1. Introduction
Given a positive number and an increasing Lévy process without drift component, an OrnsteinUhlenbeck (OU) process driven by is defined by a solution to the following stochastic differential equation
(1.1) 
We refer to Sato (1999) and Bertoin (1996) as standard references on Lévy processes. In this paper, we consider nonparametric inference of the Lévy measure of the backdriving Lévy process in (1.1) from discrete observations of . The Lévy measure is a Borel measure on such that
We will assume that is stationary. If , then a unique stationary solution of (1.1) exists (see Theorem 17.5 and Corollary 17.9 in Sato (1999)), and the stationary distribution of
is selfdecomposable with the characteristic function
(1.2) 
where .
In this paper, we focus on the case that the Lévy process in (1.1) is a compound Poisson process, that is, is of the form
where is a Poisson process with intensity and
is a sequence of independent and identically distributed positivevalued random variables with common distribution
. In this case, has a characteristic function of the formand the Lévy measure is given by . We also work with macroscopic observation set up, that is, we have discrete observations at frequency with and as .
The goal of this paper is to develop nonparametric inference on the Lévy measure of Lévydriven OrnsteinUhlenbeck process (1.1). For this, we propose a spectral (or Fourierbased) estimator of the function and derive multivariate central limit theorems for the estimator over finite design points. As extension of these results, we also derive highdimensional central limit theorems for the estimator in the case that design points over an compact interval included in increases as the sample size goes to infinity. Build upon those limit theorems, we develop methods for the implementation of confidence bands. Since confidence bands provide a simple graphical description of the accuracy of a nonparametric curve estimator quantifying uncertainties of the estimator simultaneously over design points, they are practically important in statistical analysis. Moreover, we propose a practical method for bandwidth selection inspired by Bissantz et al. (2007). As a result, this paper contributes the literature on nonparametric inference of Lévydriven stochastic differential equations, and to the best of our knowledge, this is the first paper to establish limit theorems for nonparametric estimators for the Lévy measure of Lévydriven OU processes.
Lévydriven OU processes are widely used in modeling phenomena where random events occur at random discrete times. See, for example, Albrecher et al. (2001), Kella and Stadje (2001) and Noven et al. (2014) for applications to insurance, dam theory, and rainfall models. Several authors investigate parametric inference on Lévydriven OU processes driven by subordinators. We refer to Hu and Long (2009), Masuda (2010) and Mai (2014) under the highfrequency set up (i.e., as ) and Brockwell et al. (2007) under the lowfrequency set up (i.e., is fixed). There are also a large number of studies on parametric and nonparametric estimation and inference on Lévy processes. Recent contributions include Duval and Hoffmann (2011) and Duval (2014) for parametric inference, and Duval and Kappus (2018) for nonparametric estimation on compound Poisson processes under macroscopic observations, Nickl et al. (2016) for inference on Lévy measures, FigueroaLópez (2011), Konakov and Panov (2016), and Kato and Kurisu (2017) for inference on Lévy densities, and Kappus and Reiß (2010) for estimation on Lévy densities under the highfrequency set up, van Es et al. (2007), Gugushvili (2009, 2012), Neumann and Reiß (2009), Kappus and Reiß (2010), Belomestny (2011), Duval (2013), Kappus (2014), Belomestny and Reiß (2015), and Belomestny and Schoenmakers (2016) for estimation of Lévy densities, and Nickl and Reiß (2012) and Coca (2018) for inference on Lévy measures, under the lowfrequency set up. We also refer to Belomestny et al. (2017) for nonparametric estimation on Lévy measures of moving average Lévy processes under lowfrequency observations, and Bücher and Vetter (2013), Vetter (2014), Bücher et al. (2017), and Hoffmann and Vetter (2017) for inference on Lévy measures under highfrequency observations. Jongbloed et al. (2005) and Ilhe et al. (2015) investigate nonparametric estimation of the Lévydriven OU processes. Jongbloed et al. (2005) derives consistency of their estimator for a class of Lévydriven OU processes which include compound Poissondriven OU processes. Ilhe et al. (2015) establish consistency of their estimator of the Lévy density of (1.1) with compound Poisson subordinator in uniform norm at a polynomial rate. However, they does not derive limit distributions of their estimators.
The analysis of the present paper is related to deconvolution problems for mixing sequence. Masry (1991, 1993a, 1993b)
investigate probability density deconvolution problems for
