SFB 
823 
Nonparametric tests for 
detecting breaks in the jump 
behaviour of a time-
continuous process 
D
iscussion P
aper 
 
Axel Bücher, Michael Hoffmann, 
Mathias Vetter, Holger Dette 
 
Nr. 41/2014 
 
 
 
 
 
 
 
 
 
Nonparametric tests for detecting breaks in the jump
behaviour of a time-continuous process
Axel Bu¨cher∗, Michael Hoffmann∗, Mathias Vetter† and Holger Dette∗,
Ruhr-Universita¨t Bochum & Philipps-Universita¨t Marburg
December 16, 2014
Abstract
This paper is concerned with tests for changes in the jump behaviour of a
time-continuous process. Based on results on weak convergence of a sequen-
tial empirical tail integral process, asymptotics of certain tests statistics for
breaks in the jump measure of an Ito¯ semimartingale are constructed. When-
ever limiting distributions depend in a complicated way on the unknown jump
measure, empirical quantiles are obtained using a multiplier bootstrap scheme.
An extensive simulation study shows a good performance of our tests in finite
samples.
Keywords and Phrases: Change points; Le´vy measure; multiplier bootstrap; sequen-
tial empirical processes; weak convergence.
AMS Subject Classification: 60F17, 60G51, 62G10.
1 Introduction
Recent years have witnessed a growing interest in statistical tools for high-frequency
observations of time-continuous processes. With a view on finance, the seminal paper
by Delbaen and Schachermayer (1994) suggests to model such a process using Ito¯
semimartingales, say X, which is why most research has focused on the estimation
of (or on tests concerned with) its characteristics. Particular interest has been paid
to integrated volatility or the entire quadratic variation, mostly adapting parametric
procedures based on normal distributions, as the continuous martingale part of an
Itoˆ semimartingale is nothing but a time-changed Brownian motion. For an overview
on methods in this field see the recent monographs by Jacod and Protter (2012) and
Aı¨t-Sahalia and Jacod (2014).
Still less popular is inference on the jump behaviour only, even though empirical
research shows a strong evidence supporting the presence of a jump component
within X; see e.g. Aı¨t-Sahalia and Jacod (2009b) or Aı¨t-Sahalia and Jacod (2009a).
In this work, we will address the question whether the jump behaviour of X is time-
invariant. Corresponding tests, commonly referred to as change point tests, are well
1Ruhr-Universita¨t Bochum, Fakulta¨t fu¨r Mathematik, 44780 Bochum, Germany. E-mail:
axel.buecher@rub.de, holger.dette@rub.de, michael.hoffmann@rub.de
2Philipps-Universita¨t Marburg, Fachbereich Mathematik und Informatik, 35032 Marburg, Ger-
many. E-mail: vetterm@mathematik.uni-marburg.de
1
known in the framework of discrete time series, but have recently also been extended
to time-continuous processes; see e.g. Lee et al. (2006) on changes in the drift or
Iacus and Yoshida (2012) on changes in the volatility function of X. However, to
the best of our knowledge, no procedures are available for detecting breaks in the
jump component.
Suppose that we observe an Ito¯ semimartingale X which admits a decomposition
of the form
Xt = X0 +
∫ t
0
bs ds+
∫ t
0
σs dWs +
∫ t
0
∫
R
u1{|u|≤1}(µ− µ¯)(ds, du)
+
∫ t
0
∫
R
u1{|u|>1}µ(du, dz), (1.1)
where W is a standard Brownian motion, µ is a Poisson random measure on R+×R,
and the predictable compensator µ¯ satisfies µ¯(ds, du) = dsνs(du). As a fairly general
structural assumption, we allow the characteristics of X, i.e. bt, σt and νt, to depend
deterministically on time. Recall that νt can be interpreted as a local Le´vy measure,
such that ∫
R
(1 ∧ |u|2)νt(du) <∞
for each t and νt(A) denotes the average number of jumps that fall into the set A
over a unit time interval.
Now, we assume that we have data from the process in a high-frequency setup.
Precisely, at stage n ∈ N, we are able to observe realizations of the process X at the
equidistant times i∆n for i = 1, . . . , n, where the mesh ∆n → 0, while n∆n → ∞.
In this situation we want to test the null hypothesis that the jump behaviour of
the process is the same for all n observations, i.e. there exists some measure ν such
that νt(dz) = ν(dz) for all t, against alternatives involving the non-constancy of νt.
For instance, one might consider an alternative consisting of one break point, i.e.
there exists some θ0 ∈ (0, 1) and two Le´vy measures ν1, ν2 such that the process
behind the first bnθ0c observations has Le´vy measure ν1 and the remaining n−bnθ0c
observations are taken from a process with Le´vy measure ν2. The restriction to a
deterministic drift and volatility in (1.1) is merely technical here, as it allows to use
empirical process theory for independent observations later. An argument similar to
that in Section 5.3 in Bu¨cher and Vetter (2013) proves that one might as well work
with random coefficients b and σ.
Throughout the work, we will restrict ourselves to positive jumps only. Thus,
for z > 0, let U(z) := ν([z,∞)) denote the tail integral (or spectral measure; see
Ru¨schendorf and Woerner, 2002) associated with ν, which determines the jump
measure uniquely. For `1, `2 ∈ {1, . . . , n} such that `1 < `2, define
U`1:`2(z) :=
1
(`2 − `1 + 1)∆n
`2∑
j=`1
1{∆njX≥z} (z > 0),
with ∆njX := Xj∆n −X(j−1)∆n , which serves as an empirical tail integral based on
the increments ∆n`1X, . . . ,∆
n
`2X. If X is a Le´vy process with a Le´vy measure ν not
changing in time, Figueroa-Lopez (2008) illustrated that U1:n(z) is a suitable esti-
mator for the tail integral U(z) in the sense that, under regularity conditions, Un(z)
2
is L2-consistent for U(z). Following the approach in Inoue (2001), it is therefore
likely that we can base tests for H0 on suitable functionals of the process
Dn(θ, z) := U1:bnθc(z)− U(bnθc+1):n(z),
where θ ∈ [0, 1] and z > 0. Under the null hypothesis, this expression can be
expected to converge to 0 for all θ ∈ [0, 1] and z > 0, whereas under alternatives,
for instance those involving a change at θ0 as described before, Dn(θ0, z) should
converge to an expression which is non-zero.
More precisely, we will consider the following standardized version of Dn, namely
Tn(θ, z) :=
√
n∆nλn(θ)
{
U1:bnθc(z)− U(bnθc+1):n(z)
}
(1.2)
for θ ∈ [0, 1] and z > 0, where λn(θ) =
bnθc
n
n−bnθc
n . An appropriate functional
allowing to test the hypothesis of a constant Le´vy measure is for instance given by
a Kolmogorov-Smirnov statistic of the form
T (ε)n := sup
θ∈[0,1]
sup
z≥ε
|Tn(θ, z)|, (ε > 0). (1.3)
The null hypothesis of no change in the Le´vy measure is rejected for large values
of T (ε)n . The restriction to jumps larger than ε is important, since there might be
infinitely many of arbitrary small size.
The limiting distribution of the previously mentioned test statistic will turn out
to depend in a complicated way on the unknown Le´vy measure ν. Therefore, cor-
responding quantiles are not easily accessible and must be obtained by suitable
bootstrap approximations. Following related ideas for detecting breaks within mul-
tivariate empirical distribution functions (Inoue, 2001), we opt for using empirical
counterparts based on a multiplier bootstrap scheme, frequently also referred to as
wild or weighted bootstrap. The approach essentially consists of multiplying each
indicator within the respective empirical tail integrals with an additional, indepen-
dent and standardized multiplier. The underlying empirical process theory is for
instance summarized in the monograph Kosorok (2008).
The remaining part of this paper is organized as follows: the derivation of a
functional weak convergence result for the process Tn under the null hypothesis
is the content of Section 2. The asymptotic properties of T (ε)n can then easily be
derived from the continuous mapping theorem. Section 3 is concerned with the
approximation of the limiting distribution using the previously described multiplier
bootstrap scheme. In Section 4, we discuss the formal derivation of several tests
for a time-homogeneous jump behaviour, whereas an extensive simulation study is
presented in Section 5. All proofs are deferred to the Appendix, which is Section 6.
2 Functional weak convergence of the sequential empir-
ical tail integral
In this section, we derive a functional weak convergence result for the process Tn
defined in (1.2). For that purpose, we have to introduce an appropriate function
space. We set A := [0, 1] × (0,∞) and let B∞(A) denote the space of all functions
f : A → R which are bounded on every set A ⊂ A for which the projection onto
the second coordinate, p2(A) = {z ∈ (0,∞) | ∃θ ∈ [0, 1] such that (θ, z) ∈ A}, is
3
bounded away from 0. Moreover, for k ∈ N, we define Ak := [0, 1] × [k−1,∞) ⊂ A,
and, for f, g ∈ B∞(A), we set
d(f, g) :=
∞∑
k=1
2−k(‖f − g‖Ak ∧ 1),
where ‖f − g‖Ak = sup{|f(x) − g(x)| : x ∈ Ak}. Note that d defines a metric on
B∞(A) which induces the topology of uniform convergence on all sets A such that
its projection p2(A) is bounded away from 0, i.e. a sequence of functions converges
with respect to d if and only if it converges uniformly on each Ak (Van der Vaart
and Wellner, 1996, Chapter 1.6).
Furthermore to establish our results on weak convergence under the null hypoth-
esis, we impose the following conditions.
Condition 2.1. X is an Ito¯ semimartingale with the representation in (1.1) such
that
(a) The drift bt and the volatility σt are ca`gla`d, bounded and deterministic.
(b) There exists some Le´vy measure ν such that νt ≡ ν for all t ∈ R+.
(c) X has only positive jumps, that is, the jump measure ν is supported on (0,∞).
(d) ν is absolutely continuous with respect to the Lebesgue measure on (0,∞).
Its density h = dν/dλ, called Le´vy density, is differentiable with derivative h′
and satisfies
‖h‖Mk + ‖h
′‖Mk <∞
for all k ∈ N with Mk := [k−1,∞).
The next lemma is essential for the weak convergence results. Similar statements
can be found in Figueroa-Lopez and Houdre (2009), with slightly stronger assump-
tions on h, and in Bu¨cher and Vetter (2013) in the bivariate case.
