On critical cases in limit theory for stationary increments Lévy driven moving averages

Abstract

In this paper we present some limit theorems for power variation of stationary increments Lévy driven moving averages in the setting of critical regimes. In [5] the authors derived first and second order asymptotic results for k-th order increments of stationary increments Lévy driven moving averages. The limit theory heavily depends on the interplay between the given order of the increments, the considered power, the Blumenthal-Getoor index of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0. In this work we will study the critical cases, which were not covered in the original work [5].

A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities

“…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 R d -valued random variables (Y n ) 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 (Y n , Z) d − → (Y, Z) holds for any F -measurable Z; in this case we use the notation Y n L−s − −− → Y .…”

“…using the assumption that p > β. We consider only (3.19) in the case of Theorem 2.2(i) as (ii) is very similar, see [5]. In the case of (i) then b 1/p n = n α+1/p .…”

“…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 (X t ) t≥0 , defined on a probability space (Ω, F , P), that are given by…”

“…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.…”

“…Here ∆L s = L s − L s− where L s− = lim u↑s,u<s L u . To formulate the results of [6,5], we introduce the following set of assumptions on g, g 0 and ν:…”