mixing sequences and they derive convergence rate and asymptotic distributions of deconvolution estimators. Since the Lévydriven OU process (1.1) is mixing under some condition (see Masuda (2004) for details), our analysis can be interpreted as a deconvolution problem for a mixing sequence. However, we need nontrivial analysis since we have to take account of additional structures which come from the properties of the Lévydriven OU process.The estimation problem of Lévy measures is generally illposed in the sense of inverse problems and the illposedness is induced by the decay of the characteristic function of a Lévy process as explained in Nickl and Reiß (2012). In our case, the illposedness is induced by the decay of the characteristic function of the stationary distribution of the Lévydriven OU (1.1). In this sense the problem in this paper is a (nonlinear) inverse problem. Trabs (2014a) investigates conditions that a selfdecomposable distribution is nearly ordinary smooth, that is, the characteristic function of the selfdecomposable distribution decays polynomially at infinity up to a logarithmic factor. Trabs (2014b) applies those results to nonparametric calibration of selfdecomposable Lévy models of option prices. As a refinement of a result in Trabs (2014a), we will show that the characteristic function of a selfdecomposable distribution is regularly varying at infinity with some index . This enables us to derive asymptotic distributions of the spectral estimator proposed in this paper.
Kato and Sasaki (2016) is a recent contribution in the literature on the construction of uniform confidence bands in probability density deconvolution problems for independent and identically distributed observations. They develop methods for the construction of uniform confidence bands build on applications of the intermediate Gaussian approximation theorems developed in Chernozhukov et al. (2014a, b, 2015, 2016) and provides multiplier bootstrap methods for the implementation of uniform confidence bands. Kato and Kurisu (2017) also develops confidence bands for Lévy densities based on the intermediate Gaussian and multiplier bootstrap approximation theorems. Our analysis is related to these papers but we adopt different methods for the construction of confidence bands. We derive highdimensional central limit theorems based on intermediate Gaussian approximation for
mixing process. We can show that the variancecovariance matrix of the Gaussian random vector appearing in multivariate and highdimensional central limit theorems is the identity matrix. Therefore, we do not need bootstrap methods to compute critical values of confidence bands.
The rest of the paper is organized as follows. In Section 2, we define a spectral estimator for function. We give multivariate central limit theorems of the spectral estimator in Section 3 and highdimensional central limit theorems of the estimator are also given in Section 4. Procedures for the implementation of confidence bands are described in Section 5. In Section 6, we propose a practical method for bandwidth selection and report simulation results to study finite sample performance of the spectral estimator. All proofs are collected in Appendix A.
1.1. Notation
For any nonempty set and any (complexvalued) function on , let , and for , let for . For any positive sequences , we write if there is a constant independent of such that for all , if and , and if as . For , let . For and , we use the shorthand notation . The transpose of a vector is denoted by . We use the notation as convergence in distribution. For random variables and , we write if they have the same distribution. For any measure on , its support is denoted by .
2. Estimation of function
In this section, we introduce a spectral estimator of the Lévy measure (function) of the Lévydriven OrnsteinUhlenbeck process (1.1). First, we consider a symmetrized version of the function, that is,
A simple calculation yields that
Therefore, we have that
This yields that
Let
Here, is a sequence of constants such that as (in the rest of this paper, we set ). Let be an integrable function (kernel) such that
and its Fourier transform
is supported in (i.e., for all ). Then the spectral estimator of for is defined byfor , where is a sequence of positive constants (bandwidths) such that as , and
In the following sections we develop central limit theorems for .
Remark 2.1.
Remark 2.2.
For a complex value , let be the complex conjugate of . We note that is realvalued. In fact, since and , by a change of variables, we have that
Remark 2.3.
Another natural estimator of for would be
but this estimator have large bias than . We need to symmetrize function to use global regularity of for suitable bound of the deterministic bias. Note that is continuous at the origin and if has bounded th derivative on for some , the deterministic bias of is (Lemma A.10 in Appendix A). However, the deterministic bias of is because of the discontinuity of at the origin.
3. Multivariate Central Limit Theorems
In this section we present multivariate central limit theorems for .
Assumption 3.1.
We assume the following conditions.

