Near–Integrated
Random Coefficient Autoregressive Time Series
Alexander Aue
1
Abstract: We determine the limiting behavior of near–integrated first–order random
coefficient autoregressive RCA(1) time series. It is shown that the asymptotics of the finite
dimensional distributions crucially depends on how the critical value 1 is approached,
which determines whether the process is near–stationary, has a unit–root or is mildly
explosive.
In a second part, we derive the limit distribution of the serial correlation coefficient in
the near–stationary and the mildly explosive setting under very general conditions on the
parameters. The results obtained are in accordance with those available for first–order
autoregressive time series and can hence serve as addition to existing literature in the
area.
AMS 2000 Subject Classification: Primary 62M10, Secondary 91B84.
Keywords and Phrases: near–integrated time series; random coefficient time series;
serial correlation coefficient; near–stationarity; mildly explosive models; unit–root models
1
Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, UT 84112–
0090, USA, email: aue@math.utah.edu
Partially supported by NATO grant PST.EAP.CLG 980599 and NSF–OTKA grant INT–0223262
1
Introduction
Let {Xt } be a first–order autoregressive AR(1) process, i.e. the solution of the recurrence
equations
Xt = ϕXt−1 + εt ,
−∞ < t < ∞.
(1.1)
In case the noise {εt } is an independent, identically distributed (iid) sequence, it is well–
known that the process is strictly stationary if ϕ < 1 and explosive if ϕ > 1 (cf. for
instance [6]). {Xt } is said to have a unit–root if ϕ = 1. Then, the process resembles the
behavior of a random walk. If ϕ ≈ 1 and ϕ → 1 with increasing sample size, {Xt } is
referred to as near–integrated.
A number of papers has been devoted to examining the asymptotic structure of {Xt }
if the coefficient ϕ is reparameterized to
ϕn = 1 −
β
,
nα
α ∈ [0, 1],
n ≥ 1,
(1.2)
with some fixed real number β, and if model (1.1) is transformed to
Xt (n) = ϕn Xt−1 (n) + εt ,
t = 1, . . . , n,
(1.3)
with initial value X0 (n) = 0 for all n. For instance, Chan and Wei (cf. [7], [8]) studied
the near–stationary AR(1) process (i.e. β > 0) as well as the mildly explosive case (i.e.
β < 0) if α = 1 and the noise {εt } constitutes a sequence of martingale differences.
An extensive analysis for near–integrated time series was carried out by Phillips [16].
His more general multivariate approach, which includes multiple times series in which
some component series have unit–roots, some are stationary, while others are explosive,
is in one dimension asymptotically equivalent to the setting introduced by Chan and
Wei in [7] and [8]. Related work in the above framework was done by Ling and Li [13],
who considered unstable autoregressive moving average ARMA processes with general
autoregressive heteroscedastic GARCH errors. In recent work, Phillips and Magdalinos
[17] established limit theorems in the general parameter case ϕn → 1 for what they call
moderate deviations from a unit–root.
The asymptotic theory obtained in all these papers has consequences for testing procedures for the presence of unit–roots. Usually these tests have a low power because the
behavior of the test statistic under consideration does not crucially differ between the
(null) hypothesis of a unit–root and the alternatives of near–stationary or mildly explosive processes. Unit–root tests play a fundamental role in economometrics, cf. for example
[16] and [14] and the references therein.
2
Motivated by results for GARCH(1,1) sequences obtained by Berkes, Horváth and
Kokoszka [3], we will study the finite dimensional distributions of so–called first–order
random coefficient autoregressive RCA(1) time series, which are defined by the stochastic
recurrence equations in (2.1). Originally, these processes have been applied to model
certain dynamical systems (cf. [15]), but are also used in fincance (see [18]). Surveys
on basic properties of this model, such as stationarity and existence of moments, can be
found, for instance, in [15] and [2].
Our results differ from the above approach in two directions. On one hand, the parameter ϕn in (1.3) is allowed to be random, which is enabled by changing it to the
corresponding quantity ϕn + bt (n) in (2.1) involving an independent sequence {bt (n)} of
random variables having finite second moments. On the other hand, the parameterization (1.2) can be changed. The limit results of Theorems 2.1 (near–stationary case), 2.2
(unit–root) and 2.3 (mildly explosive case) are obtained under the asymptotic rates given
in (2.6). The proofs of these theorems are based on an additive representation of the
RCA(1) process.
A great variety of the afore mentioned unit–root tests are based on the least squares
estimator
!
ϕ̂n =
n
X
t=1
2
(n)
Xt−1
−1 n
X
Xt−1 (n)Xt (n),
t=1
n ≥ 2,
of ϕn . Its limiting behavior has been examined thoroughly. However, most of the existing
literature is devoted to the special case α = 1 in (1.2). See, e.g., Chan and Wei [7],
[8]. For the general convergence
ϕn → 1, it turns out that a Gaussian limit exists under
q
a normalization of order n/|ϕn − 1| if the AR(1) process is near–stationary. On the
other hand, if Xt (n) is in the explosive regime, i.e. ϕn > 1, the limit will no longer be
Gaussian but a Cauchy random variable, while the normalization sequence has now the
exponential order en(ϕn −1) /(ϕn − 1). These results can be found in [17]. We will show
that similar statements hold true also in the random coefficient setting provided in (2.1).
Theorem 2.4 below deals with mild explosion, Theorem 2.5 gives the corresponding result
for near–stationarity.
Ultimately, it is worthwhile mentioning that, since all calculations can be accomplished
also if bt (n) ≡ 0, all results remain valid for AR(1) time series. [This applies primarily
to the finite dimensional distribution results.] So, the paper can serve as complementary
work to the existing results cited above.
The paper is organized as follows. In Section 2, we present the RCA(1) model and state
and discuss the main results. Proofs are given in Section 3.
3
2
Main Results
Consider the sequence of random coefficient autoregressive RCA(1) models
Xt (n) = [ϕn + bt (n)]Xt−1 (n) + εt ,
t = 1, . . . , n,
(2.1)
with initial value X0 (n) = 0 for all n. Therein, the random vectors
{(γt , εt )} are independent, identically distributed with
E(γ1 , ε1 ) = (0, 0) and Eγ12 = 1, Eε21 = σ 2
(2.2)
and, for all n,
bt (n) = ωn2 γt ,
t = 1, . . . , n.
(2.3)
We shall assume that
ϕn → 1 and ωn2 → 0
(n → ∞),
(2.4)
which implies immediately that here ϕ2n + ωn2 → 1, i.e. the parameter sum converges
to the critical value determining whether or not the process allows for (nonanticipative)
second–order stationary solutions. We will find out that the rate of convergence as well
as the limit distribution of the sequence of RCA(1) processes crucially depends on how
the parameter ϕn approaches 1. Hence, we define
ϕ̃n = ϕn − 1,
n ≥ 1,
(2.5)
and differ between the three cases ϕ̃n > 0, ϕ̃n = 0 and ϕ̃n < 0 for all n. Moreover, we
need the following rates
√
nϕ̃n → 0
(n → ∞)
(2.6)
n|ϕ̃n | → ∞ but
and
nωn → ∞ but
√
nωn → 0
(n → ∞).
(2.7)
Since we are actually dealing with a sequence of models, the point of view is the one of
finite dimensional distributions and we study the asymptotic behavior of the vectors
h
i
X⌊nκm ⌋ : m = 1, . . . , N ,
where N is a fixed integer, 0 < κ1 < . . . < κN ≤ 1 and ⌊·⌋ the integer part.
At first, we investigate the case ϕ̃n > 0 for all n. Hence, we study a sequence of
nonstationary models. The asymptotics is given in the following theorem.
4
Theorem 2.1 (mildly explosive case) Let {Xt (n)} be an RCA(1) time series satisfying model (2.1) with ϕ̃n > 0 for all n. If (2.2)–(2.7) hold and if
E|ε1|2+δ < ∞
the random variables
for some δ > 2,
"√
2ϕ̃n
X⌊nκm ⌋ : m = 1, . . . , N
enϕ̃n σ
(2.8)
#
are asymptotically (n → ∞) independent standard normal.
If ϕ̃n = 0 for all positive integers n, a different limiting distribution under a different
normalization is obtained. Since the random part bt (n) of the coefficients dies out for
n → ∞ by assumption (2.4), it is not surprising to see a resemblance of the random walk
here.
Theorem 2.2 (unit–root case) Let {Xt (n)} be an RCA(1) time series satisfying model
(2.1) with ϕ̃n = 0 for all n. If (2.2)–(2.5), (2.7) hold and if
E|ε1|2+δ < ∞
for some δ > 0,
(2.9)
then,
"
#
1
D
√ X⌊nκm ⌋ : m = 1, . . . , N −→ [W (κm ) : m = 1, . . . , N]
nσ
(n → ∞),
where {W (s) : s ∈ [0, 1]} denotes a standard Wiener process.
Finally, we examine the stationary
case,
√
√ i.e. ϕ̃n < 0 for all n. While the order of the
and unit–root setting,
oscillations is exponential enϕ̃n / ϕ̃n and n in the near–explosive
q
respectively, fluctuations increase with the smaller rate |ϕ̃| in case of near–stationary
models.
Theorem 2.3 (near–stationary case) Let {Xt (n)} be an RCA(1) time series satisfying model (2.1) with ϕ̃n < 0 for all n. If (2.2)–(2.7) hold and if
E|ε1|2+δ < ∞
the random variables
q
for some δ > 0,
−1
2|ϕ̃n |σ X⌊nκm ⌋ : m = 1, . . . , N
are asymptotically (n → ∞) independent standard normal.
5
(2.10)
The proofs of all three theorems are based on a common asymptotic additive representation of {Xt (n)}, which is given in Theorem 3.1 below. Using this result as a starting
point, the respective asymptotic distributions are obtained by applying and exploiting
special characteristics of the individual cases to filter out the dominating part of the
decomposition. Details are deferred to the next chapter.
We conclude the paragraph with the remark that Theorems 2.1–2.3 hold true also if
the noise sequence {εt } is allowed to depend on n, i.e. εt = εt (n) = σn2 δt , where {δt }
are independent, identically distributed and centered random variables with Eδ12 = 1 and
σn2 → σ 2 > 0 as n → ∞.
The remainder of the section is devoted to examining the serial correlation coefficient.
From now on, we assume that the initial values X0 (n) = ε0 are constant. Denote by
ϕ̂n =
n
X
t=1
2
Xt−1
(n)
!−1
n
X
Xt−1 (n)Xt (n),
t=1
n ≥ 1,
(2.11)
the (conditional) least squares estimator for ϕn . We say, that a random variable ξ is
standard Cauchy if its density is given by
1
f (x) =
,
−∞ < x < ∞.
π(1 + x2 )
We start with giving the asymptotics for the serial autocorrelation coefficient ϕ̂n − ϕn
if ϕn is larger than 1, that is, approaches 1 from the explosive region. Under a suitable
normalization this limit is standard Cauchy.
Theorem 2.4 (mild explosion) Let {Xt (n)} be an RCA(1) time series satisfying model
(2.1) with ϕ̃n > 0 for all n. If (2.2)–(2.7) hold, then
enϕ̃n
D
(ϕ̂n − ϕn ) −→ ξ
2ϕ̃n
(n → ∞),
where ϕ̂n is defined in (2.11) and ξ denotes a standard Cauchy random variable.
In contrast to the Cauchy limit for moderate deviations in the explosive direction, the
near–stationary case provides a Gaussian limit theorem.
Theorem 2.5 (near–stationarity) Let {Xt (n)} be an RCA(1) time series satisfying
model (2.1) with ϕ̃n < 0 for all n. If (2.2)–(2.7) hold, then
s
n
D
(ϕ̂n − ϕn ) −→ ζ
2|ϕ̃n |
(n → ∞),
where ϕ̂n is defined in (2.11) and ζ denotes a standard normal random variable.
6
The results are in accordance with results existing for near–integrated first–order autoregressive time series. The most recent work has been done in Phillips and Magdalinos
[17] for an equivalent general parameter case. Previous results in the literature were
mainly concerned with the special cases ϕn = 1 + β/nα with α ∈ [0, 1] and the parameter
β determininig whether the setting is mildly explosive (i.e. β > 0) or near–stationary (i.e.
β < 0). Using this particular form of ϕn , one can gain deeper insight on how to bridge
gaps between limit results for near–integration and unit–roots on one hand, and between
near–integration and stationarity, respectively, explosion on the other. For a detailed
discussion, we refer to [17].
More general limit theorems hold true also if the noise {εt } is in the domain of attraction
of a stable law. Corresponding results are due to Aue and Horváth [1] for first–order
autoregression. Adaptations of the proofs will as well work in the present model with
random coefficients.
The proof of Theorem 2.4 falls back on the representation of Xt (n) given in Theorem
3.1, while Theorem 2.5 can be established exploiting a result in [17]. Details are given in
Sections 3.5 and 3.6 below.
3
Proofs
Chapter 3 is divided into six sections. In the first, we derive the representation of {Xt (n)}
which is the basic tool for the proofs of Theorems 2.1–2.5 in the subsequent sections.
To simplify notation, we suppress the dependence on n in the following. Hence, we
abbreviate
Xt = Xt (n)
and
bt = bt (n)
as well as
ϕ = ϕn , ϕ̃ = ϕ̃n
and
ω 2 = ωn2 .
But nonetheless, we should always keep in mind, that we are actually dealing with a
triangular array of random variables. Throughout the chapter, we will use
t = ⌊κn⌋
3.1
with some 0 < κ < 1.
An asymptotic representation of RCA(1) time series
We shall need the following small lemma.
7
(3.1)
Lemma 3.1 Under the assumptions (2.2)–(2.7) and (3.1), we have
max |ϕ − 1 + bt−i | = oP (1)
(n → ∞).
0≤i<t
Proof. We can estimate
max |ϕ − 1 + bt−i | ≤ |ϕ̃| + ω max |γt−i |.
0≤i<t
(3.2)
0≤i<t
Using Corollary 3 of [9], p. 90, we see that
max |γt−i | = max |γj | = O
0≤i<t
1≤j≤t
√ t
a.s.
as n → ∞. So the claim follows from (3.2) after an application of (2.6) and (2.7).
2
A repeated application of the defining equations (2.1) yields
Xt = X0
Y
(ϕ + bt−i ) +
0≤i<t
X
Y
(ϕ + bt−i )εt−j .
0≤j<t 0≤i<j
The next theorem gives the promised additive decomposition by examining the products
in the previous equation.
Theorem 3.1 Let {Xt } follow model (2.1). If (2.2)–(2.7) and (3.1) are satisfied, then