Lemma 2.2. Let X be an Ito¯ semimartingale that satisfies Condition 2.1. Let
further be δ > 0 fixed. If X0 = 0, then there exist constants K = K(δ) > 0 and
t0 = t0(δ) > 0, depending on the bounds on the characteristics in Condition 2.1(a),
such that the inequality
|P(Xt ∈ [z,∞))− tν([z,∞))| < Kt2
holds for all z ≥ δ and all 0 < t < t0.
Remark 2.3. If X is an Ito¯ semimartingale satisfying Condition 2.1, then Lem-
ma 2.2 implies immediately that we have
sup
s∈R+
|P(Xs+t −Xs ∈ [z,∞))− tν([z,∞))| ≤ Kt2
as well. To see this note that, for each fixed s ∈ R+, the Ito¯ semimartingale
(Y (s)t )t∈R+ with Y
(s)
t := Xs+t − Xs satisfies Condition 2.1, and its characteristics
have the same bounds as the characteristics of X.
4
The limiting behaviour of the process Tn can be deduced from the next theorem,
which is a result for weak convergence of a sequential empirical tail integral process.
For θ ∈ [0, 1] and z > 0 set
Un(θ, z) :=
bnθc
n
U1:bnθc(z) =
1
kn
bnθc∑
j=1
1{∆njX≥z}, (2.1)
where kn := n∆n and denote its standardized version by
Gn(θ, z) :=
√
kn{Un(θ, z)− EUn(θ, z)}. (2.2)
Obviously, the sample paths of Un(θ, z) are elements of B∞(A).
Theorem 2.4. Let X be an Ito¯ semimartingale that satisfies Condition 2.1. Fur-
thermore, assume that the observation scheme has the properties:
∆n → 0, n∆n = kn →∞.
Then, Gn  G in (B∞(A), d), where G is a tight mean zero Gaussian process with
covariance
H(θ1, z1; θ2, z2) := E[G(θ1, z1)G(θ2, z2)] = (θ1 ∧ θ2)× ν([z1 ∨ z2,∞))
for (θ1, z1), (θ2, z2) ∈ A. The sample paths of G are almost surely uniformly contin-
uous on each Ak (k ∈ N) with respect to the semimetric
ρ(θ1, z1; θ2, z2) :=
{
(θ1 ∧ θ2)ν([z1 ∧ z2, z1 ∨ z2)) + |θ1 − θ2| ν([zI(θ1,θ2),∞))
} 1
2
with I(θ1, θ2) := 1 + 1{θ1≤θ2}.
Note that we have centered Un(θ, z) around its expectation in (2.2). In most
applications, however, we are interested in estimating functionals of the jump mea-
sure, and according to Lemma 2.2 we need stronger conditions then. Precisely, we
consider the process
G˜n(θ, z) :=
√
kn{Un(θ, z)− θν([z,∞))}
and get, as an immediate consequence of the previous two results, the following
sequential generalization of Theorem 4.2 of Bu¨cher and Vetter (2013).
Corollary 2.5. Let X be an Ito¯ semimartingale that satisfies Condition 2.1. If the
observation scheme meets the conditions
∆n → 0, n∆n = kn →∞,
√
kn∆n → 0,
then G˜n  G in (B∞(A), d), where G denotes the Gaussian process from Theo-
rem 2.4.
A further consequence of Theorem 2.4 is the desired weak convergence of the
process Tn, which was defined in (1.2), under the null hypothesis.
5
Theorem 2.6. Suppose the assumptions of Corollary 2.5 are satisfied. Then, the
process Tn defined in (1.2) converges weakly to T in (B∞(A), d), where
T(θ, z) = G(θ, z)− θG(1, z)
for (θ, z) ∈ A, and where G denotes the limit process in Theorem 2.4. T is a tight
mean zero Gaussian process with covariance function
Hˆ(θ1, z1; θ2, z2) := E{T(θ1, z1)T(θ2, z2)} = {(θ1 ∧ θ2)− θ1θ2}ν([z1 ∨ z2,∞)).
Using the continuous mapping theorem, we are now able to derive the weak conver-
gence of various statistics allowing for the detection of breaks in the jump behaviour.
The following corollary treats the statistic T (ε)n defined in (1.3).
Corollary 2.7. Under the assumptions of Corollary 2.5 we have, for each ε > 0,
T (ε)n  T
(ε) := sup
0≤θ≤1
sup
z≥ε
|T(θ, z)|,
where T is the limit process defined in Theorem 2.6.
The covariance function of the limit process in Theorem 2.6 depends on the Le´vy
measure of the underlying process, which is usually unknown in applications. If one
only wants to detect changes in the tail integral of the Le´vy measure at a fixed point
z0, the following proposition deals with the simple transformation
V(z0)n (θ) :=
Tn(θ, z0)
√
U1:n(z0)
1{U1:n(z0)>0}
of Tn which yields a pivotal limiting distribution.
Proposition 2.8. Let X be an Ito¯ semimartingale that satisfies Condition 2.1.
Moreover, let z0 > 0 be a real number with ν([z0,∞)) > 0 and suppose that the
underlying observation scheme meets the assumptions from Corollary 2.5. Then,
V(z0)n  B in `∞([0, 1]), where B denotes a standard Brownian bridge. As a conse-
quence,
V (z0)n := sup
θ∈[0,1]
|V(z0)n (θ)| sup
θ∈[0,1]
|B(θ)|,
the limiting distribution being also known as the Kolmogorov-Smirnov distribution.
Remark 2.9. We have derived the previous results under somewhat simplified
assumptions on the observation scheme in order to keep the presentation rather
simple. A more realistic setting could involve additional microstructure noise effects
or might rely on non-equidistant data. In both cases, standard techniques still yield
similar results.
For example, in case of noisy observations, Vetter (2014) has shown that a partic-
ular de-noising technique allows for virtually the same results on weak convergence
as for the plain Un(θ, z) in the case without noise. For non-equidistant data, the
limiting covariance functions H and Hˆ in general depend on the sampling scheme.
The latter effect is well-known from high-frequency statistics in the case of volatility
estimation; see e.g. Mykland and Zhang (2012).
6
3 Bootstrap approximations for the sequential empiri-
cal tail integral
We have seen in Corollary 2.7 that the distribution of the limit T of the process
Tn depends in a complicated way on the unknown Le´vy measure of the underlying
process. However, we need the quantiles of T or at least good approximations for
them to obtain a feasible test procedure. Typically, one uses resampling methods to
solve this problem.
Probably the most natural way to do so is to use U1:n(z) in order to obtain an
estimator νˆn for the Le´vy measure first, and to draw a large number of independent
samples of an Ito¯ semimartingale with Le´vy measure νˆn then, possibly with estimates
for drift and volatility as well. Based on each sample, one might then compute the
test statistic Tn, and by doing so one obtains empirical quantiles for T.
However, from a computational side, such a method is computationally expensive
since one has to generate independent Ito¯ semimartingales for each stage within
the bootstrap algorithm. Therefore we have decided to work with an alternative
bootstrap method based on multipliers, where one only needs to generate n i.i.d.
random variables with mean zero and variance one (see also Inoue, 2001, who used
a similar approach in the context of empirical processes).
Precisely, the situation now is as follows: The bootstrapped processes, say Yˆn =
Yˆn(X1, . . . , Xn, ξ1, . . . , ξn), will depend on some random variables X1, . . . , Xn and on
some random weights ξ1, . . . , ξn. The X1, . . . , Xn, that we consider as collected data,
are defined on a probability space (ΩX ,AX ,PX). The random weights ξ1, . . . , ξn
are defined on a distinct probability space (Ωξ,Aξ,Pξ). Thus, the bootstrapped
processes live on the product space (Ω,A,P) := (ΩX ,AX ,PX) ⊗ (Ωξ,Aξ,Pξ). The
following notion of conditional weak convergence will be essential. It can be found
in Kosorok (2008) on pp. 19–20.
Definition 3.1. Let Yˆn = Yˆn(X1, . . . , Xn; ξ1, . . . , ξn) : (Ω,A,P) → D be a (boot-
strapped) element in some metric space D depending on some random variables
X1, . . . , Xn and some random weights ξ1, . . . , ξn. Moreover, let Y be a tight, Borel
measurable map into D. Then Yˆn converges weakly to Y conditional on the data
X1, X2, . . . in probability, notationally Yˆn ξ Y , if and only if
(a) sup
f∈BL1(D)
|Eξf(Yˆn)− Ef(Y )|
P∗
→ 0,
(b) Eξf(Yˆn)∗ − Eξf(Yˆn)∗
P∗
→ 0 for all f ∈ BL1(D).
Here, Eξ denotes the conditional expectation over the weights ξ given the data
X1, . . . , Xn, whereas BL1(D) is the space of all real-valued Lipschitz continuous func-
tions f on D with sup-norm ‖f‖∞ ≤ 1 and Lipschitz constant 1. Moreover, f(Yˆn)∗
and f(Yˆn)∗ denote a minimal measurable majorant and a maximal measurable mi-
norant with respect to the joint data (including the weights ξ), respectively.
Remark 3.2.
(i) Note that we do not use a measurable majorant or minorant in item (a) of the
definition. This is justified through the fact that, in this work, all expressions
f(Yˆn), with a bootstrapped statistic Yˆn and a Lipschitz continuous function f ,
are measurable functions of the random weights.
7
(ii) Note that the implication “(ii)⇒ (i)” in the proof of Theorem 2.9.6 in Van der
Vaart and Wellner (1996) shows that, in general, conditional weak convergence
 ξ implies unconditional weak convergence  with respect to the product
measure P.
Throughout this paper we denote by
Gˆn = Gˆn(θ, z) = Gˆn(X∆n , . . . , Xn∆n , ξ1, . . . , ξn; θ, z)
the bootstrap approximation which is defined by
Gˆn(θ, z) :=
1
n
√
kn
bnθc∑
j=1
n∑
i=1
ξj{1{∆njX≥z} − 1{∆ni X≥z}}
=
1
√
kn
bnθc∑
j=1
ξj{1{∆njX≥z} − ηn(z)},
where ηn(z) = n−1
∑n
i=1 1{∆ni X≥z}. The following theorem establishes conditional
weak convergence of this bootstrap approximation for the sequential empirical tail
integral process Gn.
