In this paper we consider a pure jump α-stable Lévy process X = { X t } t ≥ 0 with α ∈ ( 1 , 2 ), defined on a filtered probability space ( Ω , F , { F t } t ≥ 0 , P ). The distribution law of the process X is uniquely determined by the characteristic function (1.1) E [ exp ( i ξ X t ) ] = exp ( − i ξ t η − σ t | ξ | α ( 1 − i β tan ( π α 2 ) sgn ( ξ ) ) ) for ξ ∈ R, t ≥ 0, η ∈ R, β ∈ [ − 1 , 1 ] and σ > 0. We will focus on estimation of the occupation time measure and the local time of X given high frequency data { X i / n } 1 ≤ i ≤ ⌊ n T ⌋ , with T > 0 being fixed and n → ∞. We recall that for a < b the occupation time of X at the set ( x , ∞ ) over the interval [ a , b ], denoted by O [ a , b ] ( x ), is defined as O [ a , b ] ( x ) : = ∫ a b 1 ( x , ∞ ) ( X s ) d s . The local time of X at point x ∈ R over the interval [ a , b ] denoted as L [ a , b ] ( x ) is defined implicitly via the occupation density formula: (1.2) O [ a , b ] ( x ) = ∫ x ∞ L [ a , b ] ( y ) d y P -a.s. (if it exists). Throughout the paper we use the abbreviation L t : = L [ 0 , t ] and O t : = O [ 0 , t ] . The existence and smoothness properties of local times of stochastic processes have been extensively studied in the 70s and 80s; we refer to articles [3–6, 10, 11] among many others. In particular, in the setting of pure jump α-stable Lévy processes, the local time exists only for α ∈ ( 1 , 2 ) (cf. [21]). Furthermore, there exists a version of the local time, which is continuous in space and time. Indeed, according to [4, Theorems 2 and 3] and [2, Theorem 4.3], there exists a version of the local time with property L t ( · ) is P -a.s. locally Hölder continuous with index ( α − 1 ) / 2 − ε , for any ε ∈ ( 0 , ( α − 1 ) / 2 ). Moreover, for all ε ∈ ( 0 , 1 − 1 / α ) and T > 0, there exists a deterministic constant C T > 0, such that (1.3) sup 0 ≤ s ≤ t ≤ T | L t ( x ) − L s ( x ) | ≤ C T | t − s | ( 1 − 1 / α ) − ε P -a.s. In particular, L ( · ) ( x ) is P-a.s. locally Hölder continuous with index ( 1 − 1 / α ) − ε for any ε ∈ ( 0 , 1 − 1 / α ). Throughout the paper we consider the aforementioned Hölder continuous version of the local time.

The goal of this article is to introduce the L 2 -optimal estimators of L t ( x ) and O t ( x ), and study their asymptotic properties. For this purpose we define the σ-algebra A n : = σ ( { X i / n ; i ∈ N } ) and construct the estimators via S L , t n ( x ) = E [ L t ( x ) ∣ A n ] and S O , t n ( x ) = E [ O t ( x ) ∣ A n ] . By construction, S L , t n (resp. S O , t n ) is an L 2 -optimal approximation of L t ( x ) (resp. O t ( x )). We will show a functional stable convergence for both statistics, which exhibit a mixed normal limit. Our main tool is Jacod’s stable central limit theorem for partial sums of high frequency data stated in [13].

Functional limit theorems for estimators of local times have been studied for numerous stochastic models. In the setting of α-stable Lévy processes and related models, such estimators often take the form G ( x , ϕ ) n = { G ( x , ϕ ) t n ; t ≥ 0 }, with (1.4) G ( x , ϕ ) t n = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ ϕ ( n 1 α ( X i − 1 n − x ) ) , where ϕ ∈ L 1 ( R ). Consistency and asymptotic mixed normality for statistics G ( x , ϕ ) n and related functionals have been investigated in [7, 14] in the setting of Brownian motion and continuous stochastic differential equations. Some optimality results for estimation of the local time and the occupation time measure of a Brownian motion can be found in [20]. Estimation errors for occupation time functionals of stationary Markov processes have been studied in [1]. Limit theorems for statistics of the form (1.4) in the case of the fractional Brownian motion are discussed in [15, 17, 18, 23], although the complete weak limit theory is far from being understood.

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., [9]. In a similar spirit, local times and mixed normal local times appear as fundamental limits for additive functionals of a variety of Gaussian processes (see Papanicolaou, Stroock and Varadhan [22], Jaramillo, Nourdin, Nualart and Peccati [16] as well as Minhao Hong, Heguang Liu and Fangjun Xu [12]), thus serving as simplifying probabilistic models for otherwise complex probabilistic objects. Efficient estimation of local times and occupation times is very beneficial for statistical inference in this framework.