Xt = X0 etϕ̃ 1 +
+
X
0≤j<t
where, as n → ∞,
X
0≤j<t

ej ϕ̃ 1 +

(1)
bt−j + Rt 
X
0≤i<j
(1)
(2)

(3)
|Rt | = OP t(ϕ̃2 + ω 2 ) ,
(2)
max |Rt,j | = OP (tω 2 ),
0≤j<t
(2)
|Rt,j |
= OP (ω 2 ),
max
0≤j<t j log log j
1 (3)
max |Rt,j | = OP (ϕ̃2 + ω 2).
0≤j<t j
8
bt−i + Rt,j  1 + Rt,j εt−j ,
Proof. The proof is given in three steps. In the first step, we obtain an additive repreQj−1
sentation for the product i=0
(ϕ + bt−i ), which is refined in a second and third step.
(i) Define the events
1
,
An = max |ϕ̃ + ωγt−i | ≤
0≤i<t
2
n ≥ 1.
From Lemma 3.1, we obtain [recall that t = ⌊κn⌋ by (3.1)]
lim P (An ) = 1.
n→∞
A Taylor expansion yields | log(1 + x) − x| ≤ 2x2 for |x| ≤ 21 . Hence for all j < t, on An ,
X
X
log(ϕ + bt−i ) −
(ϕ̃ + ωγt−i )
0≤i<j
0≤i<j
X
≤ 2
(ϕ̃ + ωγt−i)2
0≤i<j
≤ 4j ϕ̃2 + 4ω 2
X
2
γt−i
.
0≤i<j
Using the strong law of large numbers and the fact that by definition Eb21 = ω 2, we can
estimate the sum of the γi2 ’s as
1
1
D
2
| = max |γ12 + . . . + γj2 | = OP (1)
max |γt2 + . . . + γt−i
1≤i≤t i
1≤i≤t i
(n → ∞).
So, we get for the product of interest
Y
0≤i<j

(ϕ + bt−i ) = exp(j ϕ̃) exp ω

= exp(j ϕ̃) exp 
X
0≤i<j
X
0≤i<j
as n → ∞, where



γt−i  exp(St,j )

(3)
bt−i  1 + Rt,j

X
1
1
max |St,j | = OP  max 4j ϕ̃2 + 4ω 2
γt−i 
0≤j<t j
0≤j<t j
0≤i<j
= OP ϕ̃2 + ω 2
9
(n → ∞).
(3.3)
(3)
Hence, we can choose Rt,j such that the corresponding rate given in Theorem 3.1 is
satisfied, namely
1 (3)
(n → ∞).
max |Rt,j | = OP (ϕ̃2 + ω 2 )
0≤j<t j
(ii) Since {γi} is a sequence of independent, identically distributed random variables with
Eγ12 = 1, it follows from the weak convergence of partial sums to Brownian motion that
X
γt−i max 0≤j<t 0≤i<j
= OP
√ t
(n → ∞).
Hence, from (2.7) and (3.1),
X
max bt−i 0≤j<t 0≤i<j
=
Define the events
X
max ω
γt−i 0≤j<t 0≤i<j
Bn =
Then,
√
= OP ( tω) = oP (1)
X
bt−i max 0≤j<t 0≤i<j



1
≤
,
2
(n → ∞).
(3.4)
n ≥ 1.
lim P (Bn ) = 1.
n→∞
Using again a Taylor expansion argument, | exp(x) − (1 + x)| ≤
for all j < t, on Bn ,
with

 

X
X
exp 



b
−
1
+
b
t−i
t−i 0≤i<j
0≤i<j

max 
0≤j<t
X
0≤i<j
2
bt−i  = max ω 2 
0≤j<t
as n → ∞ using (3.4). Furthermore,


2
X
0≤i<j
√
e 2
x
2
for |x| ≤ 21 . Hence
2
√ 
e X
bt−i 
≤
2 0≤i<j
2
γt−i  = OP (tω 2 )

2
X
X
ω2
1


bt−i  = max
γt−i  = OP (ω 2 )
max
0≤j<t j log log j
0≤j<t j log log j
0≤i<j
0≤i<j
10
as n → ∞, where the law of the iterated logarithm has been applied to the sequence {γi }
to obtain the desired rate. Inserting these results into the right–hand side of (3.3), we
arrive at


Y
0≤i<j
(2)
(ϕ + bt−i ) = exp(j ϕ̃) 1 +
X
0≤i<j
(3)
(2)
(3)
bt−i + Rt,j  1 + Rt,j ,
(3.5)
where Rt,j and Rt,j satisfy the corresponding rates given in Theorem 3.1.
(iii) From the first part of the proof,
max
0≤j<t
(3)
|Rt,j |
= OP
max |St,j | = OP t(ϕ̃2 + ω 2 ) = oP (1)
0≤j<t
(n → ∞).
Thus, we obtain from (3.5)
Y
0≤i<t

(ϕ + bt−i ) = exp(tϕ̃) 1 +
X
0≤i<t

bt−i + OP (tω 2 ) 1 + OP t(ϕ̃2 + ω 2 )
.
Applying the central limit theorem to the partial sums of {γi } and taking assumption
(2.7) into account,

1 +
X
0≤i<t


= 1 + ω
= 1+
bt−i + OP (tω 2 ) 1 + OP t(ϕ̃2 + ω 2 )
X
0≤i<t
X
0≤i<t

γt−i + OP (tω 2 ) 1 + OP t(ϕ̃2 + ω 2 )
bt−i + OP t(ϕ̃2 + ω 2 )
as n → ∞. So, finally
Y
0≤i<t
(1)
where Rt

(ϕ + bt−i ) = exp(tϕ̃) 1 +
X
bt−i +
0≤i<t
satisfies the rate stated in Theorem 3.1.
11

(1)
Rt 
(n → ∞),
2
3.2
Proof of Theorem 2.1
The following lemma will be frequently used throughout the proofs of Theorems 2.1–2.3.
It has been established in [3] within the context of near–integrated GARCH(1,1) sequences
and we state it without proof. Let {an } and {bn } be two sequences of real numbers. We
say that an ∼ bn if limn→∞ an b−1
n = 1.
Lemma 3.2 For any ν ≥ 0 it holds,
X
0≤j<t
where Γ(t) =
R∞
0
j ν ej ϕ̃ ∼
Γ(ν + 1)
,
|ϕ̃|ν+1
e−x xt−1 dx denotes the Gamma function.
Proof: See [3], Lemma 4.1.
2
Theorem 2.1 is proved by an application of Theorem 3.1. Rewriting the decomposition
yields
tϕ̃