Theorem 3.3. Let X be an Ito¯ semimartingale that satisfies Condition 2.1 and
assume that the observation scheme meets the conditions from Theorem 2.4. Fur-
thermore, let (ξj)j∈N be independent and identically distributed random variables
with mean 0 and variance 1, defined on a distinct probability space as described
above. Then,
Gˆn ξ G
in (B∞(A), d), where G denotes the limiting process of Theorem 2.4.
Theorem 3.3 suggests to define the following bootstrapped counterparts of the
process Tn defined in equation (1.2):
Tˆn(θ, z) := Tˆn(X∆n , . . . , Xn∆n ; ξ1, . . . , ξn; θ, z) := Gˆn(θ, z)−
bnθc
n
Gˆn(1, z)
=
√
n∆n
bnθc
n
n− bnθc
n
[
1
bnθc∆n
bnθc∑
j=1
ξj{1{∆njX≥z} − ηn(z)}
−
1
(n− bnθc)∆n
n∑
j=bnθc+1
ξj{1{∆njX≥z} − ηn(z)}
]
,
The following result establishes consistency of Tn in the sense of Definition 3.1.
Theorem 3.4. Under the conditions and notations of Theorem 3.3, we have
Tˆn ξ T
in (B∞(A), d), with T defined in Theorem 2.6.
The distribution of the Kolmogorov-Smirnov-type test statistic T (ε)n defined in
(1.3) can be approximated with the bootstrap statistics investigated in the following
corollary. It can be proved by a simple application of Proposition 10.7 in Kosorok
(2008) on an appropriate `∞(Ak).
8
Corollary 3.5. Under the assumptions of Theorem 3.3 we have, for each ε > 0,
Tˆ (ε)n := sup
0≤θ≤1
sup
z≥ε
|Tˆn(θ, z)| ξ sup
0≤θ≤1
sup
z≥ε
|T(θ, z)| =: T (ε).
4 The testing procedures
4.1 Hypotheses
In order to derive a test procedure which utilizes the results on weak convergence
from the previous two sections, we have to formulate our hypotheses first. Under
the null hypothesis the jump behaviour of the process is constant. More precisely,
this means the following:
H0: We observe an Ito¯ semimartingale as in equation (1.1) with characteristic
triplet (bt, σt, ν) that satisfies Condition 2.1.
We want to test this hypothesis versus the alternative that there is exactly one
change in the jump behaviour. This means in detail:
H1: There exists some θ0 ∈ (0, 1) and two Le´vy measures ν1 6= ν2 satisfying Con-
dition 2.1(c) and (d) such that, at stage n, we observe an Ito¯ semimartingale
X = X(n) with characteristic triplet (b(n)t , σ
(n)
t , ν
(n)
t ) such that
ν(n)t = 1{t<bnθ0c∆n}ν1 + 1{t≥bnθ0c∆n}ν2
Furthermore, b(n)t and σ
(n)
t satisfy Condition 2.1(a) and are uniformly bounded
in n ∈ N and t > 0.
The corresponding alternative for a fixed z0 > 0 is then given through:
H(z0)1 : We have the situation from H1, but with ν1([z0,∞)) 6= ν2([z0,∞)) and
ν1([z0,∞)) ∨ ν2([z0,∞)) > 0.
4.2 The tests and their asymptotic properties
In the sequel, let B ∈ N be some large number and let (ξ(b))b=1,...,B denote in-
dependent vectors of i.i.d. random variables, ξ(b) := (ξ(b)j )j=1,...,n, with mean zero
and variance one. As before, we assume that these random variables are generated
independently from the original data. We denote by Tˆn,ξ(b) or Tˆ
(ε)
n,ξ(b)
the particu-
lar statistics calculated with respect to the data and the b-th bootstrap multipliers
ξ(b)1 , . . . , ξ
(b)
n . For a given level α ∈ (0, 1), we consider the following test procedures:
KSCP-Test1. Reject H0 in favor of H
(z0)
1 , if V
(z0)
n ≥ qK1−α, where V
(z0)
n is defined in
Proposition 2.8 and where qK1−α denotes the 1−α quantile of the Kolmogorov-
Smirnov-(KS-)distribution, that is the distribution of K = sups∈[0,1] |B(s)|
with a standard Brownian bridge B.
KSCP-Test2. Reject H0 in favor of H
(z0)
1 , if
W (z0)n := sup
θ∈[0,1]
|Tn(θ, z0)| ≥ qˆ
(B)
1−α(W
(z0)
n ),
where qˆ(B)1−α(W
(z0)
n ) denotes the (1− α)-sample quantile of Wˆ
(z0)
n,ξ(1)
, . . . , Wˆ (z0)
n,ξ(B)
,
and where Wˆ (z0)
n,ξ(b)
:= supθ∈[0,1] |Tˆn,ξ(b)(θ, z0)|.
9
CP-Test. Choose an appropriate small ε > 0 and reject H0 in favor of H1, if
T (ε)n ≥ qˆ
(B)
1−α(T
(ε)
n ),
where qˆ(B)1−α(T
(ε)
n ) denotes the (1− α)-sample quantile of Tˆ
(ε)
n,ξ(1)
, . . . , Tˆ (ε)
n,ξ(B)
.
Since ε > 0 has to be chosen prior to an application of the CP-Test, we can only
detect changes in the jumps larger than ε. From a theoretical point of view this is
not entirely satisfactory, since one is interested in distinguishing arbitrary changes in
the jump behaviour. On the other hand, in most applications only the larger jumps
are of particular interest, and at least the size of ∆n provides a natural bound to
disentangle jumps from volatility. Thus, a practitioner can choose a minimum jump
size ε first, and use the CP-Test to decide whether there is a change in the jumps
larger than ε.
The following proposition shows that three aforementioned tests keep the asymp-
totic level α under the null hypothesis.
Proposition 4.1. Suppose the sampling scheme meets the conditions of Corol-
lary 2.5. Then, KSCP-Test1, KSCP-Test2 and CP-Test are asymptotic level α tests
for H0 in the sense that, under H0, for all α ∈ (0, 1),
lim
n→∞
P(V (z0)n ≥ q
K
1−α) = α, lim
B→∞
lim
n→∞
P{W (z0)n ≥ qˆ
(B)
1−α(W
(z0)
n )} = α,
and
lim
B→∞
lim
n→∞
P{T (ε)n ≥ qˆ
(B)
1−α(T
(ε)
n )} = α,
for all ε > 0 such that ν([ε,∞)) > 0.
The next proposition shows that the preceding tests are consistent under the fixed
alternatives defined in Section 4.1. For simplicity, we only consider alternatives
involving one change point, even though the results may be extended to alternatives
involving multiple breaks or even continuous changes.
Proposition 4.2. Suppose the sampling scheme meets the conditions of Corol-
lary 2.5. Then, KSCP-Test1, KSCP-Test2 and CP-Test are consistent in the fol-
lowing sense: under H(z0)1 , for all α ∈ (0, 1) and all B ∈ N, we have
lim
n→∞
P(V (z0)n ≥ q
K
1−α) = 1 and limn→∞
P(W (z0)n ≥ qˆ
(B)
1−α(W
(z0)
n )) = 1.
Under H1, there exists an ε > 0 such that, for all α ∈ (0, 1) and all B ∈ N,
lim
n→∞
P{T (ε)n ≥ qˆ
(B)
1−α(T
(ε)
n )} = 1.
4.3 Locating the change point
Let us finally discuss how to construct suitable estimators for the location of the
change point. We begin with a useful proposition.
Proposition 4.3. Suppose the sampling scheme meets the conditions of Corol-
lary 2.5. Then, under H1, (θ, z) 7→ k
−1/2
n Tn(θ, z) converges in B∞(A) to the function
T (θ, z) :=
{
θ(1− θ0){ν1(z)− ν2(z)} if θ ≤ θ0
θ0(1− θ){ν1(z)− ν2(z)} if θ ≥ θ0,
10
in outer probability, with ν1(z) := ν1([z,∞)) and ν2(z) := ν2([z,∞)).
Since θ 7→ T (θ, z) attains its maximum in θ0, natural estimators for the position
of the change point are therefore given by
θˆ(ε)n := arg maxθ∈[0,1] sup
z≥ε
|Tn(θ, z)|
for the test problem H0 versus H1 and
θ˜(z0)n := arg maxθ∈[0,1] |Tn(θ, z0)|
in the setup H(z0)0 versus H
(z0)
1 . The next proposition states that these estimators
are consistent.
Proposition 4.4. Suppose the sampling scheme meets the conditions of Corol-
lary 2.5. If H1 is true, there exists an ε > 0 such that θˆ
(ε)
n = θ0 + oP(1) as n→∞.
In the special case of H(z0)1 , we have θ˜
(z0)
n = θ0 + oP(1).
5 Finite-sample performance
In this section, we present results of a large scale Monte Carlo simulation study,
assessing the finite-sample performance of the proposed test statistics for detecting
breaks in the Le´vy measure. Moreover, under the alternative of one single break, we
show results on the performance of the estimator for the break point from Section 4.3.
The experimental design of the study is as follows.
• We consider five different choices for the number of trading days, namely kn =
50, 75, 100, 150, 250, and corresponding frequencies ∆−1n = 450, 300, 225, 150, 90.
Note that n = kn∆n = 22, 500 for any of these choices.
• We consider two different models for the drift and the volatility : either, we
set bt = σt ≡ 0 or bt = σt ≡ 1, resulting in a pure jump process and a process
including a continuous component, respectively.
• We consider one parametric model for the tail integral, namely
Uβ(z) = νβ([z,∞)) =
(
β
piz
)1/2
, β > 0 (5.1)
(which yields a 1/2-stable subordinator in the case of bt = σt ≡ 0). For the
parameter β, we consider 51 different choices, that is β = 1 + 2j/25, with
j ∈ 0, . . . , 50, ranging from β = 1 to β = 5.
• We consider models with one single break in the tail integral at 50 different
break points, ranging form θ0 = 0 to θ0 = 0.98 (note that θ0 = 0 corresponds
to the null hypothesis). The tail integrals before and after the break point are
chosen from the previous parametric model.
