On the limit behavior of the periodogram of high-frequency sampled stable CARMA processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Fasen, Vicky²  Fuchs, Florian^1,3 

[1] Univ Ulm, Inst Math Finance, D-89081 Ulm, Germany

[2] ETH, Dept Math, CH-8092 Zurich, Switzerland

[3] Tech Univ Munich, Int Sch Appl Math, D-85748 Garching, Germany

关键词: CARMA process;    High-frequency data;    Lattice;    Levy process;    Periodogram;    Self-normalized periodogram;    Stable distribution;   

DOI  :  10.1016/j.spa.2012.08.003

来源: Elsevier

【 摘 要 】

In this paper we consider a continuous-time autoregressive moving average (CARMA) process (Y-t)(t is an element of R) driven by a symmetric alpha-stable Levy process with alpha is an element of (0, 2] sampled at a high-frequency time-grid {0, Delta(n), 2 Delta(n), ..., n Delta(n)}. where the observation grid gets finer and the last observation tends to infinity as n + infinity. We investigate the normalized periodogram I-n,I-Y Delta n (omega) = vertical bar n(-1/alpha) Sigma(n)(k=1) Y-k Delta n e(-i omega k)vertical bar(2). Under suitable conditions on Delta(n) we show the convergence of the finite-dimensional distribution of both Delta(2-2/alpha)(n)[I-n,I- Y Delta n (omega(1)Delta(n)), ..., I-n,I- Y Delta n (omega(m)Delta(n))] for (omega(1), ..., omega(m)) is an element of (R \ {0})(m) and of self-normalized versions of it to functions of stable distributions. The limit distributions differ depending on whether omega(1), ..., omega(m) are linearly dependent or independent over Z. For the proofs we require methods from the geometry of numbers. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free