We have measured the count probability distribution function (CPDF) in a series of 10 volume-limited sub-samples of a deep redshift survey of IRAS galaxies. The CPDF deviates significantly from both the Poisson and Gaussian limits in all but the largest volumes. We derive the volume-averaged 2-, 3-, 4-, and 5 point correlation functions from the moments of the CPDF and find them all to be reasonably well described by power laws. Weak systematic effects with the sample size provide evidence for stronger clustering of galaxies of higher luminosity on small scales. Nevertheless, remarkably tight relationships hold between the correlation functions of different order. In particular, the "normalized" skewness defined by the ratio S₃ ≡ barζ₃/ barζ₂² varies at most weakly with scale in the range 0.1 < ζ₂ < 10. That is, S₃ is close to constant (= 1.5±0.5) from weakly to strongly nonlinear scales. On small scales, this is consistent with previous determinations of the three-point correlation function ζ ≡ ζ₃. On larger scales, this conforms with the hypothesis of the growth of observed structures by gravitational clustering of initially Gaussian density fluctuations. We similarly find that barζ₄ is proportional to the third power of barζ₂ in the same range of barζ₂, and there is weak evidence that barζ₅ is proportional to the fourth power of barζ₂. Furthermore, we find that the void probability function obeys a scaling relation with density to great precision, in accord with the scale-invariance hypothesis ( barζN ∝ barζ₂N-1). Double-counting cluster galaxies in order to match the cluster overdensities seen in optically selected samples of galaxies increases greatly the derived value of S₃ and S₄, although the scaling between the correlations of different orders remains. Unfortunately, the relative sparseness of the IRAS sample precludes using it to make the most demanding tests of scale invariance, which rely on the overall shape of the CPDF at different scales. In this sparse limit, various models for the CPDF become degenerate and fit the IRAS data nearly equally well. Indeed, the CPDF is well fitted by both the negative binomial distribution, and the thermodynamical model of Saslaw & Hamilton, and to a somewhat lesser extent by the log-normal distribution. All three models fit the data poorly for the densest subsample of IRAS galaxies examined, but this may be more a reflection of finite volume effects than of the inadequacy of the models.
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A 1993 study studied this question.