The target values of our study are, on the one hand, the empirical rejection level
of the tests and, on the other hand, the empirical distribution of the estimators
for the change point θ0. To assess these target values, any combination of the
11
previously described settings was run 1, 000 times, with the bootstrap tests being
based on B = 250 bootstrap replications. The Ito¯ semimartingales were simulated
by a straight-forward modification of Algorithm 6.13 in Cont and Tankov (2004),
where, under alternatives involving one break point, we simply merged two paths of
independent semimartingales together.
The simulation results under these settings are partially reported in Table 1 and 2
(for the null hypothesis) and in Figures 1–4 (for various alternatives). More precisely,
Table 1 and 2 contain simulated rejection rates under the null hypothesis for various
values of kn and z0 in the KSCP-tests, for the pure jump subordinator (Table 1)
and for the process involving a continuous component (Table 2). For the CP-tests,
the suprema over z ∈ [ε,∞) were approximated by taking a maximum over a finite
grid M : we used the grids M = {j · 0.05 | j = 1, . . . , 200} in the pure jump case,
resulting in ε = 0.05, and M = {(2 + j · 0.5)
√
∆n | j = 0, . . . , 196} in the case
bt = σt ≡ 1, resulting in ε = 2
√
∆n. In the latter case, we chose ε depending on√
∆n since jumps of smaller size may be dominated by the Brownian component
resulting in a loss of efficiency of the CP-test (see also the results in Figure 3 below).
The results in the two tables reveal a rather precise approximation of the nominal
level of the tests (α = 5%) in all scenarios. In general, KSCP-Test 1 turns out to be
slightly more conservative than KSCP-Test 2.
The results presented in Figure 1 consider the CP-test for alternatives involving
one fixed break point at θ0 = 0.5 and a varying height of the jump size, as measured
through the value of β in (5.1). In contrast to the results in Tables 1 and 2, due
to computational reasons, we subsequently used smaller grids M = {j · 0.2 | j =
1, . . . , 20} for the case bt = σt ≡ 0, resulting in ε = 0.2, and M = {2.5 ·
√
∆n · j |
j = 1, . . . , 20} for the case bt = σt ≡ 1, resulting in ε = 2.5
√
∆n. The left plot
is based on the pure jump process (bt = σt ≡ 0), whereas the right one is based
on bt = σt ≡ 1. The dashed red line indicates the nominal level of α = 5%. We
observe that the rejection rate of the test is increasing in β (as to be expected)
and in kn. The latter can be explained by the fact that kn represents the effective
sample size (interpretable as the number of trading days). Finally, the rejection rates
turn out to be higher when no continuous component is involved in the underlying
semimartingale.
The next two graphics in Figure 2 show the rate of rejection of the CP-Test under
alternatives involving one break point from β = 1 to β = 2.5 within the model in
kn CP-Test Pointwise Tests z0 = 0.1 z0 = 0.15 z0 = 0.25 z0 = 1 z0 = 2
50 0.06 KSCP-Test 1 0.048 0.056 0.047 0.035 0.033
KSCP-Test 2 0.060 0.067 0.060 0.050 0.048
75 0.054 KSCP-Test 1 0.034 0.044 0.045 0.041 0.046
KSCP-Test 2 0.045 0.059 0.061 0.058 0.060
100 0.06 KSCP-Test 1 0.047 0.044 0.042 0.044 0.042
KSCP-Test 2 0.060 0.056 0.058 0.062 0.056
150 0.06 KSCP-Test 1 0.049 0.056 0.049 0.040 0.042
KSCP-Test 2 0.065 0.064 0.065 0.059 0.061
250 0.07 KSCP-Test 1 0.046 0.042 0.046 0.055 0.050
KSCP-Test 2 0.054 0.048 0.059 0.072 0.060
Table 1: Test procedures under H0. Simulated relative frequency of rejections
in the application of the KSCP-Test 1, the KSCP-Test 2 and the CP-Test to 1000
pure jump subordinator data vectors under the null hypothesis.
12
kn CP-Test Pointwise Tests z0 = 2
√
∆n z0 = 3.5
√
∆n z0 = 6.5
√
∆n z0 = 7
√
∆n
50 0.049 KSCP-Test 1 0.032 0.036 0.035 0.031
KSCP-Test 2 0.049 0.051 0.049 0.050
75 0.050 KSCP-Test 1 0.042 0.039 0.039 0.032
KSCP-Test 2 0.050 0.057 0.051 0.053
100 0.051 KSCP-Test 1 0.039 0.040 0.037 0.038
KSCP-Test 2 0.051 0.054 0.049 0.057
150 0.057 KSCP-Test 1 0.038 0.045 0.034 0.039
KSCP-Test 2 0.057 0.057 0.053 0.052
250 0.049 KSCP-Test 1 0.031 0.035 0.042 0.030
KSCP-Test 2 0.049 0.048 0.053 0.042
Table 2: Test procedures under H0. Simulated relative frequency of rejections
in the application of the KSCP-Test 1, the KSCP-Test 2 and the CP-Test to 1000
subordinator data vectors plus a drift b = 1 and plus a Brownian motion under H0.
1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Factor of Jump Size
Reje
ction
 Rat
e
k= 250
k= 150
k= 100
k= 75
k= 50
1 2 3 4 50
.0
0.2
0.4
0.6
0.8
1.0
Factor of Jump Size
Rejec
tion R
ate
Figure 1: Rejection rate of the CP-Test for pure jump subordinator data (on the
left-hand side) and a subordinator plus a drift and a Brownian motion (on the
right-hand side). β changes from 1 to the factor of jump size.
(5.1) for varying locations of the change point θ0 ∈ (0, 1). Again, the left and right
plots correspond to bt = σt ≡ 0 and ≡ 1, respectively. Additionally to the general
conclusions drawn from the results in Figure 1, we observe that break points can be
detected best if θ0 = 1/2, and that the rejection rates are symmetric around that
point.
Figure 3 shows the rejection rates of the KSCP-Test 1 and 2, evaluated at different
points z0, for one fixed alternative model involving a single change from β = 1 to
β = 2.5 at the point θ0 = 1/2. The curves in the left plot are based on a pure jump
process. We can see that the rejection rates are decreasing in z0, explainable by the
fact that there are only very few large jumps both for β = 1 and for β = 2.5. In the
right plot, involving drift and volatility (bt = σt ≡ 1), we observe a maximal value of
the rejection rates that is increasing in the number of trading days, kn. For values
of z0 smaller than this maximum, the contribution of the Brownian component
(an independent normally distributed term with variance ∆n within each increment
∆njX) predominates the jumps of that size and results in a decrease of the rejection
rate.
13
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Break point θ
0
Reje
ction
 Rat
e
k= 250
k= 150
k= 100
k= 75
k= 50
0.0 0.2 0.4 0.6 0.8 1.00
.0
0.2
0.4
0.6
0.8
1.0
Break point θ0
Rejec
tion R
ate
Figure 2: Rejection rate of the CP-Test for pure jump subordinator data (left panel)
and a subordinator with a drift plus a Brownian motion (right panel) for different
change point locations.
0 2 4 6 8 100
.0
0.2
0.4
0.6
0.8
1.0
z0
Rejec
tion R
ate
KSCP1KSCP2 k= 250k= 100k= 50
0 2 4 6 8 100
.0
0.2
0.4
0.6
0.8
1.0
z0
Rejec
tion R
ate
Figure 3: Rejection rates of the KSCP-Test 1 and 2 for different z0. Left panel: pure
jump subordinator, right panel: subordinator with a drift plus Brownian motion.
Finally, in Figure 4, we depict box plots for the estimators θ˜(z0)n and θˆ
(ε)
n of the
change point for certain values of z0 and for M as specified in the case of Tables 1
and 2. The results are based on two models, involving a change in β from 1 to 4
at time point θ0 = 0.5 (left panel) and θ0 = 0.75 (right panel) for kn = 250 and
∆−1n = 90, and with bt = σt ≡ 1. We observe a reasonable approximation of the true
value (indicated by the red line) with more accurate approximations for θ0 = 0.5.
For θ0 = 0.75, the distribution of the estimator is skewed, giving more weight to
the left tail directing to θ0 = 0.5. This might be explained by the fact that the
distribution of the argmax absolute value of a tight-down stochastic process indexed
by θ ∈ [0, 1] gives very small weight to the boundaries of the unit interval. Moreover,
as for the results presented in the right plot of Figure 3, the plots in Figure 4 reveal
that the estimator θ˜(z0)n behaves best for an intermediate choice of z0. Results for
bt = σt ≡ 0 are not depicted for the sake of brevity, since they do not transfer any
14
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ll
2 ∆n 3 ∆n 4 ∆n 10 ∆n 20 ∆n CP0
.0
0.2
0.4
0.6
0.8
1.0
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l
l
2 ∆n 3 ∆n 4 ∆n 10 ∆n 20 ∆n CP0
.0
0.2
0.4
0.6
0.8
1.0
Figure 4: Box plots for the estimators θ˜(z0)n and θˆ
(ε)
n based on a subordinator with
a drift plus Brownian motion and a change from β = 1 to β = 4 at θ0 = 0.5 (left
panel) and θ0 = 0.75 (right panel). The first five box plots in each panel correspond
to five different choices of z0.
additional insight.
6 Appendix
6.1 Proof of Lemma 2.2
Let ε < (δ/6 ∧ 1) and pick a smooth cut-off function cε : R→ R satisfying
1[−ε/2,ε/2](u) ≤ cε(u) ≤ 1[−ε,ε](u).
We also define the function c¯ε via c¯ε(u) = 1 − cε(u). We use c¯ to define the
“large” jumps of the process, that means, there exist independent processes Xε
and X˜ε such that X =d Xε + X˜ε where X˜ε is a compound Poisson process with
intensity λε =
∫
c¯ε(u)ν(du) and jump distribution ρε(du) = c¯ε(u)ν(du)/λε. See e.g.
Figueroa-Lopez and Houdre (2009). Accordingly, Xε is an Ito¯ semimartingale with
characteristics (bεs, σ
2
s , cε(u)ν(du)), where we set b

s = bs −
∫
1{|u|>1}uc¯ε(u)ν(du).
