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On limit theory for functionals of stationary increments Lévy driven moving averages. (English) Zbl 1467.60019
Summary: In this paper we present new limit theorems for variational functionals of stationary increments Lévy driven moving averages in the high frequency setting. More specifically, we will show the “law of large numbers” and a “central limit theorem”, which heavily rely on the kernel, the driving Lévy process and the properties of the functional under consideration. The first order limit theory consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we prove the associated weak limit theory, which again consists of three different cases. Our work is related to [the first author et al., Ann. Probab. 45, No. 6B, 4477–4528 (2017; Zbl 1427.60081); the first author and M. Podolskij, Stochastics 89, No. 1, 360–383 (2017; Zbl 1379.60050)], who considered power variation functionals of stationary increments Lévy driven moving averages. However, the asymptotic theory of the present paper is more complex. In particular, the weak limit theorems are derived for an arbitrary Appell rank of the involved functional.

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G22 Fractional processes, including fractional Brownian motion
60G52 Stable stochastic processes
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