In order to test for non-gaussianity, one can use traditional methods
based on the distribution of temperature anisotropies. The simplest tests
are based on the third and/or fourth order moments (skewness and kurtosis) of
the
distribution, both equal to zero for a Gaussian distribution. Another test for
non-gaussianity through the temperature distribution is based on the study of
the cumulants (Ferreira et al. 1997; Winitzki 1998). The *n*-point
correlation functions also give very valuable statistical information.
In particular, the three-point function, and
its spherical harmonic transform (the bispectrum), are appropriate tools for
the detection of non-gaussianity
(Luo & Schramm 1993; Kogut et al. 1996; Ferreira & Magueijo
1997; Ferreira et al. 1998; Heavens 1998; Spergel & Goldberg 1998). An
investigation of the detailed behaviour of each multipole of the CMB angular
power
spectrum (transform of the two-point function) is another non-gaussianity
indicator (Magueijo 1995). Finally, other tests of non-gaussianity are
based on topological pattern statistics (Coles 1988; Gott et al. 1990).

The works of Ferreira & Magueijo (1997) on the search for
non-gaussianity in Fourier space, and of Ferreira et al. (1997) on the
cumulants of wavelet transformed maps, have shown that these approaches
allow the study of characteristic scales, which is particularly interesting
when studying a combination of Gaussian and non-Gaussian signals as a
function of scale. In the present work, we study non-gaussianity in a
wavelet decomposition framework, that is, using the coefficients in the wavelet
decomposition. We
decompose the image of the studied signal into a wavelet basis and analyse the
statistical properties of the coefficient distribution. We then
search for reliable statistical diagnoses to distinguish between Gaussian and
non-Gaussian signals.
The aim of this study is to find suitable tools for demonstrating and
quantifying the non-Gaussian signature of a signal when it is combined with a
Gaussian distributed signal of similar, or higher, amplitude.

In Sect. 2, we describe the methods for wavelet decomposition and filtering. We then briefly describe the characteristics of the wavelet we use in our study. We also give the main characteristics of the test data sets used in our work. In Sect. 3, we present the statistical criteria we developed to test for non-gaussianity. We then apply the tests to sets of Gaussian test maps (Sect. 4). We also apply them, in Sect. 5, to sets of non-Gaussian maps as well as combinations of Gaussian and non-Gaussian signals with the same power spectrum. In Sect. 6, we present and apply the detection strategy we propose to quantify the detectability of the non-Gaussian signature. Finally, in Sect. 7 we conclude and present our main results.

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