Since our result is a distributional one only, it is possible to work with this par-
ticular representation of X in the following. Call N εt the number of jumps of X˜
ε up
to time t. Define f(x) = 1{x≥z}. Using the law of total expectation we have
E[f(Xt)] =
∞∑
k=0
e−λεt
(λεt)k
k!
E[f(Xt)|N εt = k]
= e−λεtE[f(Xεt )] + e
−λεtλεtE[f(Xεt + ξ1)]
+
∞∑
k=2
e−λεt
(λεt)k
k!
E
[
f
(
Xεt +
k∑
`=1
ξ`
)]
, (6.1)
where the random variables ξ` are i.i.d. with distribution ρε.
For the first summand on the right of the last display, i.e. the case of no large
jumps, we discuss drift, volatility and small jumps separately. For that purpose, we
15
write Xεt = B
ε
t +Qt+Y
ε
t where B
ε
t =
∫ t
0 b

sds, Qt =
∫ t
0 σsdWs and where Y
ε
t is a pure
jump Le´vy martingale with jump measure cε(u)ν(du). By the triangle inequality
e−λεtE[f(Xεt )] ≤ P (X
ε
t ≥ δ) ≤ P (B
ε
t ≥ δ/3) + P (Ct ≥ δ/3) + P (Y
ε
t ≥ δ/3).
Let us show that the right-hand side of this display can be bounded by Kt2 for all
0 < t < t0, with constants K = K(δ) and t0 = t0(δ). Regarding the summand
P (Y εt ≥ δ/3), we can use equation (3.3) in Figueroa-Lopez and Houdre (2009)
applied to a pure jump Le´vy process. Following their result, ε < δ/6 ensures the
existence of K and t0, both depending on δ only, such that
P(Y εt ≥ δ/3) < Kt
2 (6.2)
for all 0 < t < t0. Since b and σ are bounded, we further have E[|Bεt |
r] ≤ Ktr
and E[|Qt|r] ≤ Ktr/2 for an arbitrary integer r and with K depending on δ again.
Markov’s inequality then yields bounds similar to (6.2) when applied to the processes
involving drift, Bεt , and volatility, Ct.
Also, for the sum over k on the right-hand side of (6.1), we have
e−λεt
∞∑
k=2
(λεt)k
k!
< Kt2.
It therefore remains to focus on E[f(Xεt + ξ1)]. As a consequence of Condi-
tion 2.1(d), and observing that the distribution of ξ1 is h(u)c¯ε(u)du/λε with the
Le´vy density h, it follows that
g(x) = E[f(x+ ξ1)] = P(x+ ξ1 ≥ z)
is twice continuously differentiable with bounded derivatives. Using independence
of Xε and ξ1, it is sufficient to discuss E[g(Xεt )], for which we can use Itoˆ formula
now: for arbitrary Y we have
g(Yt) = g(Y0) +
∫ t
0
g′(Ys−)dYs +
1
2
∫ t
0
g′′(Ys−)d[Y, Y ]
c
s
+
∑
0<s≤t
(
g(Ys)− g(Ys−)− g
′(Ys−)∆Ys)
)
, (6.3)
where [Y, Y ]cs denotes the quadratic variation and ∆Ys is the jump size at time s.
Plugging in Xε for Y we discuss each of the four summands in (6.3) separately: first,
u ≥ z implies c¯ε(u) = 1 by definition of ε. Thus, with Xε0 = 0
g(Xε0) = P(ξ1 ≥ z) =
1
λε
∫
1{u≥z}h(u)c¯ε(u)du =
1
λε
∫
1{u≥z}h(u)du =
1
λε
ν([z,∞)).
Second, two of the three summands in Xεs are martingales. Therefore
∣
∣
∣
∣E
[∫ t
0
g′(Xεs−)dX
ε
s
]∣
∣
∣
∣ ≤
∫ t
0
∣
∣E[g′(Xεs−)]b
ε
s
∣
∣ ds < Kt
due to boundedness of the first derivatives of g. We may proceed similarly for
the third term in (6.3). Finally, conditioning on Xεs− and using the definition of a
16
compensator gives
∑
0<s≤t
E
[
g(Xεs )− g(X
ε
s−)− g
′(Xεs−)∆X
ε
s
]
=
∫ t
0
∫
E[g(Xεs− + u)− g(X
ε
s−)− g
′(Xεs−)u]cε(u)ν(du)ds
for the final quantity. The Taylor formula proves that the inner integrand above may
be bounded in absolute value by Ku2cε(u). Since ν is a Le´vy measure, we obtain
∣
∣E[g(Xεt )]−
1
λε
ν([z,∞))
∣
∣ < Kt.
From |1− exp(−λεt)| < Kt for 0 < t < t0 the conclusion follows.
6.2 Proof of Theorem 2.4
Due to Theorem 1.6.1 in Van der Vaart and Wellner (1996), it suffices to prove weak
convergence Gn  G in `∞(Ak) for any fixed k ∈ N.
To this end, we use Theorem 11.16 of Kosorok (2008). Note that Gn can be
written as
Gn(θ, z) =
1
√
kn
bnθc∑
j=1
{1{∆njX≥z} − P(∆
n
jX ≥ z)} =
n∑
j=1
{fnj(ω; θ, z)− Efnj(·; θ, z)}
with the triangular array {fnj(ω; θ, z) | n ≥ 1; j = 1, . . . , n; (θ, z) ∈ Ak} consisting
of the processes
fnj(θ, z) := fnj(ω; θ, z) :=
1
√
kn
1{j≤bnθc}1{∆njX(ω)≥z},
which are independent within rows since we assume a deterministic drift and volatil-
ity. By Theorem 11.16 in Kosorok (2008), the proof is complete if the following six
conditions for {fnj} can be established:
(1) {fnj} is almost measurable Suslin (AMS);
(2) the {fnj} are manageable with envelopes {Fnj | n ∈ N, j = 1, . . . , n} given
through Fnj := k
−1/2
n 1{∆njX≥k−1}, which are also independent within rows;
(3) H(θ1, z1; θ2, z2) = lim
n→∞
E{Gn(θ1, z1)Gn(θ2, z2)} for all (θ1, z1), (θ2, z2) ∈ Ak;
(4) lim sup
n→∞
n∑
j=1
E∗F 2nj <∞;
(5) lim
n→∞
n∑
j=1
E∗F 2nj1{Fnj>ε} = 0 for all ε > 0;
(6) ρ(θ1, z1; θ2, z2) = lim
n→∞
ρn(θ1, z1; θ2, z2) for every (θ1, z1), (θ2, z2) ∈ Ak, where
ρn(θ1, z1; θ2, z2) :=
{ n∑
j=1
E |fnj(·; θ1, z1)− fnj(·; θ2, z2)|
2
} 1
2
.
17
Moreover, ρn(θ
(n)
1 , z
(n)
1 ; θ
(n)
2 , z
(n)
2 ) → 0 for all sequences (θ
(n)
1 , z
(n)
1 )n∈N and
(θ(n)2 , z
(n)
2 )n∈N ⊂ Ak such that ρ(θ
(n)
1 , z
(n)
1 ; θ
(n)
2 , z
(n)
2 )→ 0.
Proof of (1). By Lemma 11.15 in Kosorok (2008), the triangular array {fnj} is AMS
provided it is separable, i.e., provided for every n ∈ N, there exists a countable
subset Sn ⊂ Ak, such that
P∗
(
sup
(θ1,z1)∈Ak
inf
(θ2,z2)∈Sn
n∑
j=1
{fnj(ω; θ2, z2)− fnj(ω; θ1, z1)}
2 > 0
)
= 0.
Define Sn := Q2 ∩ Ak for all n ∈ N. Then, for every element ω of the underlying
probability space and for every (θ1, z1) ∈ Ak, there exists an (θ2, z2) ∈ Sn such that
n∑
j=1
{fnj(ω; θ2, z2)− fnj(ω; θ1, z1)}
2 = 0.
Proof of (2). The {Fnj} are independent within rows since we assume deterministic
characteristics of the underlying process. Therefore, according to Theorem 11.17 in
Kosorok (2008), it suffices to prove that the triangular arrays
{f˜nj(ω; z) := k
−1/2
n 1{∆njX≥z} | n ∈ N; j = 1, . . . , n; z ∈ [k
−1,∞)},
and
{g˜nj(ω; θ) := 1{j≤bnθc} | n ∈ N; j = 1, . . . , n; θ ∈ [0, 1]}
are manageable with envelopes {F˜nj(ω) := k
−1/2
n 1{∆njX≥k−1} | n ∈ N; j = 1, . . . , n}
and {G˜nj(ω) :≡ 1 | n ∈ N; j = 1, . . . , n}, respectively.
Concerning the first triangular array {f˜nj} define, for ω ∈ Ω and n ∈ N,
Fnω :=
{
(k−1/2n 1{∆n1X(ω)≥z}, . . . , k
−1/2
n 1{∆nnX(ω)≥z}) | z ∈ [k
−1,∞)
}
⊂ Rn.
For any j1, j2 ∈ {1, . . . , n}, the projection pj1,j2(Fnω) of Fnω onto the j1-th and the
j2-th coordinate is an element of the set
{
{(0, 0)}, {(0, 0), (k−1/2n , 0)}, {(0, 0), (0, k
−1/2
n )}, {(0, 0), (k
−1/2
n , k
−1/2
n )},
{(0, 0), (k−1/2n , 0), (k
−1/2
n k
−1/2
n )}, {(0, 0), (0, k
−1/2
n ), (k
−1/2
n , k
−1/2
n )}
}
.
Hence, for every t ∈ R2, no proper coordinate projection of Fnω can surround t
in the sense of Definition 4.2 of Pollard (1990). Thus, Fnω is a subset of Rn of
pseudodimension at most 1 (Definition 4.3 in Pollard, 1990). Additionally, Fnω
is a bounded set, whence Corollary 4.10 in Pollard (1990) yields the existence of
constants A and W , depending only on the pseudodimension, such that
D2(x‖α F˜n(ω)‖2, αFnω) ≤ Ax
−W =: λ(x),
for all 0 < x ≤ 1, for every rescaling vector α ∈ Rn with non-negative entries
and for all ω ∈ Ω and n ∈ N. Therein, ‖ · ‖2 denotes the Euclidean distance, D2
denotes the packing number with respect to the Euclidean distance and F˜n(ω) :=
18
(F˜n1(ω), . . . , F˜nn(ω)) ∈ Rn is the vector of envelopes. Since
∫ 1
0
√
log λ(x)dx < ∞,
the triangular array {f˜nj} is indeed manageable with envelopes {F˜nj}.
