A novel theoretical result on estimation of the local time and the occupation time measure of an

In this paper we consider a pure jump

The goal of this article is to introduce the

Functional limit theorems for estimators of local times have been studied for numerous stochastic models. In the setting of

While estimation of the local time and the occupation time measure has an interest in its own right, accurate estimation of these objects can be useful for related statistical problems. For example, nonparametric estimators of the diffusion coefficient in a continuous stochastic differential equation often involve local times in the mixed normal limit, see, e.g., [

Our main result about the asymptotic theory for local times is mostly related to the articles [

In [

We will use stable convergence theorems for high frequency statistics introduced in [

The rest of the paper is organised as follows. Section

In this section we present the spectral representation of local times, which will be useful in the sequel. Furthermore, we introduce the notion of stable convergence and establish the asymptotic theory for the estimators

The analysis of occupation times and local times is an integral part of the theory of stochastic processes, which found manifold applications in probability during the past decades. We recall that, for

Recall furthermore that, by the definition of the occupation time

In what follows we will use the notion of

The following proposition provides an explicit expression of the statistics

The proof of Proposition

By the Dambis–Dubins–Schwarz theorem (cf. [

The statement above can be directly used to construct confidence regions for

Asymptotic confidence sets for

The setting of

From the statistical point of view Theorem

This section is devoted to the proof of the central limit theorem stated in the previous section. Throughout the proofs we denote by

Regarding the well-posedness of the elements mentioned above, it is worth noting that

Our argument is based on a martingale approach. We first deal with the estimation of the local time. Recall that

Due to Proposition

In the next step we will apply Theorem 3-2 of [

We start by showing condition (

To apply Theorem

In the next step we show condition (

Finally, we show condition (

Now we proceed to the analysis of the occupation time. As for the local time case, the proof is based on a martingale approach. The definition of

Due to Proposition

We start by proving (

Condition (

Finally, we show condition (

This section is devoted to the proof of technical lemmas and proposition we have previously stated and used in order to obtain our main result, gathered in Theorem

The Fourier representation is proven in [

Recall that

First of all show that

After that we prove that

We are left with the problem of showing (

Due to (

Next we deal with identity (

Equation (

The change of variables

The definition of the occupation time provides

Acting as in the proof of part (a) it is then easy to check that

According to (

We remark that, from the definition of

It follows by arguments analogous to part (ii) in Proposition