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

In this paper we consider a continuous-time autoregressive moving average (CARMA) process (Yt)t∈R driven by a symmetric α-stable Lévy process with α∈(0,2] sampled at a high-frequency time-grid {0,Δn,2Δn,…,nΔn}, where the observation grid gets finer and the last observation tends to infinity as n→∞. We investigate the normalized periodogram In,YΔn(ω)=|n−1/α∑k=1nYkΔne−iωk|2. Under suitable conditions on Δn we show the convergence of the finite-dimensional distribution of both Δn2−2/α[In,YΔn(ω1Δn),…,In,YΔn(ωmΔn)] for (ω1,…,ωm)∈(R∖{0})m and of self-normalized versions of it to functions of stable distributions. The limit distributions differ depending on whether ω1,…,ωm are linearly dependent or independent over Z. For the proofs we require methods from the geometry of numbers.