239 Detection of Change Points in Discrete-Valued Time Series
If this limit matrix
A
is positive denite, then this is no additional restriction on which
alternatives can be detected. Obviously, (iii) implies (ii). The following additional assump-
tion is also often fullled and yields the additional assertions in (b) and (c).
B.5 F
t
(·) is continuous in θ
1
and F
t
(θ
1
) = F
λ
(θ
1
)g(t) with
1
t, t λ,
g(t) =
λ
1
(1 − t), t λ.
1 − λ
10.7.2 Proofs
Proof of Theorem 10.1 By Assumption A.1, we can replace S(k, θ
n
) in all three statistics by
S(k, θ
0
) without changing the asymptotic distribution, where—for the statistics in (a)—one
needs to note that by (10.3)
k
k n − k
sup
w =
O(1).
1�k�n
n n n
For a bounded and continuous weight function, the assertion then immediately follows by
the functional central limit theorem. For an unbounded weight function, note that for any
0 < τ < 1/2, it follows from the Hájek–Rényi inequality and (10.3) that
max
w(k/n)
S(k, θ
0
) −
k
S(n, θ
0
)
T
−1
S(k, θ
0
) −
k
S(n, θ
0
)
1�k�τn
n n n
w(t)t
α
S(k, θ
0
)
2
P
1−τ�t<1
1�k�n
n
1−α
k
α
sup
−1/2
max
1
→ 0
as τ → 0 uniformly in n. By the backward inequality and the fact that S(k, θ
0
) −
n
k
S(n, θ
0
) =
−((S(n, θ
0
) − S(k, θ
0
)) −
n−
n
k
S(n, θ
0
)), an analogous assertion holds for max
(1−τ)n�k�n
as
well as for the corresponding maxima over the limit Brownian bridges. Since the functional
central limit theorem implies the claimed distributional convergence for max
τn�k�(1−τ)n
,
careful arguments yield the assertion. The result for estimated is immediate.
To prove (b), rst note that by the invariance principle in A.2 and the law of iterated
logarithm, we get
n
k
T
−1
k
b
d
(log n)
2
max
S(k, θ
0
) −
S(n, θ
0
)
S(k, θ
0
) −
S(n, θ
0
) = o
P
.
1�k�log n
k(n − k) n n a(log n)
By the invariance principle and Theorem 2.1.4 in Schmitz [33] (the theorem used is for the
univariate case but the rates immediately carry over to the multivariate situation here), we
also get
240 Handbook of Discrete-Valued Time Series
n
k
T
−1
k
max
S(k, θ
0
) −
S(n, θ
0
)
S(k, θ
0
) −
S(n, θ
0
)
n/ log n�k�n/2
k(n − k) n n
b
d
(log n)
2
= o
P
.
a(log n)
The invariance principle in combination with Horvath [15], Lemma 2.2 (in addition to
analogous arguments as given earlier), implies that
P a(log n) max
n
S(k,
θ
n
)
T
−1
S(k,
θ
n
) − b
d
(log n) t → exp(−e
−t
).
log n�k�n/ log n
k(n − k)
By Assumption A.2, the exact same arguments lead to analogous assertions for k n/2,
which imply the assertion by the asymptotic independence guaranteed by Assumption A.3.
√
n
From this, we also get that
k(n−k)
(S(k, θ
0
) −
n
k
S(n, θ
0
))
= O
P
( log log n), which implies
n
k
T
k
log log n
S(k, θ
0
) −
S(n, θ
0
)
−1
S(k, θ
0
) −
S(n, θ
0
)
k(n − k) n n
n
k
T
k
−
log log n
S(n, θ
0
)
S(k, θ
0
) −
S(n, θ
0
)
S(k, θ
0
) −
−1
k(n − k) n n
n
k
−1/2
−
−1/2
log log n ( )
S(k, θ
0
) −
S(n, θ
0
)
k(n − k) n
= O
P
(log log n)
−1/2
−
−1/2
= o
P
(1),
showing that the statistic with estimated covariance matrix has the same asymptotics.
Proof of Proposition 10.1 Using the subsequence principle, it sufces to prove the following
deterministic result: Let sup
x
G
n
(x) − G(x)→0(as n →∞). Then it holds for G
n
(x
n
) =
0and x
1
is the unique zero of G(x) in the strict sense of B.2 that x
n
→ x
1
. To this end,
assume that this is not the case. Then there exists ε > 0 and a subsequence α(n) such that
|x
α(n)
− x
1
| ε. But then since x
1
is a unique zero in the strict sense, G(x
α(n)
) δ for
some δ > 0. However, this is a contradiction as
G(x
α(n)
)=G(x
α(n)
) − G
α(n)
(x
α(n)
) sup G
α(n)
(x) − G(x)→0.
