The Statistics of Radio Astronomical Polarimetry: Disjoint, Superposed, and Composite Samples

A statistical framework is presented for the study of the orthogonally polarized modes of radio pulsar emission via the covariances between the Stokes parameters. To accommodate the typically heavy-tailed distributions of single-pulse radio flux density, the fourth-order joint cumulants of the electric field are used to describe the superposition of modes with arbitrary probability distributions. The framework is used to consider the distinction between superposed and disjoint modes, with particular attention to the effects of integration over finite samples. If the interval over which the polarization state is estimated is longer than the timescale for switching between two or more disjoint modes of emission, then the modes are unresolved by the instrument. The resulting composite sample mean exhibits properties that have been attributed to mode superposition, such as depolarization. Because the distinction between disjoint modes and a composite sample of unresolved disjoint modes depends on the temporal resolution of the observing instrumentation, the arguments in favor of superposed modes of pulsar emission are revisited, and observational evidence for disjoint modes is described. In principle, the four-dimensional covariance matrix that describes the distribution of sample mean Stokes parameters can be used to distinguish between disjoint modes, superposed modes, and a composite sample of unresolved disjoint modes. More comprehensive and conclusive interpretation of the covariance matrix requires more detailed consideration of various relevant phenomena, including temporally correlated subpulse modulation (e.g., jitter), statistical dependence between modes (e.g., covariant intensities and partial coherence), and multipath propagation effects (e.g., scintillation and scattering).