Concerning the triangular array {g˜ni}, we proceed similar and consider the set
Gnω := {(g˜n1(ω; θ), . . . , g˜nn(ω; θ)) | θ ∈ [0, 1]}
= {(0, . . . , 0), (1, 0, . . . , 0), (1, 1, 0, . . . , 0), . . . , (1, . . . , 1)}.
Then, for any j1, j2 ∈ {1, . . . , n}, the projection pj1,j2(Gnω) of Gnω onto the j1-th and
the j2-th coordinate is either {(0, 0), (1, 0), (1, 1)} or {(0, 0), (0, 1), (1, 1)}. Therefore,
the same reasoning as above shows that Gnω is a set of pseudodimension at most
one, whence the triangular array {g˜nj} is manageable with envelopes {G˜nj}.
Proof of (3). For any (θ1, z1), (θ2, z2) ∈ Ak, by independence of {fnj} within rows,
we can write
E{Gn(θ1, z1)Gn(θ2, z2)}
=
n∑
j=1
E
[
{fnj(ω; θ1, z1)− Efnj(·; θ1, z1)}{fnj(ω; θ2, z2)− Efnj(·; θ2, z2)}
]
=
1
kn
bn(θ1∧θ2)c∑
j=1
{P(∆njX ≥ z1 ∨ z2)− P(∆
n
jX ≥ z1)P(∆
n
jX ≥ z2)} (6.4)
By Remark 2.3 and the choice K = K(k−1) and t0 = t0(k−1) > 0, we have
P(∆njX ≥ z) = ∆nν([z,∞)) +O(∆
2
n), n→∞ (6.5)
for all z ≥ k−1 and all j = 1, . . . , n, whence the right-hand side of equation (6.4)
can be written as
bn(θ1 ∧ θ2)c
n
{ν([z1 ∨ z2,∞)) +O(∆n)} = H(θ1, z1; θ2, z2) + o(1), n→∞.
Proof of (4). Again from Remark 2.3, we have
n∑
j=1
E∗F 2nj =
1
n∆n
n∑
j=1
P(∆njX ≥ k
−1) = ν([k−1,∞)) +O(∆n)→ ν([k
−1,∞)) <∞
as n→∞.
Proof of (5). For ε > 0 define N := min{n ∈ N | k−1/2m ≤ ε for all m ≥ n}. Choose
K = K(k−1) and t0 = t0(k−1) as in Lemma 2.2. Then, for any sufficiently large n
such that ∆n < t0, we have
n∑
j=1
E∗F 2nj1{Fnj>ε} ≤
N∑
j=1
E∗F 2nj =
1
n∆n
N∑
j=1
P(∆njX ≥ k
−1)
≤
N
n
{ν([k−1,∞)) +K∆n} → 0, n→∞.
19
Proof of (6). For (θ1, z1), (θ2, z2) ∈ Ak, we can write
ρ2n(θ1, z1; θ2, z2)
=
n∑
j=1
E |fnj(·; θ1, z1)− fnj(·; θ2, z2)|
2
=
1
n∆n
{ bn(θ1∧θ2)c∑
j=1
(P(∆njX ≥ z1 ∧ z2)− P(∆
n
jX ≥ z1 ∨ z2))
+
bn(θ1∨θ2)c∑
j=bn(θ1∧θ2)c+1
P(∆njX ≥ zI(θ1,θ2))
}
=
{
(θ1 ∧ θ2) +O(n
−1)
}
×
{
ν([z1 ∧ z2, z1 ∨ z2)) +O(∆n)
}
+
{
|θ1 − θ2|+O(n
−1)
}
×
{
ν([zI(θ1,θ2),∞)) +O(∆n)
}
as n → ∞, where the O-terms are uniform in (θ1, z1), (θ2, z2) ∈ Ak for the same
reason as in equation (6.5). Thus ρ2n converges even uniformly on each Ak × Ak
to ρ2. Consequently, for any sequences (θ(n)1 , z
(n)
1 )n∈N, (θ
(n)
2 , z
(n)
2 )n∈N ⊂ Ak such that
ρ(θ(n)1 , z
(n)
1 ; θ
(n)
2 , z
(n)
2 )→ 0, it follows ρn(θ
(n)
1 , z
(n)
1 ; θ
(n)
2 , z
(n)
2 )→ 0.
Finally, ρ is a semimetric: applying first the triangle inequality in Rn and then the
Minkowski inequality, one sees that each ρn satisfies the triangle inequality. Thus
the triangle inequality also holds for ρ.
6.3 Proof of Corollary 2.5
For k ∈ N, choose K = K(k−1) and t0 = t0(k−1) as in Lemma 2.2. Then, for any
(θ, z) ∈ Ak and for sufficiently large n, we have
|Gn(θ, z)− G˜n(θ, z)| =
√
kn|EUn(θ, z)− θν([z,∞))|
≤
√
kn
∣
∣
∣
∣
∣
∣
1
n
bnθc∑
j=1
{∆−1n P(∆
n
jX ≥ z)− ν([z,∞))}
∣
∣
∣
∣
∣
∣
+
√
knν([z,∞))
∣
∣
∣
∣
bnθc
n
− θ
∣
∣
∣
∣
≤K
√
kn∆n + ν([k
−1,∞))
√
∆n
n
→ 0.
because of equation (6.5). Since the convergence is uniform in (θ, z) ∈ Ak, we obtain
that d(Gn, G˜n) → 0 in probability. Lemma 1.10.2(i) in Van der Vaart and Wellner
(1996) yields the assertion.
6.4 Proof of Theorem 2.6
We are going to use the extended continuous mapping theorem (Theorem 1.11.1 in
Van der Vaart and Wellner, 1996). For n ∈ N0, define gn : B∞(A)→ B∞(A) through
gn(f)(θ, z) = f(θ, z)−
bnθc
n
f(1, z), for n ∈ N
and
g0(f)(θ, z) = f(θ, z)− θf(1, z)}.
20
Note that gn is Lipschitz continuous for any n ∈ N0.
Obviously, Tn = gn(Gn) + ETn for each n ∈ N and T = g0(G). We have
ETn(θ, z) =
√
knλn(θ)
{
n
bnθc
EUn(θ, z)−
n
n− bnθc
[EUn(1, z)− EUn(θ, z)]
}
and the proof of Corollary 2.5 shows that ETn converges to 0 in B∞(A). Thus, by
Slutsky’s theorem (Van der Vaart and Wellner, 1996, Example 1.4.7), it suffices to
verify gn(Gn) g0(G).
Due to Theorem 1.11.1 in Van der Vaart and Wellner (1996) (note that G is
separable as it is tight; see Lemma 1.3.2 in the last-named reference) this weak
convergence is valid, if we can show that, for any sequence (fn)n∈N ⊂ B∞(A) with
fn → f0 for some f0 ∈ B∞(A), we have
gn(fn)→ g0(f0).
Let (fn)n∈N be such a sequence with limit point f0. Convergence in (B∞(A), d) is
equivalent to uniform convergence on each Ak with k ∈ N. The latter is true since
‖gn(fn)− g0(f0)‖Ak = ‖fn(θ, z)− (bnθc/n)fn(1, z)− f0(θ, z) + θf0(1, z)‖Ak
≤ n−1‖f0‖Ak + 2‖fn − f0‖Ak .
Obviously, T is a tight, mean-zero Gaussian process. Moreover, from Theorem 2.4,
Cov{T(θ1, z1),T(θ2, z2)} = H(θ1, z1; θ2, z2)− θ1H(1, z1; θ2, z2)
− θ2H(θ1, z1; 1, z2) + θ1θ2H(1, z1; 1, z2)
= {(θ1 ∧ θ2)− θ1θ2}ν([z1 ∨ z2,∞))
for any (θ1, z1), (θ2, z2) ∈ A.
6.5 Proof of Proposition 2.8
Because of Corollary 2.5 (and the continuous mapping theorem) U1:n(z0) = Un(1, z0)
converges to ν([z0,∞)) > 0 in probability. Therefore, it follows easily that the ran-
dom variable {Un(1, z0)}−1/21{Un(1,z0)>0} converges to {ν([z0,∞))}
−1/2 in probabil-
ity. Hence, by Slutsky’s theorem (Van der Vaart and Wellner, 1996, Example 1.4.7)
we obtain
V (z0)n (θ) 
1
√
ν([z0,∞))
T(θ, z0).
By Theorem 2.6 the process on the right-hand side of this display is a tight mean
zero Gaussian with covariance function k(θ1, θ2) = θ1 ∧ θ2 − θ1θ2. Thus, the law of
that process is the law of a standard Brownian bridge on `∞([0, 1]).
6.6 Proof of Theorem 3.3
Due to Lemma 6.2 below it suffices to prove conditional weak convergence on
`∞(Ak) for any fixed k ∈ N. Recall the triangular array {fnj(ω; θ, z) | n ≥ 1; j =
1, . . . , n; (θ, z) ∈ Ak} consisting of the processes
fnj(ω; θ, z) := k
−1/2
n 1{j≤bnθc}1{∆njX≥z}.
21
Set µnj(θ, z) := Efnj(·; θ, z) = k
−1/2
n 1{j≤bnθc}P(∆njX ≥ z) and let
µˆnj(θ, z) := µˆnj(ω; θ, z) := k
−1/2
n 1{j≤bnθc}ηn(z)
be an estimator for µnj(θ, z). Then, Gˆn can be written as
Gˆn(θ, z) = Gˆn(ω; θ, z) =
n∑
j=1
ξj{fnj(ω; θ, z)− µˆnj(ω; θ, z)}.
Due to Theorem 3 in Kosorok (2003) the proof is complete, if we show the following
properties for the triangular array {µˆnj(ω; θ, z) | n ≥ 1; j = 1, . . . , n; (θ, z) ∈ Ak}:
(i) {µˆnj} is almost measurable Suslin.
(ii) sup
(θ,z)∈Ak
n∑
j=1
{µˆnj(ω; θ, z)− µnj(θ, z)}2
P∗
→ 0.