Xt = X0 e 1 +
+
X
0≤j<t
+
X
X
bt−j +
0≤j<t

ej ϕ̃ 1 +
j ϕ̃
e

(1)
Rt 
(2)
X
0≤i<j
(2)
Rt,j εt−j

(3)
bt−i + Rt,j  Rt,j εt−j
0≤j<t
+
X
0≤j<t
(1)

ej ϕ̃ 1 +
(2)
X
0≤i<j
(3)

bt−i  εt−j
(4)
= Xt + Xt + Xt + Xt
(4)
(3.6)
It turns out, that the dominating part of Xt is Xt . Its limiting behavior is derived in
(1)
(2)
Lemma 3.3. In the subsequent Lemmas 3.4–3.6 it is proved that the terms Xt , Xt and
(3)
Xt are negligible. That is, Theorem 2.1 follows readily from these auxiliary results.
Lemma 3.3 Under the assumptions of Theorem 2.1,
q
−nϕ̃
lim ϕ̃e
n→∞
(4) D
Xt = n→∞
lim
q
2ϕ̃σ
Z
0
t
e(x−n) d[W (t) − W (x)],
where {W (t) : t ≥ 0} denotes a standard Wiener process.
12
√
Proof: From the central limit theorem, γt + γt−1 + . . . + γ1 = OP ( t) as n → ∞. Hence,
X
X
bt−j = ω
0≤j<t
γt−j = oP (1)
(n → ∞),
0≤j<t
(3.7)
by (2.7). This yiels
(4)
Xt
X
=
ej ϕ̃ εt−j (1 + oP (1))
(n → ∞)
0≤j<t
and we have to study the sum on the right–hand side of the latter equation. Write
X
0≤j<t
ej ϕ̃ εt−j =
Z
0
t
exϕ̃ d[S(t) − S(x)] = S(t) −
Z
0
t
[S(t) − S(x)]ϕ̃exϕ̃ dx
with
S(0) = 0,
S(x) =
X
εj ,
x > 0.
0≤j<x
By (2.8) and the Komlós, Major and Tusnády approximations (see [11] and [12]), there
is a Wiener process {W (t) : t ≥ 0} such that
S(x) − σW (x) = o x1/(2+δ)
a.s.
as x → ∞, where δ > 2. Hence,
√
Z κn
ϕ̃
xϕ̃
sup
([S(x)
−
S(κn)]
−
[W
(x)
−
W
(κn)])
ϕ̃e
dx
n
ϕ̃
e 0≤κ≤1 0
√ 1/(2+δ)
Z κn
ϕ̃n
xϕ̃
= OP (1)
ϕ̃e dx
sup n
ϕ̃
e
0≤κ≤1 0
q
= OP (1) ϕ̃n1/(2+δ)
= oP (1)
as n → ∞ by (2.6), since δ > 2. Also,
√ 1/(2+δ)
√
ϕ̃
ϕ̃n
sup
|S(κn)
−
σW
(κn)|
=
O
(1)
= oP (1)
P
enϕ̃ 0≤κ≤1
enϕ̃
(n → ∞).
Using partial integration in the opposite direction for {W (t) : t ≥ 0}, we see that
q
2ϕ̃σ
has the same limit distribution as
Z
t
0
√
e(x−n)ϕ̃ d[W (t) − W (x)]
(4)
ϕ̃e−nϕ̃ Xt .
13
2
Lemma 3.4 Under the assumptions of Theorem 2.1,
√ −tϕ̃
ϕ̃e
(1)
Xt = oP (1)
(n → ∞),
σ
(1)
where Rt
satisfies the corresponding rate in Theorem 3.1.
Proof: Using Theorem 3.1, (2.6) and (2.7), we obtain that
(1)
Rt = OP t(ϕ̃2 + ω 2) = oP (1)
(n → ∞).
Moreover, bt + bt−1 + . . . + b1 = oP (1) by (3.7) and clearly also
√
√ −tϕ̃
ϕ̃e
ϕ̃X0
tϕ̃
X0 e =
= oP (1)
(n → ∞)
σ
σ
by the fact that ϕ̃ → 0. Hence, the proof is complete.
2
Lemma 3.5 Under the assumptions of Theorem 2.1,
√ −tϕ̃
ϕ̃e
(2)
Xt = oP (1)
(n → ∞),
σ
(2)
(3)
where Rt,j and Rt,j satisfy the corresponding rates in Theorem 3.1.
Proof: Note that by Theorem 3.1 and (2.7)
(2)
max |Rt,j | = OP (tω 2 ) = oP (1)
(n → ∞).
0≤j<t
Thus, by (3.7) and Theorem 3.1 it holds,
√ −tϕ̃
√ −tϕ̃
X
ϕ̃e
ϕ̃e
(3)
(2)
ej ϕ̃ Rt,j εt−j
Xt
= OP (1)
σ
σ
0≤j<t
q
= OP (1) ϕ̃e−tϕ̃ (ϕ̃2 + ω 2)
= OP (1)
−tϕ̃
e
2
2
s X
j 2 e2j ϕ̃
0≤j<t
(ϕ̃ + ω )
√ 3
2 ϕ̃
= oP (1)
as n → ∞, where Lemma 3.2 has been used to obtain the third equality sign, while the
final one follows from (2.6) and (2.7).
2
14
Lemma 3.6 Under the assumptions of Theorem 2.1,
√ −tϕ̃
ϕ̃e
(3)
Xt = oP (1)
(n → ∞),
σ
(2)
where Rt,j satisfies the corresponding rate in Theorem 3.1.
Proof: It holds,
√
ϕ̃e−tϕ̃ (3)
Xt
=
σ
√
ϕ̃e−tϕ̃ X j ϕ̃ (2)
e Rt,j εt−j
σ
0≤j<t
√ −tϕ̃
X
ϕ̃e
ej ϕ̃ εt−j
tω 2
= OP (1) √
tσ
0≤j<t
q
= OP (1) ϕ̃e−tϕ̃ tω 2
q
= OP (1) ϕ̃e−tϕ̃ tω 2
X
e2j ϕ̃
0≤j<t
1
2ϕ̃
e−tϕ̃ tω 2
= OP (1) √
2 ϕ̃
= oP (1)
as n → ∞, where the respective equality sign have been obtained by (1) definition, (2)
Theorem 3.1, (3) the central limit theorem for the {εi}, (4) Lemma 3.2, (5) simple algebra,
and (6) by (2.6) and (2.7).
2
Proof of Theorem 2.1: Putting together Lemmas 3.3–3.6, it remains only to check the
asymptotic independence of the finite dimensional distributions. Choose ti = ⌊κi n⌋ >
⌊κj n⌋ = tj . Then,
E
Z
ti
xϕ̃
e
0
=
Z
tj
d[W (t) − W (x)]
Z
0
tj
xϕ̃
e
d[W (t) − W (x)]
e2xϕ̃ dx
0
2tj ϕ̃
e
(1 + o(1)).
2ϕ̃
Thus, using the normalization provided in Theorem 2.1, the finite dimensional distributions of the limiting process have exponentially decreasing covariances
=
e2(κj −1)nϕ̃ → 0
(n → ∞),
since κj − 1 < 0 and nϕ̃ → ∞ by (2.6), completing the proof.
15
2
3.3
Proof of Theorem 2.2
We follow the arguments of the previous section using the decomposition (3.6). It will
(4)
be shown in a series of lemmas that Xt is determining the limit distribution, while
the other terms are asymptotically negligible. The main auxiliary result is the following
lemma. Recall that t = ⌊κn⌋ with some κ ∈ (0, 1) by (3.1).
Lemma 3.7 Under the assumptions of Theorem 2.2,
1
D
√ Xt(4) −→ W (κ)
nσ
(n → ∞),
where {W (s) : s ∈ [0, 1]} denotes a standard Wiener process.
Proof. As derived in (3.7),
X
0≤j<t
so
bt−j = ω
X
γt−j = oP (1)
(n → ∞),
0≤j<t
1
1 X
√ Xt(4) = √
εt−j (1 + oP (1))
nσ
nσ 0≤j<t
(n → ∞).
By assumption (2.9) there is a Wiener process {W (1) (s) : s ≥ 0} such that
X
1≤j≤s
εj − σW (1) (s) = o s1/(2+δ)
a.s.
(s → ∞)
for some δ > 0. Using this strong approximation and the scale transformation for Wiener
processes, we obtain on recalling once more (3.1) that
1 X
D
√
εt−j −→ W (κ)
nσ 0≤j<t
(n → ∞),
