CUMULANTS AS NON-GAUSSIAN QUALIFIERS

We discuss the requirements of good statistics for quantifying non-Gau ssianity in the cosmic microwave background. The importance of rotatio nal invariance and statistical independence is stressed, but we show t hat these are sometimes incompatible. It is shown that the first of th ese requirements prefers a real space (or wavelet) formulation, wherea s the latter favors quantities defined in Fourier space. Bearing this in mind we decide to be eclectic and define two new sets of statistics to quantify the level of non-Gaussianity. Both sets make use of the c oncept of cumulants of a distribution. However, one set is defined in real space, with reference to the wavelet transform, whereas the other is defined in Fourier space. We derive a series of properties concern ing these statistics for a Gaussian random field and show how one can relate these quantities to the higher order moments of temperature map s. Although our frameworks lead to an infinite hierarchy of quantities we show how cosmic variance and experimental constraints give a natur al truncation of this hierarchy. We then focus on the real space stati stics and analyze the non-Gaussian signal generated by point sources o bscured by large scale Gaussian fluctuations. We conclude by discussin g the practical implementations of these techniques. [S0556-2821(97)02 920-2].