(iii) The triangular array {µˆnj} is manageable with envelopes {Fˆnj} given through
Fˆnj(ω) := k
−1/2
n n−1
∑n
i=1 1{∆ni X≥k−1}.
(iv) There exists a constant M <∞ such that M ∨
n∑
j=1
Fˆ 2nj
P∗
→M .
Proof of (i). As in the proof of (1) in Theorem 2.4, it suffices to verify that the
triangular array {µˆnj} is separable. This can be seen by taking Sn := Ak∩Q2 again.
Proof of (ii). We have
sup
(θ,z)∈Ak
n∑
j=1
{µˆnj(ω; θ, z)− µnj(θ, z)}
2
= sup
z≥1/k
n−3∆−1n
n∑
j=1
[ n∑
i=1
{1{∆ni X≥z} − P(∆
n
jX ≥ z)}
]2
= n−1 sup
z≥1/k
{Gn(1, z)}2 +OP(∆2n),
where the final approximation error is a consequence of equation (6.5) in the proof
of Theorem 2.4. The last quantity in the above display converges to 0 in probability
by Theorem 2.4.
Proof of (iii). In the proof of Theorem 2.4 we have already shown that the triangular
array
{g˜nj(θ) := g˜nj(ω; θ) := 1{j≤bnθc} | n ∈ N; j = 1, . . . , n; θ ∈ [0, 1]}
is manageable with envelopes {G˜nj(ω)
def
≡ 1 | n ∈ N; j = 1, . . . , n}. Therefore, due
to Theorem 11.17 in Kosorok (2008), it suffices to prove that the triangular array
{
h˜nj(ω; z) :=
1
n
√
kn
n∑
i=1
1{∆ni X≥z} | n ∈ N; j = 1, . . . , n; z ∈ [k
−1,∞)
}
is manageable with envelopes {Fˆnj(ω) | n ∈ N; j = 1, . . . , n}. But h˜nj(ω; z) does not
depend on j at all, such that every projection of Hnω := {(h˜n1(ω; z), . . . , h˜nn(ω; z)) |
z ≥ k−1} onto two coordinates lies in the straight line {(x, y) ∈ R2 | x = y}. Conse-
quently, the set Hnω has a pseudodimension of at most 1 (Definition 4.3 in Pollard,
22
1990) and is bounded. Hence, the same arguments as in the proof of Theorem 2.4
show the desired manageability.
Proof of (iv). A straight forward calculation yields
E



n∑
j=1
Fˆ 2nj



= n−2∆−1n
n∑
i1=1
n∑
i2=1
E
{
1{∆ni1X≥k
−1}1{∆ni2X≥k
−1}
}
= O(∆n).
Here we used equation (6.5) again and the fact, that the increments of X are inde-
pendent since we assume deterministic characteristics. Thus
n∑
j=1
Fˆ 2nj is oP(1).
6.7 Proof of Theorem 3.4
Again by Lemma 6.2 it suffices to prove the convergence in the spaces `∞(Ak). Let
therefore k ∈ N be fixed for the rest of the proof.
By definition, Tˆn = gn(Gˆn) in `∞(Ak), with gn defined in the proof of Theo-
rem 2.6. Now, (`∞(Ak), ‖ · ‖Ak) is a Banach space and the mapping g0 : `
∞(Ak) →
`∞(Ak) defined in the proof of Theorem 2.6 is Lipschitz continuous. Hence, Propo-
sition 10.7(i) in Kosorok (2008) yields the convergence
g0(Gˆn) ξ g0(G) = T
in `∞(Ak). Furthermore, due to the definition of the mappings gn, g0 and the defi-
nition of the process Gˆn we obtain that
‖gn(Gˆn)− g0(Gˆn)‖Ak ≤
1
n
sup
z≥ 1k
|Gˆn(1, z)|
By Theorem 3.3, the right-hand side converges to 0 in probability. Another appli-
cation of Lemma 6.1 shows that Tn = gn(Gˆn) ξ g0(G) = T as asserted.
6.8 Proof of Proposition 4.1
The assertion that limn→∞ P(V
(z0)
n ≥ qK1−α) = α under H0 is a simple consequence
of Proposition 2.8 and the fact that the KS-distribution has a continuous cumulative
distribution function.
With respect to the assertion regarding W (z0)n note that, under H0, Proposition 6.3
and the continuous mapping theorem imply that, for any fixed B ∈ N,
(W (z0)n , Wˆ
(z0)
n,ξ(1)
, . . . , Wˆ (z0)
n,ξ(B)
) (W (z0),W (z0),(1), . . . ,W (z0),(B))
in RB+1, where W (z0) := supθ∈[0,1] |T(θ, z0)| with the limit process T of Theorem 2.6
and where W (z0),(1), . . . ,W (z0),(B) are independent copies of W (z0). According to the
corollary to Proposition 3 in Lifshits (1984), W (z0) has a continuous c.d.f. under H0.
Thus, Proposition F.1 in the supplement to Bu¨cher and Kojadinovic (2014) implies
that
lim
B→∞
lim
n→∞
P{W (z0)n ≥ qˆ
(B)
1−α(W
(z0)
n )} = α
23
for all α ∈ (0, 1), as asserted. Observing that, under H0 and for ε > 0 with
ν([ε,∞)) > 0, the distribution of T (ε) has a continuous c.d.f., essentially the same
reasoning also implies that
lim
B→∞
lim
n→∞
P{T (ε)n ≥ qˆ
(B)
1−α(T
(ε)
n )} = α
for all α ∈ (0, 1).
6.9 Proof of Proposition 4.2
In order to prove consistency of the CP-Test, choose ε > 0 as in Proposition 6.4
such that limn→∞ P(T
(ε)
n ≥ K) = 1 for any K > 0. By Proposition 6.5, for given
δ > 0 and fixed B ∈ N, we may choose K0 > 0 such that
sup
n∈N
P
(
max
b=1,...,B
Tˆ (ε)
n,ξ(b)
> K0
)
≤
δ
2
.
For this K0, we can now take N ∈ N such that
P(T (ε)n ≥ K0) ≥ 1−
δ
2
holds for all n ≥ N . Then, for any n ≥ N ,
1− δ ≤ P(T (ε)n ≥ K0)− P
(
max
b=1,...,B
Tˆ (ε)
n,ξ(b)
> K0
)
≤ P
(
T (ε)n ≥ K0, max
b=1,...,B
Tˆ (ε)
n,ξ(b)
≤ K0
)
≤ P
{
T (ε)n ≥ qˆ
(B)
1−α(T
(ε)
n )
}
.
This proves the assertion for the CP-Test, and the claim for KSCP-Test2 follows
along the same lines. The assertion for KSCP-Test1 is a direct consequence of
Proposition 6.4.
6.10 Proof of Proposition 4.3
Let X(1)(n) and X(2)(n) denote two independent Ito¯ semimartingales with charac-
teristics (b(n)t , σ
(n)
t , ν1) and (b
(n)
t , σ
(n)
t , ν2), respectively. For n ∈ N and j = 0, . . . , n,
set Yj(n) = X
(1)
j∆n(n) and Zj(n) = X
(2)
j∆n(n). Let U
(1)
n and U
(2)
n denote the quantity
defined in (2.1), based on the observations Yj(n) and Zj(n), respectively, instead on
Xj∆n . Moreover, define a random element Sn with values in B∞(A) through
Sn(θ, z) :=
n− bnθc
n
U (1)n (θ, z)−
bnθc
n
{U (1)n (θ0, z)− U
(1)
n (θ, z)}
−
bnθc
n
{U (2)n (1, z)− U
(2)
n (θ0, z)},
for (θ, z) ∈ A with θ ≤ θ0, whereas for (θ, z) ∈ A with θ ≥ θ0,
Sn(θ, z) :=
n− bnθc
n
U (1)n (θ0, z) +
n− bnθc
n
{U (2)n (θ, z)− U
(2)
n (θ0, z)}
−
bnθc
n
{U (2)n (1, z)− U
(2)
n (θ, z)}.
24
According to Theorem II.4.15 in Jacod and Shiryaev (2002), we have the distribu-
tional equality
(
∆n1X(n), . . . ,∆
n
bnθ0cX(n),∆
n
bnθ0c+1X(n), . . . ,∆
n
nX(n)
)
D
=
(
∆n1X
(1)(n), . . . ,∆nbnθ0cX
(1)(n),∆nbnθ0c+1X
(2)(n), . . . ,∆nnX
(2)(n)
)
.
Hence, for any (θ1, z1), . . . , (θp, zp) ∈ A and p ∈ N, we also have that
(
k−1/2n Tn(θ1, z1), . . . , k
−1/2
n Tn(θg, zg)
) D
=
(
Sn(θ1, z1), . . . , Sn(θg, zg)
)
.
By Theorem 1.6.1 in Van der Vaart and Wellner (1996), we have to show uniform
convergence of k−1/2n Tn to T on any Ak with k ∈ N, in probability. Now, from the
previous display, and from the fact that the function T is continuous in (θ, z) and
that the functions Tn(θ, z) depend only through bnθc on θ and are left-continuous
in z, we immediately get that
sup
(θ,z)∈Ak
∣
∣
∣k−1/2n Tn(θ, z)− T (θ, z)
∣
∣
∣ = sup
(θ,z)∈Ak∩Q2
∣
∣
∣k−1/2n Tn(θ, z)− T (θ, z)
∣
∣
∣
D
= sup
(θ,z)∈Ak∩Q2
|Sn(θ, z)− T (θ, z)|
This expression is in fact oP(1) as a consequence of Corollary 2.5 and the continuous
mapping theorem. Note that the proofs of Lemma 2.2, Theorem 2.4 and Corol-
lary 2.5 show that Corollary 2.5 is in fact applicable in this setup, because the char-
acteristics b(n)t and σ
(n)
t have a uniform bound in n ∈ N and the resulting constants
of Lemma 2.2 depend only on the bound of the characteristics and on δ.
6.11 Proof of Proposition 4.4
Under H1, choose ε > 0 such that there exists a z0 ≥ ε with ν1(z0) 6= ν2(z0). Then,
according to Proposition 4.3 and the continuous mapping theorem, the random
functions θ 7→ supz≥ε |k
−1/2
n Tn(θ, z)| converge weakly in `∞([0, 1]) to the continuous
function θ 7→ supz≥ε |T (θ, z)|, which has a unique maximum at θ0.