where {W (t) : t ∈ [0, 1]} denotes a standard Wiener process. This completes the proof.
2
It remains to show that all other terms in the representation (3.6) are negligible. This
will be done in the following lemmas.
Lemma 3.8 Under the assumptions of Theorem 2.2,
1
√ Xt(1) = oP (1)
nσ
(1)
where Rt
(n → ∞),
satisfies the corresponding rate in Theorem 3.1.
16
√
(1)
Proof. By (2.7) and Theorem 3.1, Rt = oP (1). Moreover, ( nσ)−1 X0 = oP (1) and,
as in (3.7), bt + . . . + b1 = oP (1) after an application of the central limit theorem to the
partial sums of the γi ’s and the claim follows.
2
Lemma 3.9 Under the assumptions of Theorem 2.2,
1
√ Xt(2) = oP (1)
nσ
(2)
(n → ∞),
(3)
where Rt,j and Rt,j satisfy the corresponding rates in Theorem 3.1.
Proof. Theorem 3.1, (2.7) and (3.7) imply that
X
γt−j
max ω
0≤j<t 0≤i<t
Hence,
(2) + Rt,j = oP (1)
(n → ∞).
1 X (3)
1
√ Xt(2) = OP (1) √
R εt−j
nσ
nσ 0≤i<t t,j
ω2 X
jεt−j
= OP (1) √
nσ 0≤i<t
ω2 s X 2
j
= OP (1) √
n 0≤i<t
ω 2t3/2
= OP (1) √
n
= oP (1)
(3)
as n → ∞, where we have applied (1) the rate established for Rt,j in Theorem 3.1, (2)
the central limit theorem for the sequence {iεi }, (3) that the order of the sum of squares
of the first t positive integers is t3 , and finally (4) assumptions (2.7) and (3.1).
2
Lemma 3.10 Under the assumptions of Theorem 2.2,
1
√ Xt(3) = oP (1)
nσ
(2)
(n → ∞),
where Rt,j satisfies the corresponding rate in Theorem 3.1.
17
Proof. By similar arguments as in the previous proofs,
1
1 X (2)
1
√ Xt(3) = √
Rt,j εt−j = OP (1) √ t3/2 ω 2 = oP (1)
nσ
nσ 0≤j<t
n
(n → ∞),
using Theorem 3.1, (2.7) and (3.1).
2
Proof of Theorem 2.2: The assertion follows directly from Lemmas 3.7–3.10.
2
3.4
Proof of Theorem 2.3
Finally, we repeat the steps accomplished in the previous two subsections. First, we will
(4)
derive the limiting behavior of the term Xt and then show that all other terms of the
decomposition do not contribute to the asymptotics. Let
tm = ⌊κm n⌋
with some 0 < κm < 1,
m = 1, . . . , N.
(3.8)
Our first auxiliary result is the following lemma.
Lemma 3.11 Let ϕ̃ < 0. If the second part of assumption (2.6) is satisfied, then
q
D
2|ϕ̃|[τ1 , . . . , τN ] −→ σ[ξ1 , . . . , ξN ]
(n → ∞),
where, for m = 1, . . . , N,
τm =
X
0≤i<tm
eiϕ̃ εtm −i
and ξ1 , . . . , ξN are idependent, identically distributed standard normal random variables.
Proof. The proof is given in two steps. First, the Cramér–Wold device (see e.g. [4],
Theorem 29.4) is applied, i.e. we are investigating linear combinations of τ1 , . . . , τN . Then,
we show that the central limit theorem is satisfied by verifying Liapunov’s condition (see
e.g. [4], Theorem 27.3).
(i) Choose real numbers λ1 , . . . , λN and write
λ1 τ1 + . . . +h λN τN
i
X
=
λ1 e(t1 −i)ϕ̃ + λ2 e(t2 −i)ϕ̃ + . . . + λN e(tN −i)ϕ̃ εi
1≤i≤t1
+
X
λ2 e(t2 −i)ϕ̃ + . . . + λN e(tN −i)ϕ̃ εi
t1 <i≤t2
+...+
i
h
X
λN e(tN −i)ϕ̃ εi
tN−1 <i≤tN
= S1 + . . . + SN .
(3.9)
18
Clearly, S1 , . . . , SN have expectation zero. It holds for the variance of S1 ,
X h
ES12 = σ 2
λ1 e(t1 −i)ϕ̃ + . . . + λN e(tN −i)ϕ̃
1≤i≤t1
= σ
2
X
λ2j
X
e2(tj −i)ϕ̃ + σ 2
1≤i≤t1
1≤j≤N
e2(t1 −i)ϕ̃ =
X
0≤i<t1
1≤i≤t1
X
λj λl
e2iϕ̃ ∼
1
2|ϕ̃|
X
e(tj +tl −2i)ϕ̃ .
1≤i≤t1
1≤j6=l≤N
By Lemma 3.2,
X
i2
(n → ∞).
On the other hand, for 2 ≤ j ≤ N, using Lemma 3.2 also
X
2(tj −i)ϕ̃
e
2(tj −t1 )ϕ̃
=e
1≤i≤t1
X
2(t1 −i)ϕ̃
e
1≤i≤t1
1
=o
|ϕ̃|
!
(n → ∞),
since (tj − t1 )ϕ̃ < 0. Similar arguments apply to the second part of ES12 . In detail,
X
e(tj +tl −2i)ϕ̃ = e(tj −t1 )ϕ̃ e(tl −t1 )ϕ̃
1≤i≤t1
X
e2(t1 −i)ϕ̃
1≤i≤t1
1 tj −t1 )ϕ̃ (tl −t1 )ϕ̃
∼
e
e
2|ϕ̃|
!
1
= o
|ϕ̃|
as n → ∞. Hence, putting together the previous calculations, we obtain
ES12
1
λ2 σ 2
= 1 +o
2|ϕ̃|
|ϕ̃|
!
(n → ∞).
Repeating the above arguments for any of the remaining ESj2 (j = 2, . . . , N), we arrive
at
!
σ2 2
1
2
2
E [λ1 τ1 + . . . + λN τN ] =
(n → ∞),
(λ + . . . + λN ) + o
2|ϕ̃| 1
|ϕ̃|
finishing the first part of the proof.
(ii) It remains to verify Liapunov’s condition. Observe that by an rearrangement of (3.9),
we can write
λ1 τ1 + . . . + λN τN = µ1 ε1 + . . . + µtN εtN
19
with appropriately chosen coefficients µi , (i = 1, . . . , tN ). In the following it will be shown
that
h
|µ1 |2+δ E|ε1 |2+δ + . . . + |µtN |2+δ E|εtN |2+δ
h
µ21 Eε21 + . . . + µ2tN Eε2tN
i1/2
i1/(2+δ)
= o(1)
(n → ∞)
(3.10)
for some δ > 0. First, we determine the asymptotic rate of the denominator. It holds,
µ21 Eε21 + . . . + µ2tN Eε2tN
1
= σ 2 (µ21 + . . . + µ2tN ) = O
|ϕ̃|
!
(n → ∞).
To check the numerator, select the summation range i = 1, . . . , t1 and observe that by
Jensen’s inequality
2+δ
|µi|2+δ = λ1 e(t1 −i)ϕ̃ + . . . + λN e(tN −i)ϕ̃ h
i
≤ C1 (N) |λ1 |2+δ e(t1 −i)ϕ̃(2+δ) + . . . + λN |2+δ e(tN −i)ϕ̃(2+δ) .
Hence,
|µ1|
2+δ
+ . . . + |µt1 |
C1 (N)|λ1 |2+δ
1
∼
+o
(2 + δ)|ϕ̃|
|ϕ̃|
2+δ
!
≤
C2 (N)
|ϕ̃|
(n → ∞).
Using similar calculations for the remaining summation ranges, we find that the asymptotic rate in (3.10) can be estimated by
"
1
O
|ϕ̃|
#1/(2+δ)−1/2 

