  A spectral representation for regularly varying L\'evy processes with index
between one and two is established and the properties of the resulting random
noise are discussed in detail giving also new insight in the $L^2$-case where
the noise is a random orthogonal measure.
  This allows a spectral definition of multivariate regularly varying
L\'evy-driven continuous time autoregressive moving average (CARMA) processes.
It is shown that they extend the well-studied case with finite second moments
and coincide with definitions previously used in the infinite variance case
when they apply.
