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 ( Y t ) t ∈ R driven by a symmetric α -stable Lévy process with α ∈ ( 0 , 2 ] sampled at a high-frequency time-grid { 0 , Δ n , 2 Δ n , … , n Δ n } , where the observation grid gets finer and the last observation tends to infinity as n → ∞ . We investigate the normalized periodogram I n , Y Δ n ( ω ) = | n − 1 / α ∑ k = 1 n Y k Δ n e − i ω k | 2 . Under suitable conditions on Δ n we show the convergence of the finite-dimensional distribution of both Δ n 2 − 2 / α [ I n , Y Δ n ( ω 1 Δ n ) , … , I n , Y Δ n ( ω m Δ n ) ] for ( ω 1 , … , ω m ) ∈ ( R ∖ { 0 } ) m and of self-normalized versions of it to functions of stable distributions. The limit distributions differ depending on whether ω 1 , … , ω m are linearly dependent or independent over Z . For the proofs we require methods from the geometry of numbers.