Similarly, under H(z0)1 , the random functions θ 7→ |k
−1/2
n Tn(θ, z0)| converge weakly
in `∞([0, 1]) to the continuous function θ 7→ |T (θ, z0)|, which also has a unique
maximum in θ0.
Thus, the asserted convergences follow from the argmax-continuous mapping the-
orem (Theorem 2.7 in Kim and Pollard, 1990).
6.12 Additional auxiliary results
The following two auxiliary results are needed for validating the bootstrap proce-
dures defined in Section 3. The first lemma is proved in Bu¨cher (2011), Lemma A.1.
Lemma 6.1. Consider two bootstrapped statistics Gˆn = Gˆn(X1, . . . , Xn, ξ1, . . . , ξn)
and Hˆn = Hˆn(X1, . . . , Xn, ξ1, . . . , ξn) in a metric space (D, d) with d(Gˆn, Hˆn)
P∗
→ 0.
Then, for a tight Borel measurable process G in D, we have Gˆn ξ G if and only if
Hˆn ξ G.
25
For the second auxiliary lemma, let T1 ⊂ T2 ⊂ . . . be arbitrary sets and set
T :=
⋃∞
k=1 Tk. Let (B∞(T ), d) be defined as the complete metric space of all real-
valued functions on T that are bounded on each Tk, equipped with the metric
d(f1, f2) =
∞∑
k=1
2−k(‖f1 − f2‖Tk ∧ 1),
where ‖·‖Tk denotes the sup-norm on Tk (Van der Vaart and Wellner, 1996, Chapter
1.6). Bootstrap variables on such spaces converge weakly conditionally in probability
if and only if the same holds true in (`∞(Tk), ‖ · ‖Tk) for all k ∈ N.
Lemma 6.2. Let Gˆn = Gˆn(X1, . . . , Xn, ξ1, . . . , ξn) be a bootstrapped statistic with
values in B∞(T ) and let G be a tight Borel measurable process taking values in
B∞(T ). Then, Gˆn ξ G in (B∞(T ), d) if and only if Gˆn ξ G in (`∞(Tk), ‖ · ‖Tk)
for all k ∈ N.
The proof of this lemma can be found in Bu¨cher (2011), Lemma A.5, for a special
choice of the Tk. The proof, however, is independent of this choice.
The proof of Proposition 4.1 is based on the following auxiliary result, establishing
unconditional weak convergence of the vector of processes (Tn, Tˆn,ξ(1) , . . . , Tˆn,ξ(B)).
Proposition 6.3. Suppose the conditions from Theorem 3.3 are met. Then, under
H0, for all B ∈ N, we have
(Tn, Tˆn,ξ(1) , . . . , Tˆn,ξ(B)) (T,T
(1), . . . ,T(B))
in (B∞(A), d)B+1, where  denotes (unconditional) weak convergence (with respect
to the probability measure P), and where T(1), . . . ,T(B) are independent copies of T.
Proof. We are going to apply Corollary 1.4.5 in Van der Vaart and Wellner (1996).
Therefore, let f (0), f (1), . . . , f (B) ∈ BL1(B∞(A)). Since Tn, Tˆn,ξ(1) , . . . , Tˆn,ξ(B) are
independent conditional on the data, we have
Eξ
{
f (0)(Tn) · f
(1)(Tˆn,ξ(1)) · . . . · f
(B)(Tˆn,ξ(B))
}
= f (0)(Tn) · Eξf (1)(Tˆn,ξ(1)) · . . . · Eξf
(B)(Tˆn,ξ(B)) =: Sn.
By Definition 3.1 and Theorem 3.4, Eξf (b)(Tˆn,ξ(b)) converges in outer probability to
E(f (b)(T(b))) =: cb for each b ∈ {1, . . . , B}. Therefore,
Sn  c1 · . . . · cB · f
(0)(T) =: S
by using the continuous mapping theorem, Slutsky’s Lemma and Lemma 1.10.2 in
Van der Vaart and Wellner (1996) several times.
Choose an M > 0 with |Sn| ∨ |S| ≤ M for all ω ∈ Ω, n ∈ N and let g : R −→ R
be a bounded and continuous function with g(x) = x on [−M,M ]. Then
E∗X
[
E∗ξ
{
f (0)(Tn) · f (1)(Tˆn,ξ(1)) · . . . · f
(B)(Tˆn,ξ(B))
}]
= E∗XSn = E
∗
Xg(Sn)
(1)
−→ E(g(S)) = ES
(2)
= E
{
f (0)(T) · f (1)(T(1)) · . . . · f (B)(T(B))
}
. (6.6)
26
Note that (1) uses the fact that a coordinate projection on a product probability
space is perfect (Lemma 1.2.5 in Van der Vaart and Wellner, 1996). Moreover, (2)
holds because the limit processes are independent.
By Theorem 2.6, Remark 3.2(ii), Theorem 3.4 and Lemma 1.3.8 and Lemma 1.4.4
in Van der Vaart and Wellner (1996) the vector of processes (Tn, Tˆn,ξ(1) , . . . , Tˆn,ξ(B))
is (jointly) asymptotically measurable. Consequently, Equation (6.6), Fubini’s the-
orem (Lemma 1.2.6 in Van der Vaart and Wellner, 1996) and Corollary 1.4.5 in
Van der Vaart and Wellner (1996) yield the desired weak convergence. Note that
the limit process (T,T(1), . . . ,T(B)) is separable because it is tight (Lemma 1.3.2 in
the previously mentioned reference).
Proposition 6.4. Suppose the sampling scheme meets the conditions from Corol-
lary 2.5. Then, under H1, there exists an ε > 0 such that, for all K > 0,
lim
n→∞
P(T (ε)n ≥ K) = 1.
If H(z0)1 is true, the same assertion holds for V
(z0)
n and W
(z0)
n .
Proof. Choose ε > 0 such that there exists a zˆ ≥ ε with ν1(zˆ) 6= ν2(zˆ). Then c :=
supθ∈[0,1] supz≥ε |T (θ, z)| ∈ (0,∞), with the function T defined in Proposition 4.3.
But Proposition 4.3 and the continuous mapping theorem show that k−1/2n T
(ε)
n =
c+ oP(1) and this yields the assertion for T
(ε)
n .
The same argument implies the claim for W (z0)n , using the fact that ν1(z0) 6= ν2(z0)
and consequently supθ∈[0,1] |T (θ, z0)| > 0 under H
(z0)
1 .
Finally, let us prove the claim for V (z0)n . As in the proof of Proposition 4.3,
let X(1)(n) and X(2)(n) be independent Ito¯ semimartingales with characteristics
(b(n)t , σ
(n)
t , ν1) and (b
(n)
t , σ
(n)
t , ν2), respectively. For n ∈ N and j = 0, . . . , n, set Yj(n) =
X(1)j∆n(n) and Zj(n) = X
(2)
j∆n(n). Let U
(1)
n and U
(2)
n denote the quantity defined in
(2.1), based on the observations Yj(n) and Zj(n), respectively, instead on Xj∆n .
Then the quantities V (z0)n and W
(z0)
n differ only by a factor A
−1/2
n 1{An>0}, with An
being equal in distribution to (Theorem II.4.15 in Jacod and Shiryaev, 2002)
U (1)n (θ0, z0) + U
(2)
n (1, z0)− U
(2)
n (θ0, z0).
This expression converges to θ0ν1(z0) + (1− θ0)ν2(z0) > 0, in probability, which in
turn implies the assertion regarding V (z0)n .
Proposition 6.5. Suppose the sampling scheme meets the conditions from Corol-
lary 2.5. Then, under H1, for all ε > 0 and all b ∈ {1, . . . , B},
Tˆ (ε)
n,ξ(b)
= OP(1), that is lim
K→∞
lim sup
n→∞
P(Tˆ (ε)
n,ξ(b)
> K) = 0.
Moreover, under H(z0)1 , for all b ∈ {1, . . . , B},
Wˆ (z0)
n,ξ(b)
= OP(1), that is lim
K→∞
lim sup
n→∞
P(Wˆ (z0)
n,ξ(b)
> K) = 0.
Proof. Since the results are independent of b, we omit this index throughout the
proof. Also note that, for both assertions, it suffices to show that, for any k ∈ N,
supθ∈[0,1] supz≥1/k |Gˆn(θ, z)| = OP(1) under H1.
27
For n ∈ N and j = 0, . . . , n, let Yj(n) = X
(1)
j∆n(n) and Zj(n) = X
(2)
j∆n(n) be
defined as in the proof of Proposition 4.3. Let U (1)n , η
(1)
n and U
(2)
n , η
(2)
n denote the
corresponding quantities, based on the observations Yj(n) and Zj(n), respectively,
instead on Xj∆n .
Then, for θ ≤ θ0, we can write Gˆn(θ, z) as
1
√
kn
bnθc∑
j=1
ξj{1{∆nj Y≥z} − η
(1)
n (z)}+



1
√
n
bnθc∑
j=1
ξj



×
{
∆−1/2n
(
η(1)n − ηn
)}
.
The first term of this display is OP(1), uniformly in θ ≤ θ0 and z ≥ 1/k, by The-
orem 3.3 and Remark 3.2 (ii). By the classical Donsker theorem, the first term in
curly brackets on the right-hand side is also OP(1) uniformly in θ ≤ θ0. The quantity
∆−1/2n η
(1)
n (z) = ∆
1/2
n U
(1)
n (1, z) is oP(1) by Corollary 2.5. Finally, the same argument
as in the proof of Proposition 4.3 yields
∆−1/2n sup
z≥1/k
|ηn(z)| =
√
∆n sup
z≥1/k
∣
∣
∣U (1)n (θ0, z) + U
(2)
n (1, z)− U
(2)
n (θ0, z)
∣
∣
∣ = oP(1).
To conclude,
sup
θ≤θ0
sup
z≥1/k
|Gˆ(θ, z)| = OP(1).
The supremum over θ > θ0 and z ≥ 1/k can be treated similarly.
Acknowledgements. This work has been supported by the Collaborative Research
Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt
A1, A7, C1) of the German Research Foundation (DFG) which is gratefully acknowl-
edged.
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