Our main result about the asymptotic theory for local times is mostly related to the articles [17–20, 26]. The paper [17] addresses consistency in the case of the linear fractional stable motions, a class which in particular includes stable Lévy processes.

In [18, 19, 26] the authors prove the asymptotic mixed normality for continuous version of the functional G ( x , ϕ ) n , but only in the zero energy setting, i.e. ∫ R ϕ ( y ) d y = 0. From the statistical point of view, this case is a less interesting one as we would like to use statistics of the type (1.4) as an estimator of the local time L t ( x ). More importantly, in the setting ∫ R ϕ ( y ) d y ≠ 0 the methods developed in [18, 19] do not apply.

We will use stable convergence theorems for high frequency statistics introduced in [13] to show asymptotic mixed normality of standardised versions of S L , t n ( x ) and S O , t n ( x ). These results are strongly related to an earlier work [20], which considers the same problem in the setting of the Brownian motion. While their limit theorems are also based on the general results of [13], the technical aspects of the proof are more involved in the case of pure jump Lévy processes. The details will be highlighted in the proof section.

The rest of the paper is organised as follows. Section 2 presents some preliminaries, main results and an application. Proofs of the main results are collected in Section 3. Some technical statements are proved in Section 4.

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 t > 0 and x ∈ R, the local time of the α-stable Lévy process X, α ∈ ( 1 , 2 ), up to time t at x can be formally defined as L t ( x ) : = ∫ 0 t δ 0 ( X s − λ ) d s, where δ 0 denotes the Dirac delta function. A rigorous definition of the local time is obtained by replacing δ 0 by the Gaussian kernel ϕ ϵ ( x ) : = ( 2 π ϵ ) − 1 2 exp ( − 1 2 ϵ x 2 ) and taking the limit in probability for ϵ → 0. For our purposes we will systematically use the following spectral representation for L t ( x ). The proof of the representation gathered in the lemma below can be found in, e.g., [17, Proposition 11], for completeness we give the proof of the boundedness of the moments. It can be found in Section 4.

Recall furthermore that, by the definition of the occupation time O t ( x ), for any p ∈ N, E [ ( O t ( x ) ) p ] < t p .

In what follows we will use the notion of stable convergence, which is originally due to Renyi [24]. Let ( S , δ ) be a Polish space. Let { Y n } n ≥ 1 be a sequence of S-valued and F-measurable random variables defined on ( Ω , F , P ) and Y a random variable defined on an enlarged probability space ( Ω ′ , F ′ , P ′ ). We say that Y n converges stably to Y, if and only if for any continuous bounded function g : S → R and any bounded F-measurable random variable F, it holds that lim n → ∞ E [ F g ( Y n ) ] = E ′ [ F g ( Y ) ] . In this case we write Y n → Stably Y. In this paper we deal with the space of cádlág functions D ( [ 0 , T ] ) equipped with the Skorokhod J 1 -topology.

The following proposition provides an explicit expression of the statistics S L , t n and  S O , t n . The proof of Proposition 2.2 is based upon the Markov property of X and the linearity in time of our objects.

The proof of Proposition 2.2 will be given in Section 4. The next theorem is a functional limit result for the error of the approximation of the occupation and local times by their L 2 -optimal estimators.

From the statistical point of view Theorem 2.3 provides lower bounds for estimation of the path functionals L t ( x ) and O t ( x ). In particular, it shows that statistics G ( x , ϕ ) n introduced in (1.4) do not produce L 2 -optimal estimates. On the other hand, G ( x , ϕ ) n can be computed from data even when the exact law of X is unknown (use, e.g., ϕ = 1 [ − 1 , 1 ] ) in contrast to L 2 -optimal statistics (recall that the functions f and F depend on the parameters of the stable distribution). We remark however that the limit theory for statistics G ( x , ϕ ) n is expected to be much more involved; see the proofs in [14] for more details in the case of the Brownian motion. Hence, we postpone the discussion to future research.

Due to Proposition 2.2 and Lemma 3.2 we can write (3.6) as W L , t n = ∑ i = 1 ⌊ n t ⌋ Z i n L + o P ( 1 ) . We introduce here the notation E i − 1 n [ · ] for E [ · ∣ F i − 1 n ], which will be useful in the sequel.