This completes the proof.
= o(1)
(n → ∞).
2
Lemma 3.11 immediately yields the limit distribution on noticing that
q
(4)
2|ϕ̃|Xt
∼
q
2|ϕ̃|
X
ej ϕ̃ εt−j
0≤j<t
(n → ∞)
along the lines of Lemmas 3.3 and 3.7.
Lemma 3.12 Under the assumptions of Theorem 2.3,
q
(1)
2|ϕ̃|σ −1 Xt
(1)
where Rt
= oP (1)
(n → ∞),
satisfies the corresponding rate in Theorem 3.1.
20
(1)
Proof. By (2.6), (2.7) and Theorem 3.1, Rt = oP (1). Moreover,
q
2|ϕ̃|σ −1 X0 etϕ̃ = oP (1)
(n → ∞),
since tϕ̃ → −∞ by assumption (2.6). Finally, (3.7) yields the assertion.
2
Lemma 3.13 Under the assumptions of Theorem 2.3,
q
(2)
2|ϕ̃|σ −1 Xt
(2)
= oP (1)
(n → ∞),
(3)
where Rt,j and Rt,j satisfy the corresponding rate in Theorem 3.1.
Proof. Following the lines of the proofs of Lemmas 3.5 and 3.9, we get
q
2|ϕ̃|σ
−1
q
(2)
Xt
= OP (1) 2|ϕ̃|σ −1
q
X
(3)
ej ϕ̃ Rt,j εt−j
0≤j<t
−1
= OP (1) 2|ϕ̃|σ (ϕ̃2 + ω 2)
q
= OP (1) 2|ϕ̃|(ϕ̃2 + ω 2 )
X
ej ϕ̃ jεt−j
0≤j<t
s X
e2j ϕ̃ j 2
0≤j<t
1
(ϕ̃2 + ω 2 )
= OP (1) q
2|ϕ̃|
= oP (1)
as n → ∞, where we have used Lemma 3.2 to obtain the fourth equality sign. The final
equality is verified on observing that by (2.6) and (2.7), ω 2 |ϕ̃|−1/2 = (ω 2 n1/2 )(|ϕ̃|n)−1/2 →
0 as n → ∞.
2
Lemma 3.14 Under the assumptions of Theorem 2.3,
q
(3)
2|ϕ̃|σ −1 Xt
= oP (1)
(n → ∞),
(2)
where Rt,j satisfies the corresponding rate in Theorem 3.1.
Proof. Applying Lemma 3.2 yields
q
(3)
2|ϕ̃|σ −1 Xt
=
q
2|ϕ̃|σ −1
X
0≤j<t
21
(2)
ej ϕ̃ Rt,j εt−j
q
= OP (1) 2|ϕ̃|σ −1 tω 2
q
= OP (1) 2|ϕ̃|tω 2
= OP (1)tω
= oP (1)
2
X
ej ϕ̃ εt−j
0≤j<t
s X
e2j ϕ̃
0≤j<t
as n → ∞ by (2.7) and (3.1), finishing the proof.
2
Proof of Theorem 2.3: It follows from Lemmas 3.11–3.14.
3.5
2
Proof of Theorem 2.4
The proof is based on the decomposition (3.1). So, we will study the limiting behavior of
(4)
(i)
the leading term Xt first, and thereafter show that terms involving Xt , i = 1, 2, 3, do
not contribute to the asymptotics.
(4) 2
In a first part, we shall study the partial sums of Xk
(4)
Xt
=
t−1
X
j ϕ̃
e εt−j (1 + oP (1)) =
t
X
. Since by assumption ε0 = X0 ,
e(t−j)ϕ̃ εj (1 + oP (1)),
t = 1, . . . , n,
j=1
j=0
n ≥ 1.
From the preceding, we obtain
n
X
(4) 2
Xt
=

