the Hilbert
space is called linearly stationary if there exists a stationary process in another Hilbert space
n
X
) , , (
2
0 Ω Ρ
=
Η L β
n
W
κ and an invertible transformation T:κ→H
such that Xn =TWn, for all . A linearly stationary process is called bounded linearly stationary if the transformation can be chosen to be bounded.
Z n∈
n
X
T
It is clear that linearly stationary processes are in general non-stationary. Nevertheless prediction properties of linearly stationary processes can easily be investigated. Because one can transfer a prediction problem concerning a linearly stationary process to one about its stationary counterpart , we find the solution for this stationary process and then transfer the result back to the original process For more detail, one can see [15].
n
X
n
W
n
W
n
X
In what follows, we will use the following notations and terminologies.
Let be a process with period p in
and in , be its associated stationary process introduced in Lemma 3.3, namely:
n
X PC
) , , (
2
0 Ω Ρ
=L β
H Zn H p
1
1⊕ ⊕ + −
⊕
= Xn Xn+ Xn p
n K
Z
Now let denote the orthogonal projection which maps any vector in to its first coordinate, i.e.
p p P:H →H
p
1 2
1 )
(X X X X
P ⊕ ⊕K⊕ p = ,
for any X1,X2,K,Xp∈H . We denote by
κ
the subspace of spanned by all and Q to stand for the restriction ofp
H Zn's
P to κ.
Theorem 3.6. For any process with period , the following statements are equivalent.
PC Xn p
(a) Xn has a bounded shift V . (b) Xn is bounded linearly stationary.
(c) The operator Q:κ →H defined above is boundedly invertible.
Proof. We prove (a) ⇒ (b) ⇒ (c).
(c)
(a) ⇒ : Take a finite linear combination . We can write:
∑
anXn. )
1
( 2 2( 1) 2
2 1
2 2
2 1
2 1 2
2
∑
∑
∑
∑
∑
∑
∑
∑
− −
− +
+
+ + + ≤
+ +
+ =
+ +
+ =
n n p
n n p
n n n
n
p n n
n n n
n n
n
X a V
V
X a V
X a V X a
X a
X a X
a a
K K
K
Z
which shows the inverse of Q exists and is bounded.
) (b
(c) ⇒ : By Lemma 3.3, where is the stationary process associated to . Since each is clearly in
n
n Ρ
X = Z Zn
n
X Zn
κ and Q is the restriction of Ρ to
κ
then we get for all and this completes the proof.n
n Q
X = Z n∈Z
) (a
(b) ⇒ : Suppose there is a stationary process and boundedly invertible operator
such that
n
W T:H(W) )
(X
H
→
) ( n
n T W
X = for all n∈Z .
Let be the well-known unitary shift of the stationary process and define by
U
n
W V:H(X)→H(X)
1
−
=TUT
V .
One can check that V is the shift of our process . Now since T and U are bounded is bounded.
n
X
1
−
=TUT V
Theorem 3.7. Let be a process with period . The following statements are equivalent.
n
X PC p
(a) Xn has a shift.
(b) Xn is linearly stationary.
(c) The operator Q:κ → H defined above is invertible.
Proof. : Suppose has a shift, say V and suppose a finite linear combination of is zero, i.e.
(c)
(a) ⇒ Xn
n
X
0
=
∑
anXn . Applying V to both sides of thisequation p−1 times, we get
. 0 ,
,
0 1
1=
∑
=∑
anXn+ K anXn+p−Thus
0
1
0
2 2
= =
∑ ∑
∑
−=
+
p
i
i n n n
n a X
a Z
which means
∑
anZn =0. Hence Q is invertible.(b)
(c) ⇒ : By Lemma 3.3, where is the stationary process associated to . Since each is clearly in
n
n Ρ
X = Z Zn
n
X Zn
κ and Q is the restriction of Ρ to κ then we get Xn =QZn for all and this completes the proof.
Z n∈
) (a
(b) ⇒ : Let be the stationary process and be the linear transformation with
n
W
) ( ) (
:LW L X
T →
n
n TW
X = and be the standard unitary shift operator of the stationary process , then
the linear transformation clearly serves the desirable shift for .
) ( ) (
:H W H W
U →
n
W
1
−
=TUT V n
X
Next theorem gives our spectral characterization for a PC process to have a shift.
Theorem 3.8. Let be a process with period whose spectral measure is concentrated on
n
X PC p
) (⋅
line segments θ−λ=2πk/p, k=1−p,K,0,K,p−1
of
[
0,2π) (
× 2π,0]
with the measure on the diagonal being dF0.(a) has a bounded shift if and only if there exists a positive number
n
X
K such that
∫ ∫
∫
πφ θ θ ≤ 2π π φθ φ λ θ λ0 2
0 2
0 0
2
) , ( ) ( ) ( )
( )
( dF K dF
for any trigonometric polynomial φ(θ)=
∑
ane−inθ . (b) Xn has a shift if and only if, 0 )
(
0 ) , ( ) ( ) (
2
0 0
2 2
0 2
0
= ⇒
=
∫
∫ ∫
π π π
θ φ
λ θ λ φ θ φ
dF dF
for any trigonometric polynomial function φ(θ)=
.
∑
−inθne
a
Proof.(a) If has a bounded shift then by Theorem 3.6, the operator is boundedly invertible. This means there exists some such that
n
X
Q
0
>
M
∑
∑
anZn ≤M anXnfor every finite sequence of complex numbers. Squaring both sides and rewriting it in terms of the spectral measure we get
n
a
∫ ∫
∫
πφ θ ≤ 2π πφθ φ λ θ λ0 2
0 2 2
0 0
2
) , ( ) ( ) ( )
( dF M dF
p
where . This shows that (1) holds
with . Now assume that inequality (1) holds. We can rewrite it as
∑
−= θ
θ
φ in ne
a
) (
p M K = 2/
2 2
) /
(l p
∑
anZn ≤K∑
anXn ,which means the operator in part (c) of Theorem 3.6 is boundedly invertible. This in virtue of Theorem 3.6 completes the proof of part (a).
Q
Proof of part (b) is similar to the proof of part (a).
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