On limit theory for Lévy semi-stationary processes

Abstract

In this paper we present some limit theorems for power variation of Lévy semistationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in [14], where the authors derived the limit theory for kth order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power p > 0, the Blumenthal-Getoor index β ∈ (0, 2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α. In this paper we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.

On the Divergence and Vorticity of Vector Ambit Fields

“…with F a vector valued function, R a compact set in R 2 and L a real-valued homogeneous Lévy basis, we may in principle try to follow the same reasoning as in the temporal case and expect to recover similar results. Thus, we may try to decompose X as X(p) = ∂X(p) + X(p), (10) for some fields ∂X and X whose trajectories are totally determined by the behavior of L and F on ∂R(p) and R(p), respectively. Unfortunately, to our knowledge, there is no a general identification of the fields ∂X and X appearing in (10), except in very special cases (see for instance [17]).…”

“…Thus, we may try to decompose X as X(p) = ∂X(p) + X(p), (10) for some fields ∂X and X whose trajectories are totally determined by the behavior of L and F on ∂R(p) and R(p), respectively. Unfortunately, to our knowledge, there is no a general identification of the fields ∂X and X appearing in (10), except in very special cases (see for instance [17]). However, our proof is based on an analogous decomposition to that in (7).…”

“…As further references to theory and applications of LSS, see for instance [33], [6] and [35] and therein references. For recent results on limit theorems see [19] and [10].…”