n
X

t=1
t=1
t
X
j=1
2
e(t−j)ϕ̃ εj  (1 + oP (1))
and it suffices to investigate the sum on the right–hand side of the latter equation. Let
T > 0. Then
n
X
t=1


t
X
j=1
2
e(t−j)ϕ̃ εj 
=
n
X
t=1
=
e2tϕ̃ 
⌊T /ϕ̃⌋
X
t=1
+

t
X
j=1

e2tϕ̃ 
n
X
e−j ϕ̃ εj 
t
X
j=1
t=⌊T /ϕ̃⌋+1
2
2
e−j ϕ̃ εj  +

e2tϕ̃ 
⌊T /ϕ̃⌋
X
j=1
22
n
X
t=⌊T /ϕ̃⌋+1
2
e−j ϕ̃ εj 

e2tϕ̃ 
t
X
j=⌊T /ϕ̃⌋+1
2
e−j ϕ̃ εj 
= ST,1 + ST,2 + ST,3 .
(3.11)
The next lemma identifies ST,3 as leading term by showing that ST,1 and ST,2 are asymptotically small.
Lemma 3.15 Let the assumptions of Theorem 2.4 be satisfied.
(i) For every T > 0 and ε > 0,
lim P
n→∞
(
ϕ̃2
ST,1 ≥ ε = 0
e2nϕ̃
)
where ST,1 is defined in (3.11).
(ii) For every ε > 0,
lim lim sup P
T →∞
n→∞
(
ϕ̃2
ST,2 ≥ ε = 0,
e2nϕ̃
)
where ST,2 is defined in (3.11).
Proof. The proof is based on an application of Markov’s inequality. Throughout the
proof C denotes a universal constant, which may vary from line to line.
(i) Note that

⌊T /ϕ̃⌋
 X
EST,1 = E 
≤ C
t=1

e2tϕ̃ 
t
X
j=1
⌊T /ϕ̃⌋
σ 2 X 2tϕ̃
e
ϕ̃ t=1
2 

ρn−j ϕ̃ εj  
≤ C
σ2
exp(2ϕ̃⌊T /ϕ̃⌋)
ϕ̃2
≤ C
σ2
exp(2T )
ϕ̃2
for any fixed T > 0. Now Markov’s inequality yields
P
(
ϕ̃2
ϕ̃2
S
≥
ε
≤
EST,1
T,1
e2nϕ̃
εe2nϕ̃
)
23
for any ε > 0. The quantity on the right–hand side of the latter inequality becomes
arbitrarily small as n → ∞, hence part (i) is proved.
(ii) By similar arguments

