where P ( n ; r,N ) is the normalised probability of getting n out of N points as neighbours inside a radius r of any of the points. The correlation dimension D 2 of the distribution of points can be defined via the power law scaling of correlation integral,i.e., C 2 ( r ) ∝ r $^{D$_{2}$}$. Since the scaling behaviour of C 2 can be different at different scales, we expect the correlation dimension to be a function of scale. For the special case of a homogeneous distribution we see that the correlation dimension equals the ambient dimension for an infinite set of points. C 2 ( r ) defined in the manner given by equation 4 provides the average of the distribution function of P ( n ; r,N ) . In order to char- acterise the distribution of points, we need information about all the moments of the distribution function. This leads us to the gen- eralised dimension D $_{q}$, also known as Minkowski-Bouligand di- mension. This is a generalisation of the correlation dimension D 2 . The correlation integral can be generalised to define C q ( r ) as which is used to define the Minkowski-Bouligand dimension We expect the generalised dimension to be scale dependent in gen- eral. If the value of D q ( r ) is independent of both q as well as r then the point distribution is called a mono-fractal. If we are dealing with a finite number of points in a finite volume then we can always define an average density. This al- lows us to relate the generalised correlation integral with correla- tion functions. For q > 1 , the generalised correlation integral and the Minkowski-Bouligand dimension can be related to a combina- tion of n -point correlation functions with the highest n equal to q and the smallest n equal to 2 . The contribution to C q is dominated by regions of higher number density of points for q ≫ 1 whereas for q ≪ 0 the contribution is dominated by regions of very low number density. This clearly implies that the full spectrum of gen- eralised dimension gives us information about the entire distribu- tion: regions containing clusters of points as well as voids that have few points. This allows us to connect the concept of fractal dimensions with the statistical measures used to quantify the distribution of matter at large scales in the universe. In the situation where the galaxy distribution is homogeneous and isotropic on large scales, we intuitively expect D q ≃ D = 3 independent of the value of q . Whereas at smaller scales we expect to see a spectrum of values for the fractal dimension, all different from 3 . It is of considerable interest to find out the scale where we can consider the univer se to be homogeneous. We address this issue using theoretical models in this paper. In an earlier paper we have derived a leading order expres- sion for generalised correlation integral for homogeneous as well as weakly clustered distributions of points (Bagla et al. 2008). We found that even for a homogeneous distribution of finite numb er of points the Minkowski-Bouligand dimension, D q ( r ) , is not ex- actly equal to the ambient dimension, D . The difference in these two quantities arises due to discreteness effects. Hence for a finite sample of points, the correct benchmark is not D but the estimated value of D q for a homogeneous distribution of same number of The third term on the right hand side in fact encapsulates the con- tribution of clustering. This differs from the last term in the corre- sponding expression for a homogeneous distribution as in that case the “clustering” is only due to cells being centred at points whereas in this case the locations of every pair of points has a weak correla- tion. It is worth noting that the highest order term of order O ( ¯ ξ 2 ) has a factor O ( q 3 ) and hence can become important for suffi- ciently large q . This may be quantified by stating that q ¯ ξ ≪ 1 is the more relevant small parameter for this expansion. The Minkowski- Bouligand dimension for this system can be expressed as: is the average number of particles in a randomly placed sphere and the volume averaged two point correlation function respectively. The generalised correlation integral (eq :6) for this distribution can now be written as where Bagla et al. (2008) have demonstrated that the general expres- sion for m th order moment of a weakly clustered distribution of points is given by: points in the same volume. An interesting aside is that the correc- tion due to a finite size sample always leads to a smaller valuefor D q ( r ) than D . As expected, this correction is small if the average number of points in spheres is large, i.e., ¯ N ≫ 1 . The scale of homogeneity 3 For a weakly clustered distribution we note that • For hierarchical clustering, both terms have the same sign and lead to a smaller value for D q as compared to D . • Unless the correlation function has a feature at some scale, smaller correlation corresponds to a smaller correction to the Minkowski-Bouligand dimension. • If the correlation function has a feature then it is possible to have a small correction term (Δ D q )$_{clus}$ for a relatively large ξ . In such a situation, the relation between ξ and (Δ D q )$_{clus}$ is no longer one to one. In such a case (Δ D q )$_{clus}$ does not vary monotonically with scale. D q ( r ) ≃ D ( 1 − ( q − 2) 2 ¯ N ) + q 2 ∂ ¯ ξ ∂ log r = D ( 1 − ( q − 2) 2 ¯ N ) + qr 2 ∂ ¯ ξ ∂r = D ( 1 − ( q − 2) 2 ¯ N − q 2 ( ¯ ξ ( r ) − ξ ( r ) ) = D − (Δ D q ) ¯ N − (Δ D q )$_{clus}$ Δ D q ( r ) = − (Δ D q ) ¯ N − (Δ D q )$_{clus}$ (11) NC q ( r ) = 〈N q$_{−}$ $^{1}$〉 p = ¯ N q$_{−}$ 1 [ 1 + ( q − 1) ( q − 2) 2 ¯ N + q ( q − 1) 2 ¯ ξ + O ( ¯ ξ $^{2}$) + O ( ¯ ξ ¯ N ) + O ( 1 ¯ N 2 ) ] (10) ¯ N = nV & ¯ ξ ( r ) = 3 r 3 ∫ 0 ξ ( x ) x $^{2}$dx (9) 〈N $^{m}$〉 p = ¯ N m [ 1 + ( m ) ( m − 1) 2 ¯ N + m ( m + 1) 2 ¯ ξ + O ( ¯ ξ $^{2}$) + O ( ¯ ξ ¯ N ) + O ( 1 ¯ N 2 ) ] (8) D 2 ( r ) = ∂ log C 2 ( r ) ∂ log r (5) C q ( r ) = 1 N N ∑ n =0 n q$_{−}$ $^{1}$P ( n ) ≡ 1 N 〈N q$_{−}$ $^{1}$〉 p (6) D q ( r ) = 1 q − 1 d log C q ( r ) d log r (7)