x
Proof of Theorem 10.2 B.1, B.3, (10.5), and B.5 (i) imply
1
S(k
0
, θ
n
)
−1
S(k
0
, θ
n
) c + o
P
(1);
n
2
241 Detection of Change Points in Discrete-Valued Time Series
hence,
max
w(k/n)
S(k,
θ
n
)
T
−1
S(k,
θ
n
) nw(λ)(c + o
P
(1)) →∞,
1�k�n
n
a(log n)
max
n
S(k,
θ
n
)
T
−1
S(k,
θ
n
)
a(log n)
n
1
(c + o
P
(1)) →∞,
b
d
(log n)
1�k�n
k(n − k) b
d
(log n) λ(1 − λ)
which implies assertion (a). If additionally, B.5 holds, we get the assertion for the
maximum-type statistic analogously if we replace k
0
by ϑn with w(ϑ)> 0. For the
sum-type statistics, we similarly get
n
1
1
w(k/n)
1
S(k,
θ
n
)
T
−1
S(k,
θ
n
) = nc
w(t)g
2
(t) dt + o
P
(1)
→∞,
n n
j=1
0
since the assumptions on w(·) guarantee the existence of
1
2
(t) dt and
1
0
w(t)g
0
w(t)g
2
(t) dt = 0. The second assertion follows by standard arguments since λ is the
unique maximizer of the continuous function g and by Assumptions B.1 and B.5, it holds
sup
1
S(nt,
θ
n
)
T
−1
S(nt,
θ
n
) − F
λ
(θ
1
)
−1
F
λ
(θ
1
) g
2
(t)
→ 0,
0�t�1
n
2
where B.4 guarantees that the limit is not zero. The proofs show that the assertions remain
true if is replaced by
n
under the stated assumptions.
Proof of Theorem 10.3 Assumption A.1 can be obtained by a Taylor expansion, the ergodic
√
theorem, and the
n-consistency of the estimator β
n
. The arguments are given in detail in
Fokianos et al. [11] (Proof of Proposition 3), where by the stationarity of {Z
t
}their arguments
go through in our slightly more general situation for k n/2. For k > n/2, analogous
arguments give the assertion on noting that (with the notation of Fokianos et al. [11])
k n
k
Z
(i)
Z
(
t−
j)
1
π
t
(β)(1 − π
t
(β)) − Z
(i)
Z
(
t−
j)
1
π
t
(β)(1 − π
t
(β))
t−i t−i
n
t=1 t=1
n n
=−
Z
(
t−
i)
i
Z
t
(
−
j)
1
π
t
(β)(1 − π
t
(β)) +
n − k
Z
(
t−
i)
i
Z
t
(
−
j)
1
π
t
(β)(1 − π
t
(β)).
n
t=k+1
t=1
Assumption A.3 (i) follows from the strong invariance principle in Proposition 2 of
Fokianos et al. [11]. Assumption A.3 (ii) does not follow by the same proof techniques as
an autoregressive process in reverse order has different distributional properties than an
autoregressive process. However, if the covariate (Y
t
, Z
t−1
, ... , Z
t−p
)
T
is α-mixing, the same
holds true for the summands of the score process (with the same rate). Since the mixing
property also transfers to the time-inverse process, the strong invariance principle fol-
lows from the invariance principle for mixing processes given by Kuelbs and Philipp [23],
Theorem 4. The mixing assumption then also implies A.3 (iii).
242 Handbook of Discrete-Valued Time Series
Proof of Theorem 10.4 First note that
sup
S
BAR
(k, β) − ES
BAR
(k, β)=o
p
(n) (10.13)
θ∈
uniformly in k k
0
=nλby the uniform ergodic theorem of Ranga Rao [30], Theorem 6.5.
For k > k
0
,itholds
Z
k
Y
k
=
Z
˜
k
Y
˜
k
+
Z
˜
k
R
1
(t) +
Y
˜
k
R
2
(t) + R
1
(t)R
2
(t),
Z
k
π
k
(β) =
Z
˜
k
π
k
(β) + R
1
(k)π
k
(β) =
Z
˜
k
π˜
k
(β) + O(
Z
˜
k
β
T
R
1
(k)) + O(R
1
(k)),
where g(π˜
k
(β)) = β
T
Z
˜
t−1
and the last line follows from the mean value theorem. An
application of the Cauchy–Schwarz inequality together with (iii) and the compactness of
shows that the remainder terms are asymptotically negligible, implying that
sup
S
BAR
(k, β) − S
BAR
(k
0
β) − E
S
ˇ
BAR
(k, β) − ES
BAR
(k
0
, β)=o
p
(n)
β
uniformly in k > k
0
, where
min(k
0
,k)
k
S
ˇ
BAR
(k, β) = Z
t−1
(Y
t
− π
t
(β)) +
Z
˜
t−1
(
Y
˜
t
−˜π
t
(β))
t=1
t=k
0
+1
Together with (10.13), this implies B.1 with
F
t
(λ) = min(t, λ) EZ
0
(Y
1
− π
1
(β)) + (t − λ)
+
E
Z
˜
n−1
(
Y
˜
n
−˜π
n
(β)).
Since F
t
(β) is continuous in β, B.2 follows from (iv). B.5 follows since by denition of β
1
it
holds E
Z
˜
n−1
(
Y
˜
n
−˜π
n
(β
1
)) =−λ/(1 − λ)EZ
0
(Y
1
− π
1
(β
1
)).
Proof of Theorem 10.5 It sufces to show that the assumptions of Theorem 10.1 are fullled.
Assumption A.1 follows for (a) and (b) analogously to the proof of Lemma 1 in Franke
et al. [13]. The invariance principles in A.2 and A.3 then follow from the strong mixing
assumption and the invariance principle of Kuelbs and Philipp [23]. The asymptotic inde-
pendence of A.3 also follows from the mixing condition.
Proof of Theorem 10.6 This is analogous to the proof of Theorem 10.4, where the mean
value theorem is replaced by the Lipschitz assumption, which also implies that |f
γ
(x)|
|f
γ
(0) + x|.
Acknowledgments
Most of this work was done while the rst author was at Karlsruhe Institute of Technology
(KIT), where her position was nanced by the Stifterverband für die Deutsche Wissenschaft
243 Detection of Change Points in Discrete-Valued Time Series
by funds of the Claussen-Simon-trust. The second author was partly supported by the DFG
grant SA 1883/4-1. This work was partly supported by DFG grant KI 1443/2-2.
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