EST,2 = E 

n
X
t=⌊T /ϕ̃⌋+1
≤ Cσ 2
σ2
≤ C
ϕ̃
≤ C

e2tϕ̃ 
n
X
t
X
j=⌊T /ϕ̃⌋
t
X
e2tϕ̃
t=⌊T /ϕ̃⌋+1
n
X
2 

e−j ϕ̃ εj  
e−2j ϕ̃
j=⌊T /ϕ̃⌋
exp(2tϕ̃) exp(−2ϕ̃⌊T /ϕ̃⌋)
t=⌊T /ϕ̃⌋+1
σ2
exp(−2T ) exp(2nϕ̃).
ϕ̃2
So, Markov’s inequality gives part (ii) of the lemma and the proof is complete.
2
Lemma 3.15 tells us that it suffices to derive the limit distribution of the remaining
(4)
term ST,3 . The second part of the proof deals with the partial sums of Xt−1 εt . Let again
be T > 0. Then
n
X
t=1
(4)
Xt−1 εt
=
⌊T /ϕ̃⌋
X
t=1
+
n
X
(4)
Xt−1 εt +
e(t−1)ϕ̃ εt
t=⌊T /ϕ̃⌋+1
n
X
e(t−1)ϕ̃ εt
⌊T /ϕ̃⌋
X
t
X
e−j ϕ̃ εj
j=⌊T /cn ⌋
e−j ϕ̃ εj
j=1
k=⌊T /ϕ̃⌋+1
= RT,1 + RT,2 + RT,3 .
(3.12)
It will be shown next that RT,3 is the leading term, while RT,1 and RT,2 do not contribute
to the limit distribution.
Lemma 3.16 Let the assumptions of Theorem 2.4 be satisfied.
(i) For every T > 0 and ε > 0,
lim P
n→∞
ϕ̃
|RT,1 | ≥ ε = 0,
enϕ̃
24
where RT,1 is defined in (3.12).
(ii) For every ε > 0,
lim lim sup P
T →∞
n→∞
ϕ̃
|RT,2 | ≥ ε = 0,
enϕ̃
where RT,2 is defined in (3.12).
Proof. The proof is based on an application of Chebyshev’s inequality. Since ERT,i = 0,
2
we have VarRT,i = ERT,i
for i = 1, 2, 3 and it is enough to give estimates for the second
moments. Let C denote a universal but varying constant.
(i) Observe that

2
ERT,1
= E
≤ C
≤ C
⌊T /ϕ̃⌋
X
t=1
2
(4)
Xt−1 εk 
⌊T /ϕ̃⌋
σ 4 X 2tϕ̃
e
ϕ̃ t=1
σ4
exp(2T ),
ϕ̃2
(4)
where the second line is obtained using the independence of εk and Xt−1 . On recognizing
that by Chebyshev’s inequality
P
ϕ̃2
ϕ̃
2
|R
|
≥
ε
≤
ERT,1
,
T,1
n
ϕ̃
2
e
ε exp(2nϕ̃)
the probability on the left–hand side gets arbitrarily small for any fixed T > 0 if n → ∞.
So, part (i) is readily proved.
(ii) Note that,

2
= E
ERT,2
≤ C
n
X
e(t−1)ϕ̃ εt
k=⌊T /ϕ̃⌋+1
t−1
X
j=⌊T /ϕ̃⌋
σ4
exp(−2T ) exp(2nϕ̃).
ϕ̃2
Again, Chebyshev’s inequality gives claim (ii).
25
2
e−j ϕ̃ εj 
2
So, it suffices again to investigate the term RT,3 to obtain the limit distribution. Checking the remaining summation ranges in the corresponding double sum shows us that RT,3
is a product of two independent factors, whose limit can be calculated separately.
Lemma 3.17 Let the assumptions of Theorem 2.4 be satisfied. Then
2ϕ̃2
ϕ̃
D
ST,3 , 2 nϕ̃ RT,3 −→ (N12 , N1 N2 )
2
2n
ϕ̃
σ e
σ e
!
(n → ∞)
for any fixed T > 0, where N1 and N2 denote independent zero mean normal random
variables with EN12 = 1 − e−T and EN22 = 1.
Proof. The proof is given in two steps, each of them investigating one of the components
of the vector of interest. The joint behavior is obtained on combining both results.
(i) Observe that
ST,3 =