for some .

and .

Let , and let be the integer such that . The function is times differentiable, and is Hölder continuous, that is,

and , where is the Fourier transform of .

Let be an integrable function such that
where is the Fourier transform of .

, , and
for some as .
Remark 3.1.
Conditions (i) and (ii) imply that the stationary distribution has a bounded continuous density (we also denote the density by ) such that and (see Lemma A.1). In this case, the stationary Lévydriven OrnsteinUhlenbeck process defined by (1.1) is exponentially mixing (Theorem 4.3 in Masuda (2004)), that is, the mixing coefficients for the stationary continuoustime Markov process
( this representation follows from Proposition 1 in Davydov (1973)) satisfies for some . Here, is the transition probability of the Lévydriven OU (1.1) and is the total variation norm.
Condition (iii) is concerned with smoothness of . Condition (iv) is satisfied if is twotimes continuously differentiable on and . In fact, by Condition (i), we have that for , and by integrationbyparts and the RiemannLebesgue theorem, we also have that
as .
Condition (v) is concerned with the kernel function . We assume that is a th order kernel, but allow for the possibility that . Note that since the Fourier transform of has compact support, the support of the kernel function is necessarily unbounded (see Theorem 4.1 in Stein and Weiss (1971)).
Condition (vi) is concerned with the sampling frequency, bandwidth, and the sample size. The condition implies that we work with macroscopic observation scheme and this is a technical condition for the inference on . We note that for the estimation of uniformly on an interval , we do not need the condition and we can work with lowfrequency set up (i.e., is fixed). Masry (1993b) investigates deconvolution problems for mixing sequences and assume that for a mixing sequence , joint densities of and are uniformly bounded for any and to show the asymptotic independence of their estimators at different design points. Although we also observe mixing sequence , we cannot assume such a condition in the paper directly in our situation since the transition probability of has a point mass at , does not have a transition density function. Moreover, for each fixed , the absolutely continuous part of is uniformly bounded for any , however, it would be difficult to check uniform boundedness of for any and . Zhang et al. (2011) investigates the transition law of compound Poissondriven OU processes. Define the function
and let be its th convolution. Note that
is a probability density function (Lemma 1 in
Zhang et al. (2011)). They derives the concrete form of the transition probability , that is,This implies that does not have a Lebesgue density and it has a point mass at . In this case, can be written as
Suppose for fixed . Then we can show that for any and we have that
Therefore, in this case, we have that . In particular,
(3.1) 
for a wide class of compound Poissondriven OU processes.
Lemma 3.1.
Suppose that the jump distribution has a probability density function such that