In the next step we will apply Theorem 3-2 of [13]. In our setting, E i − 1 n [ Z i n L ] = 0 because of Equation (4.18) below. Then, it suffices to show the following conditions: (3.7) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 2 ] → P k L 2 L t ( x ) ∀ t ∈ [ 0 , 1 ] , (3.8) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ Z i n L Δ i n M ] → P 0 ∀ t ∈ [ 0 , 1 ] , (3.9) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 2 1 { | Z i n L | > ϵ } ] → P 0 ∀ ϵ > 0 , where the condition (3.8) should hold for all square integrable continuous martingales M. We emphasise that we have summarised the two conditions (3.12) and (3.14) from [13, Theorem 3-2] into our constraint (3.8). Indeed, conditions (3.12) and (3.14) in [13, Theorem 3-2] are formulated with respect to some continuous martingale M which has to be chosen by the user according to the problem at hand. The convergence in (3.8) holding for all bounded continuous martingales implies them both. It entails, in particular, that the continuous process G of [13, Theorem 3-2] is in our case identically zero.

We start by showing condition (3.7). Due to the identity f ( x , y ) = E [ L [ 0 , 1 ] ( x ) ∣ X 1 = y ] we can check that (3.10) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 2 ] = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ ψ ( n 1 α ( x − X i − 1 n ) ) , where ψ is given by ψ ( q ) : = E [ ( E [ L [ 0 , 1 ] ( q ) ∣ X 1 ] − L [ 0 , 1 ] ( q ) ) 2 ] . Indeed, from the definition of Z i n L we have ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 2 ] = ∑ i = 1 ⌊ n t ⌋ [ n 1 α − 1 E i − 1 n [ f 2 ( n 1 α ( x − X i − 1 n ) , n 1 α Δ i n X ) ] + n 1 − 1 α E i − 1 n [ L [ i − 1 n , i n ] 2 ( x ) ] − 2 E i − 1 n [ f ( n 1 α ( x − X i − 1 n ) , n 1 α Δ i n X ) L [ i − 1 n , i n ] ( x ) ] ] = : I 1 + I 2 + I 3 . We start studying I 1 . It is easy to see that I 1 = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ f 2 ( n 1 α ( x − X i − 1 n ) , n 1 α Δ i n X ) ] = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ g 1 ( n 1 α ( x − X i − 1 n ) ) , with g 1 ( q ) : = E [ f 2 ( q , X 1 ) ]. From Lemma 3.1(a) the following is straightforward: I 2 = n 1 − 1 α ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ L [ i − 1 n , i n ] 2 ( x ) ] = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ g 2 ( n 1 α ( x − X i − 1 n ) ) , where g 2 ( q ) : = E [ L [ 0 , 1 ] ( q ) 2 ]. We are left to study I 3 . From Lemma 3.1(c) we directly obtain I 3 = n 1 α − 1 ∑ i = 1 ⌊ n t ⌋ g 3 ( n 1 α ( x − X i − 1 n ) ) , with g 3 ( q ) : = − 2 E [ L [ 0 , 1 ] ( q ) E [ L [ 0 , 1 ] ( q ) ∣ X 1 ] ]. Putting everything together we get (3.10) with ψ ( q ) = g 1 ( q ) + g 2 ( q ) + g 3 ( q ).

To apply Theorem 1.1 we need to check that ψ , ψ 2 ∈ L 1 ( R ). By the Minkowski and Jensen inequalities, ψ ( q ) ≤ 4 E [ L [ 0 , 1 ] 2 ( q ) ]. We therefore aim at showing that the function E [ L [ 0 , 1 ] 2 ( · ) ] belongs to L 1 ( R ). Let τ q denote the first passage time of X over the level q. Conditioning over τ q , using the additivity of the local time, we deduce the inequality E [ L [ 0 , 1 ] 2 ( q ) ] ≤ E [ L [ 0 , 1 ] 2 ( 0 ) ] P ( τ q < 1 ) . Moreover, by the Fourier representation of the local time (as stated in Lemma 2.1), E [ L [ 0 , 1 ] 2 ( 0 ) ] is bounded. On the other hand, (3.11) ∫ 0 ∞ P [ τ q < 1 ] d q = ∫ 0 ∞ P [ sup s ≤ 1 X s > q ] d q = E [ sup s ≤ 1 X s ] . We recall that E [ ( sup s ≤ 1 X s ) p ] is bounded for any p ∈ ( 0 , α ) (see Corollary II.1.6 and Theorem II.1.7 in [25]). As α ∈ ( 1 , 2 ), this implies the boundedness of E [ sup s ≤ 1 X s ]. Hence, by symmetry, ψ ∈ L 1 ( R ) and k L 2 < ∞. Applying the same argument it is easy to show that also ψ 2 ∈ L 1 ( R ), thanks to (3.11) and the boundedness of E [ L [ 0 , 1 ] 4 ( q ) ]. By Theorem 1.1 we conclude that ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 2 ] → P k L 2 L t ( x ) , where k L 2 = ∫ R ψ ( y ) d y < ∞.