⌊T /ϕ̃⌋
n
X
t=⌊T /ϕ̃⌋

e2tϕ̃ 
X
j=0
2
e−j ϕ̃ εj 
2
⌊T /ϕ̃⌋
e2nϕ̃ q X −j ϕ̃ 
=
e εj (1 + o(1))
ϕ̃
2ϕ̃2
j=0
as n → ∞. Let S(x) = ε0 + . . . + ε⌊x⌋ . We need to determine the limit of
UT,1 =
=
q
ϕ̃
⌊T /ϕ̃⌋
X
e−j ϕ̃ εj
(3.13)
j=0
q Z
ϕ̃
T /ϕ̃
0
exp(−xϕ̃)dS(x)
!
x
ϕ̃
exp(−x)dS
=
ϕ̃
0
!
! #
"
Z T
q
x
T
=
exp(−x)S
− ε0 +
dx
ϕ̃ exp(−T )S
ϕ̃
ϕ̃
0
√
using integration by parts. The weak convergence of ϕ̃S(t/ϕ̃) to σW (t) on [0, T ], with
{W (t) : t ∈ [0, T ]} denoting a standard Wiener process, implies
q Z
q
ϕ̃
⌊T /ϕ̃⌋
X
j=0
T
D
exp(−j ϕ̃)εj −→ σ
Z
T
0
exp(−x)dW (x)
26
(n → ∞),
where we have also used integration by parts for the Wiener process.
(ii) For the second component we have,
ϕ̃
RT,3 = UT,1 UT,2 ,
enϕ̃
where
UT,2 =
√
n
X
ϕ̃
enϕ̃
e(t−1)ϕ̃ εt .
t=⌊T /ϕ̃⌋+1
The second sum contains noise terms starting with index t = ⌊T /ϕ̃⌋ + 1, while these
terms stop at j = ⌊T /ϕ̃⌋ in the first one. Hence, UT,1 and UT,2 are independent, yielding
the independence of N1 and N2 . The limit of UT,1 is already established in (3.13). So, it
remains to determine the limit distribution of
√
n
n
q
X
ϕ̃ X
(t−1)ϕ̃
ϕ̃
exp (−(n − t + 1)ϕ̃) εt
e
ε
=
t
enϕ̃ t=⌊T /ϕ̃⌋+1
t=⌊T /ϕ̃⌋+1
D
=
q
n−⌊T /ϕ̃⌋
ϕ̃
X
exp(−j ϕ̃)εj + oP (1).
j=0
Let {W̃ : t ≥ 0} be a standard Wiener process independent of {W (t) : t ∈ [0, T ]}. Then,
arguing as in part (i) of the proof,
√
Z ∞
n
ϕ̃ X
D
(t−1)ϕ̃
e
εt −→ σ
exp(−x)dW̃ (x)
(n → ∞).
enϕ̃ t=⌊T /ϕ̃⌋+1
0
The proof is complete after combining the results of parts (i) and (ii).
It is enough to show that the squares of the remaining terms in (3.1) are negligible.
Lemma 3.18 Let the assumptions of Theorem 2.4 be satisfied. Then,
n
ϕ̃2 X
(i) 2
(n) = oP (1)
X
t
e2nϕ̃ k=1
(n → ∞)
for i = 1, 2, 3.
Proof. (i) Let i = 1. Then
n
n
ϕ̃2 X
ϕ̃2 X
(1) 2
2 2tϕ̃
2
= oP (1)
X
=
X
e
=
O
n
ϕ̃
P
t
0
e2nϕ̃ t=1
e2nϕ̃ t=1
27
(n → ∞).
2
(ii) Let i = 2. Then,
n
ϕ̃2 X
(2) 2
Xt = OP
2n
ϕ̃
e
t=1
n(ϕ̃2 + ω 2 )2
ϕ̃e2nϕ̃
!
= oP (1)
(n → ∞).
(iii) Let i = 3. Then,
n
ϕ̃2 X
(3) 2
= OP
X
t
e2nϕ̃ t=1
n2 ω 2
e2nϕ̃
!
= oP (1)
(n → ∞).
Throughout, we have implicitly used Lemmas 3.4–3.6. The proof is complete.
2
Lemma 3.19 Let the assumptions of Theorem 2.4 be satisfied. Then,
n
ϕ̃ X
(i)
Xt−1 εt = oP (1)
n
ϕ̃
e t=1
(n → ∞)
for i = 1, 2, 3.
Proof. Similar arguments as in Lemma 3.18 lead to the conclusions paraphrasing the
proofs of Lemmas 3.4–3.6.
2
Proof of Theorem 2.4. Collecting the results of the previous lemmas, we arrive at
lim lim sup P
T →∞
n→∞
(
enϕ̃n
2ϕ̃n
σ 2 e2nϕ̃ 1
ϕ̃
(ϕ̂n − ϕn ) −
R
T,3 > ε = 0
2ϕ̃2 ST,3 σ 2 enϕ̃
)
for all ε > 0. By Lemma 3.17, for any T > 0 we have that
ϕ̃
σ 2 e2nϕ̃ 1
D N1 N2
RT,3 −→
2
2
n
ϕ̃
2ϕ̃ ST,3 σ e
N12
We note
D
(N1 , N2 ) =
q
1−
e−T M1 , M2
(n → ∞).
,
where M1 and M2 are independent standard normal random variables. So, we get
1
M2
N1 N2 D
=√
,
2
N1
1 − e−T |M1|
which converges to a standard Cauchy random variable as T → ∞.
28
2
3.6
Proof of Theorem 2.5
We follow the pattern provided in the previous subsection.
Lemma 3.20 Let the assumptions of Theorem 2.5 be satisfied. Then
(i)
(ii)
n
2|ϕ̃| X
(4) 2 P
Xt−1 −→ 1
2
σ n t=1
s
(n → ∞).
n
2|ϕ̃| X
D
(4)
Xt−1 εt −→ ζ
4
σ n t=1
(n → ∞),
where ζ denotes a standard normal random variable.
Proof. (i) An application of Markov’s inequality yields
P
(
n
2|ϕ̃| X
(4) 2
Xt−1
2
σ n
t=1
− 1
n
1 2|ϕ̃| X
(4) 2
Xt−1 − 1 .
≥ε ≤ E 2
ε σ n t=1
)
It is therefore enough to consider the expectation on the right–hand side. It holds,
n
2|ϕ̃| X
(4) 2
Xt−1
E 2
σ n t=1
"
#
=
=
=
t−1
n X
2|ϕ̃| X
e2j ϕ̃ σ 2
σ 2 n t=1 j=0
n
2|ϕ̃| X
1 − e2tϕ̃
n t=1 1 − e2ϕ̃
1 − e2nϕ̃ (1 + o(1))
as n → ∞. So, the claim follows since nϕ̃ → −∞.
(ii) See Theorem 3.2(b) in Phillips and Magdalinos (2005).
2
Lemma 3.21 Let the assumptions of Theorem 2.5 be satisfied. Then,
n
ϕ̃2 X
(i) 2
Xt = oP (1)
2n
ϕ̃
e
k=1
(n → ∞)
for i = 1, 2, 3.
Proof. (i) Let i = 1. Then
n
n
2|ϕ̃| X
2|ϕ̃| X
(1) 2
Xt =
X02 e2tϕ̃ = OP |ϕ̃|e2nϕ̃ = oP (1)
n t=1
n t=1
29
(n → ∞).
(ii) Let i = 2. Then,
n
2|ϕ̃| X
(2) 2
Xt = OP
n t=1
(ϕ̃2 + ω 2 )2
|ϕ̃|2
!
= oP (1)
(n → ∞).
(iii) Let i = 3. Then,
n
2|ϕ̃| X
(3) 2
Xt = OP n2 ω 4 = oP (1)
n t=1
(n → ∞).
The proof is complete.
2
Lemma 3.22 Let the assumptions of Theorem 2.5 be satisfied. Then,
n
2|ϕ̃| X
(i)
Xt−1 εt = oP (1)
n t=1
(n → ∞)
for i = 1, 2, 3.
Proof. Similar arguments as in Lemma 3.18 lead to the conclusions.
2
Proof of Theorem 2.5. Collecting the results of the previous lemmas, we arrive at
s
n
n
2|ϕ̃| X
(4) 2
Xt−1
(ϕ̂n − ϕn ) =
2
2|ϕ̃n |
σ n t=1
"
#−1 s
n
2|ϕ̃| X
(4)
Xt−1 εt + oP (1) → ζ
4
σ n t=1
as n → ∞, where ζ denotes a standard normal random variable.
2
References
[1] Aue, A., and Horváth, L. (2005). A limit theorem for mildly explosive autoregression
with stable errors. Preprint, University of Utah.
[2] Aue, A., Horváth, L., and Steinebach, J. (2004). Estimation in random coefficient
autoregressive models. Preprint, University of Utah.
[3] Berkes, I., Horváth, L., and Kokoszka, P. (2005). Near–integrated GARCH sequences.
Ann. Appl. Probab. 15, 890–913.
[4] Billingsley, P. (1995). Probability and Measure (3rd ed.). Wiley, New York.
30
[5] Bollerslev, T. (1986). Generalized autoregressive conditional hetereoscedasticity. J.
Econ. 31, 307–327.
[6] Brockwell P.J., and Davis R.A. (1991). Time Series: Theory and Methods (2nd ed.).
Springer–Verlag, Berlin.
[7] Chan, N.H., and Wei, C.Z. (1987). Asymptotic inference for nearly nonstationary
AR(1) processes. Ann. Statist. 15, 1050–1063.
[8] Chan, N.H., and Wei, C.Z. (1988). Limiting distributions of least squares estimates
of unstable autoregressive processes. Ann. Statist. 16, 367–401.
[9] Chow, Y.S., and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales (2nd ed.). Springer–Verlag, New York.
[10] Feigin, P.D., and Tweedie, R.L. (1985). Random coefficient autoregressive processes:
a Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal.
6, 1–14.
[11] Komlós, J., Major, P., and Tusnády, G. (1975). An approximation of partial sums of
independent r.v.’s and the sample d.f. I. Z. Wahrsch. Verw. Gebiete 32, 111–131.
[12] Komlós, J., Major, P., and Tusnády, G. (1976). An approximation of partial sums of
independent r.v.’s and the sample d.f. II. Z. Wahrsch. Verw. Gebiete 34, 33–58.
[13] Ling, S., and Li, W.K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving–average time series with general autoregressive heteroscedastic errors. Ann. Statist. 26, 84–125.
[14] Nelson, C.R., and Plosser, C. (1982). Trends and random walks in macroeconomic
time series: some evidence and implications. Journal of Monetary Economics 10,
139–162.
[15] Nicholls, D.F., and Quinn, B.G. (1982). Random Coefficient Autoregressive Models:
an Introduction. Springer–Verlag, New York.
[16] Phillips, P.C.B. (1988). Regression theory for near–integrated time series. Econometrica 56, 1021–1043.
[17] Phillips, P.C.B., and Magdalinos, T. (2005). Limit theory for moderate deviations
from a unit root. Preprint, Yale University.
31
[18] Tong, H. (1990). Non–Linear Time Series: a Dynamical System Approach. Clarendon, Oxford.
32