In recent years limit theorems and statistical inference for high frequency observations of stochastic processes have received a great deal of attention. The most prominent class of high frequency statistics are power variations that have been proved to be of immense importance for the analysis of the fine structure of an underlying stochastic process. The asymptotic theory for power variations and related statistics has been intensively studied in the setting of Itô semimartingales, fractional Brownian motion and Brownian semi-stationary processes, to name just a few; see for example [2–4, 7, 9] among many others.

In the recent work [6, 5] power variations of stationary increments Lévy moving average processes have been investigated in details. These are continuous-time stochastic processes (Xt)t≥0 , defined on a probability space (Ω,F,P) , that are given by (1.1) Xt=∫−∞t(g(t−s)−g0(−s))dLs, where L=(Lt)t∈R is a symmetric Lévy process on R with L0=0 and without Gaussian component. Moreover, g,g0:R→R are deterministic functions vanishing on (−∞,0) . The most prominent subclasses include Lévy moving average processes, which correspond to the setting g0=0 , and the linear fractional stable motion, which is obtained by taking g(s)=g0(s)=s+α and L being a symmetric β-stable Lévy process with β∈(0,2) . The latter is a self-similar process with index H=α+1/β ; see [12].

We introduce the kth order increments Δi,knX of X, k∈N , that are defined by (1.2) Δi,knX:= ∑j=0k(−1)jkjX(i−j)/n,i≥k. For example, we have that Δi,1nX=Xin−Xi−1n and Δi,2nX=Xin−2Xi−1n+Xi−2n . The main statistic of interest is the power variation computed on the basis of kth order increments: (1.3) V(X,p;k)n:= ∑i=kn|Δi,knX|p,p>0. 0.\]]]> A variety of asymptotic results has been shown for the statistic V(p;k)n in [6, 5]. The mode of convergence and possible limits heavily depend on the interplay between the power p, the form of the kernel function g and the Blumenthal–Getoor index of L. We recall that the Blumenthal–Getoor index is defined via (1.4) β:=inf{r≥0:∫−11|x|rν(dx)<∞}∈[0,2], where ν denotes the Lévy measure of L. It is well known that ∑s∈[0,1]|ΔLs|p is finite when p>β \beta $]]>, while it is infinite for p<β . Here ΔLs=Ls−Ls− where Ls−=limu↑s,u<sLu . To formulate the results of [6, 5], we introduce the following set of assumptions on g, g0 and ν:

Assumption (A): The function g:R→R satisfies the condition (1.5) g(t)∼c0tαast↓0for someα>0andc0≠0, 0\hspace{2.5pt}\text{and}\hspace{2.5pt}{c_{0}}\ne 0,\]]]> where g(t)∼f(t) as t↓0 means that limt↓0g(t)/f(t)=1 . For some w∈(0,2] , lim supt→∞ν(x:|x|≥t)tw<∞ and g−g0 is a bounded function in Lw(R+) . Furthermore, g is k-times continuous differentiable on (0,∞) and there exists a δ>0 0$]]> such that |g(k)(t)|≤Ktα−k for all t∈(0,δ) , |g(k)| is decreasing on (δ,∞) and g(j)∈Lw((δ,∞)) for j∈{1,k} .

Assumption (A-log): In addition to (A) suppose that ∫δ∞|g(k)(s)|w|log(|g(k)(s)|)|ds<∞. Intuitively speaking, Assumption (A) says that g(k) may have a singularity at 0 when α is small, but it is smooth outside of 0. The theorem below has been proved in [6, 5]. We recall that a sequence of Rd -valued random variables (Yn)n≥1 is said to converge stably in law to a random variable Y, defined on an extension of the original probability space (Ω,F,P) , whenever the joint convergence in distribution (Yn,Z)→d(Y,Z) holds for any F -measurable Z; in this case we use the notation Yn→L−sY . We refer to [1, 11] for a detailed exposition of stable convergence.

We remark that the first order asymptotic theory of [6, Theorem 1.1] includes two more regimes: an ergodic type limit theorem in the setting p<β , α<k−1/β and convergence in probability to a random integral in the setting p≥1 , α>k−1/max{p,β} k-1/\max \{p,\beta \}$]]>. However, in this paper we concentrate ourselves on results of Theorem 1.1, which are quite non-standard in the literature. More specifically, our aim is to extend the theory of Theorem 1.1 to kernels g that exhibit multiple singularities. We call a point x∈R+ a singularity point when the kth derivative g(k) of g explodes at x. Note that under Assumption (A) and condition α≤k−1/p the function g has only one singularity point at x=0 . In practical applications a singularity point x∈R+ leads to a strong feedback effect stemming from the past jumps around the time t−x . Such effects has been discussed in the context of turbulence modelling in [8].