In the next step we show condition (3.8). Let M be any continuous square integrable martingale and denote by μ (resp. μ ˜) the random measure (resp. compensated random measure) associated with the pure jump Lévy process X. The martingale representation theorem for jump measures investigated in Lemma 3 (ii) and Theorem 6 of [8] implies that Z i n L has an integral representation Z i n L = ∫ i − 1 n i n ∫ R η i n ( x , t ) μ ˜ ( d x , d t ) for some predictable square integrable process η i n . Using that the covariation between any continuous martingale and any pure jump martingale is zero, we conclude that E i − 1 n [ Z i n L Δ i n M ] = 0 . Thus, we obtain (3.8).

Finally, we show condition (3.9). The Cauchy–Schwarz inequality ensures that E i − 1 n [ | Z i n L | 2 1 { | Z i n L | > ϵ } ] ≤ ϵ − 2 E i − 1 n [ | Z i n L | 4 ] . Then, by the Markov inequality, it suffices to prove that ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 4 ] → P 0 . We have that E i − 1 n [ ( Z i n L ) 4 ] ≤ C ( n 2 ( 1 α − 1 ) E i − 1 n [ f 4 ( n 1 α ( x − X i − 1 n ) , n 1 α Δ i n X ) ] + n 2 ( 1 − 1 α ) E i − 1 n [ L [ i − 1 n , i n ] 4 ( x ) ] ) . From Lemma 3.1(a) we conclude that ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n L | 4 ] ≤ C n 2 ( 1 α − 1 ) ∑ i = 1 ⌊ n t ⌋ h ( n 1 α ( X i − 1 n − x ) ) , where h ( y ) = E [ f 4 ( y , X 1 ) + L [ 0 , 1 ] ( y ) 4 ]. It is easy to check that h ∈ L 1 ( R ). Indeed, similarly as in the proof of ψ ∈ L 1 ( R ), the Minkowski and Jensen inequalities imply h ( y ) ≤ C E [ L [ 0 , 1 ] 4 ( y ) ] ≤ C E [ L [ 0 , 1 ] 4 ( 0 ) ] P [ τ y < 1 ] . Then the boundedness of moments of the local time in 0 together with (3.11) provides h ∈ L 1 ( R ). As h 2 ( y ) ≤ C E [ L [ 0 , 1 ] 8 ( y ) ], following the same route it is easy to see that h 2 ∈ L 1 ( R ). Since α > 1, we deduce by Theorem 1.1 that n 2 ( 1 α − 1 ) ∑ i = 1 ⌊ n t ⌋ h ( n 1 α ( X i − 1 n − x ) ) → P 0 . This concludes the proof of Theorem 2.3 for the local time.

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 W O , t n together with the approximation of S O , t n as in Proposition 2.2 provides (3.12) W O , t n = n 1 2 ( 1 + 1 α ) ( S O , t n − O [ 0 , t ] ( x ) ) = ∑ i = 1 ⌊ n t ⌋ Z i n O − n 1 2 ( 1 + 1 α ) N O , t n + E O , t n ( x ) , where Z i n O is the principal term, given by Z i n O : = n 1 2 ( 1 α − 1 ) F ( n 1 α ( x − X i − 1 n ) , n 1 α Δ i n X ) − n 1 2 ( 1 + 1 α ) O [ i − 1 n , i n ] ( x ) , while N O , t n : = O [ ⌊ n t ⌋ n , t ] ( x ) . In a similar way as for N L , t n we can show that N O , t n is negligible. Indeed, the following lemma holds true. Its proof can be found in the next section and is a direct consequence of (1.3). Lemma 3.3.

The process N O n = { N O , t n ; t ≥ 0 } satisfies the following convergence in probability, uniformly over compact sets: lim n n 1 2 ( 1 + 1 α ) N O n = 0 .

Due to Proposition 2.2 and Lemma 3.3 we can write (3.12) as W O , t n = ∑ i = 1 ⌊ n t ⌋ Z i n O + o P ( 1 ) . We are dealing with martingale differences. Indeed, E i − 1 n [ Z i n O ] = 0 because of Lemma 3.1(b) with p = 1, the definition of F and the independence of the increments of the process X. Therefore, similarly as before, our proof is based on [13, Theorem 3-2]. In particular, we want to show the following convergence statements: (3.13) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n O | 2 ] → P k O 2 L [ 0 , t ] ( x ) ∀ t ∈ [ 0 , 1 ] , (3.14) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ Z i n O Δ i n M ] → P 0 ∀ t ∈ [ 0 , 1 ] , (3.15) ∑ i = 1 ⌊ n t ⌋ E i − 1 n [ | Z i n O | 2 1 { | Z i n O | > ϵ } ] → P 0 ∀ ϵ > 0 . The condition expressed in (3.14) should hold for all square integrable continuous martingales.