Sampling distributions of random electromagnetic fields in mesoscopic or dynamical systems

We derive the sampling probability density function (pdf) of an ideal localized random electromagnetic field, its amplitude, and intensity in an electromagnetic environment that is quasistatically time-varying statistically homogeneous or static statistically inhomogeneous. The results allow for the estimation of field statistics and confidence intervals when a single spatial or temporal stochastic process produces randomization of the field. Sampling distributions are particularly significant when the number of degrees of freedom ν is relatively small (typically, ν<40), e.g., in mesoscopic systems when the sample set size N is relatively small by choice or by force. Results for both coherent and incoherent detection methods are derived for Cartesian, planar, and full-vectorial fields. We show that the functional form of the sampling pdf depends on whether the random variable is dimensioned (e.g., the sampled electric field proper) or is expressed in dimensionless standardized or normalized form (e.g., the sampled electric field divided by its sample standard deviation or sample mean). For dimensioned quantities, the electric field, its amplitude, and intensity exhibit different types of Bessel K sampling pdfs, which differ significantly from the asymptotic Gauss normal and χ_2p⁽²⁾ ensemble pdfs when ν is relatively small. By contrast, for the corresponding standardized quantities, Student t, Fisher-Snedecor F, and root-F sampling pdfs are obtained that exhibit heavier tails than comparable Bessel K pdfs. Statistical uncertainties obtained from classical small-sample theory for dimensionless quantities are shown to be overestimated compared to dimensioned quantities. Differences in the sampling pdfs arising from denormalization versus destandardization are identified.