We will show that the limits in Theorem 1.1(i) and (ii) will be affected by the presence of multiple singularity points. More precisely, we will see that the increments Δi,knX can be heavily influenced by the jumps of L that happened in the past, and the time delay is determined by the singularity points of g. The obtained result is similar in spirit to the work [8] that studied quadratic variation of Brownian semi-stationary processes under multiple singularities of the kernel g. Furthermore, we will prove that in general the stable convergence in Theorem 1.1(i) only holds along a subsequence.

The paper is structured as follows. Section 2 presents the main results of the article. Proofs are collected in Section 3.

We consider stationary increments Lévy moving average processes as defined at (1.1) and recall that the driving motion L is a pure jump Lévy process with Lévy measure ν. Now, we introduce the condition on the kernel function g:

Assumption (B): For some w∈(0,2] , lim supt→∞ν(x:|x|≥t)tw<∞ and g−g0 is a bounded function in Lw(R+) . Furthermore, there exist points 0=θ0<θ1<⋯<θl such that the following properties hold: (i)

g(t)∼c0tα0 as t↓0 for some α0>0 0$]]> and c0≠0 .

It turns out that only the minimal powers among {α0,…,αl} determine the asymptotic behaviour of the statistic V(X,p;k)n . Thus, we define (2.2) α:=min{α0,…,αl}andA:={z:αz=α}. Furthermore, we introduce the notation hk,0:=hk and (2.3) hk,z(x)= ∑j=0k(−1)jkj|x−j|αzforz=1,…,l. In the main result below we consider a subsequence (nj)j∈N such that the following condition holds: (2.4) limj→∞{njθz}=ηz∈[0,1]for allz∈A, where {x} denotes the fractional part of x∈R . Obviously, such a subsequence always exists since {nθz} is a bounded sequence. Sometimes we will require a stronger condition, which is analogous to Assumption (A-log):

Assumption (B-log): Condition (B) holds and we have that ∫θl+δ′∞|g(k)(t)|w|log(|g(k)(t)|)|dt<∞. The main result of the paper is the following theorem.

We remark that the stable convergence in Theorem 2.2(i) only holds along the subsequence (nj)j≥1 , which is seen from the form of the limit in (2.5) that depends on (ηz) . The original statistic nαpV(X,p;k)n is tight, but does not converge except when θz∈N for all z∈A . On the other hand, in Theorem 2.2(ii) we do not require to consider a subsequence.

Notice that the interval [−θz,1−θz] , which appears in Theorem 2.2, is the set [0,1] shifted by θz to the left. Given the discussion of Remark 2.1, such a shift in the limit is not really surprising. We recall that a similar phenomenon has been discovered in [8] in the context of Brownian semi-stationary processes. These are stochastic processes (Yt)t≥0 defined by Yt=∫−∞tg(t−s)σsdWs, where W is a two-sided Brownian motion and (σt)t∈R is a cádlág process. When the kernel function g satisfies conditions (B)(i) and (B)(ii) along with some further assumptions, which in particular ensure the existence of Yt , the authors have shown the following convergence in probability (see [8, Theorem 3.2]): (2.7) 1nτn2V(Y,2;k)n⟶P∑z∈Aπz∫−θz1−θzσs2ds where τn2=E[(Δk,knG)2] with Gt=∫−∞tg(t−s)dWs , and the probability weights (πz)z∈A are given by πz=cz2‖hk,z‖L2(R)2∑z∈Acz2‖hk,z‖L2(R)2. Hence, we observe the same shift phenomenon in the integration region as in Theorem 2.2.

Throughout this section all positive constants are denoted by C although they may change from line to line. We will divide the proof of Theorem 2.2 into several steps. First, we will show the statements (2.5) and (2.6) for a compound Poisson process. In the second step we will decompose the jump measure of L into jumps that are bigger than ϵ and jumps that are smaller than ϵ. The big jumps form a compound Poisson process and hence the claim follows from the first step. Finally, we prove negligibility of small jumps when ϵ→0 .

We start with an important proposition.

Now, we introduce the notation (3.5) gi,n(x)= ∑j=0k(−1)jkjg((i−j)/n−x), and observe the identity Δi,knX=∫Rgi,n(s)dLs. The next lemma presents some estimates for the function gi,n . Its proof is a straightforward consequence of Assumption (B) and the Taylor expansion. Lemma 3.2.

Suppose that Assumption (B) holds and let z=1,…,l . Then there exists an N∈N such that for all n≥N and i∈{k,…,n} the following hold: (i)

|gi,n(x)|≤C(|i/n−x−θz|αz+n−αz) for all x∈[i−2kn−θz,i+2kn−θz] .