,

for .
Then (3.1) is satisfied.
Remark 3.2.
For example, the density functions of exponential, halfnormal, and half distributions satisfy the assumption of Lemma 3.1. In general, probability density functions which is nonincreasing on satisfy the assumptions.
Lemma 3.2.
A compound Poissondriven OU process with a gamma jump distribution with shape parameter and rate parameter satisfies (3.1).
From a theoretical point of view, we could assume and in that case, can be fixed, that is, we can work with lowfrequency setup. However, it would be difficult to check the condition directly or to give some sufficient conditions which can be checked for a given compound Poissondriven OU process. We can avoid the problem if we assume the condition since we do not need the uniform bound of under the condition to show the asymptotic validity of (defined by (3.5)) and the asymptotic independence of and for with (see the proof of Lemma A.8, Proposition A.2, and Proposition A.3). From a practical point of view, our methods can be applied to lowfrequency data and would work well if we suitably rescale the time scale of the data and the sample size is sufficiently large. In our simulation study, we consider the case when and our method works well in this case. We also need the condition (vi) to derive the lower bound of for the uniform consistency of for , with . We need the upper bound of for the undersmoothing condition. See also Remark A.1 after the proof of Proposition A.2 in Appendix A for the further discussion on , and Remark 3.5 in the present paper for a comment on the condition on .
To state a multivariate central limit theorem for , we introduce the notion of regularly varying functions.
Definition 3.1 (Regularly varying function).
A measurable function is regularly varying at with index (written as ) if for ,
We say that a function is slowly varying if . We refer to Resnick (2007) for details of regularly varying functions. The following lemma plays an important role in the proof of Theorem 3.1.
Lemma 3.3.
Assume Condition (ii) in Assumption 3.1. There exist a function slowly varying at and a constant such that
Remark 3.3.
Condition (ii) in Assumption 3.1 is concerned with smoothness of the stationary distribution of the Lévydriven OU process. Condition (ii) implies that the stationary distribution is nearly ordinary smooth, that is, the characteristic function (1.2) decays polynomially fast as (Lemma 3.3) up to a logarithmic factor. Since , the finiteness of is equivalent to the finiteness of the total mass of the Lévy measure of the Lévy process and this means that the Lévy process is of finite activity, i.e., it has only finitely many jumps in any bounded time intervals. It is known that a Lévy process with a finite Lévy measure is a compound Poisson process. If , the Lévy process is of infinite activity, i.e., it has infinitely many jumps in any bounded time intervals. In this case, the characteristic function (1.2) decays faster than polynomials. In particular, it decays exponentially fast as if the BlumenthalGetoor index of is positive, that is, if
Inverse Gaussian, tempered stable, normal inverse Gaussian processes are included in this case, for example. Condition (ii) rules out those cases since we could not construct confidence bands based on Gaussian approximation under our observation scheme (see also the comments after Assumption 10 in Kato and Sasaki (2016)). Therefore, we need the nearly ordinary smoothness of in our situation to drive practical asymptotic theorems for the inference on .
Remark 3.4.
Lemma 3.3 implies that is a regularly varying function at with index . A slowly varying function may go to as but it does not glow faster than any power functions, that is,
for any . In fact, if , from Proposition 1 in Trabs (2014a), we have that
for any . Such a tail behavior of is related to Condition (vi) in Assumption 3.1. If the stationary distribution is ordinary smooth, that is, satisfies the relation
for some , we can set in Condition (vi). However, we need to introduce to take into account the effect of the slowly varying function .
Remark 3.5.
As shown in (A.8) and the comments below, without the condition
we have that
where the second term of the right hand side comes from the deterministic bias. For central limit theorems to hold, we have to choose bandwidth so that the bias term is asymptotically negligible relative to the first term or “variance” term. The right hand side is optimized if . Since for any by Lemma 3.3 (see Remark 3.4), we have that if we take to satisfy Condition (vi) in Assumption 3.1.
Theorem 3.1.
Remark 3.6.
In Theorem 3.1, we also need for . However, this condition is automatically satisfied under Assumption 3.1. Let be the Lévy measure of . As explained in the introduction of the present paper, the stationary distribution is a selfdecomposable distribution with Lévy density , which is a Lebesgue density of . By the definition of function, we have that and . In this case, by Theorem 24.10 in Sato (1999). Moreover, the density of the stationary distribution satisfies that for any by Theorem 2 in Yamazato (1978) since and the density is continuous on (see Lemma A.1).
Remark 3.7.
Under Assumption 3.1, we can show that can be approximated by
(3.3) 
where . By a change of variables, we may rewrite the right hand side of (3.3) as
(3.4) 
where is a function defined by
Note that
is welldefined and realvalued. To construct confidence interval of
, we estimate the variance of , which is , by(3.5) 
where
Remark 3.8.
Now we can show the next feasible multivariate central limit theorem.
Theorem 3.2.
4. Highdimensional Central Limit Theorems
In Section 3, we give multivariate (or finitedimensional) central limit theorems of . In this Section, we give highdimensional central limit theorems as refinements of Theorems 3.1 and 3.2. As an application of those results, we propose some methods for the construction of confidence bands for function in Section 4.2.
4.1. Highdimensional central limit theorems for
For and , let
and let an interval with finite Lebesgue measure , , , . We assume that
(4.1) 
and this implies that . Therefore, can go to infinity as .
Remark 4.1.
Theorem 4.1.
Remark 4.2.
Theorem 4.1 can be shown in two steps. As a first step, we approximate the distribution of by that of . Here, is a centered normal random vector with covariance matrix where is a sequence of integers with and as , and
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