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Appendix A: Equivalence of the simulation methods

In this appendix, we want to prove analytically that the two simulation methods mentioned in are in fact equivalent.

The established method calculates the correlation function components ξ_m, m = 0,..., N − 1 directly from the field components g_n, n = 0,..., N − 1. Since we impose periodic boundary conditions on the field, this can be done using the estimator (A.1)The real-space field components are calculated from the Fourier components by (A.2)Here, we set , which is equivalent to the field having zero mean in real space. Discretization and periodicity already imply – still, in order not to give double weight to this mode, we set it to zero as well. Of course, we then have to do the same in our new method, i.e. P_N/2 ≡ 0. Note that for the sake of readability, we still include this term in the formulae of this work.

Our new method draws a realization of the power spectrum and Fourier transforms it to obtain the correlation function: (A.3)In both methods, the variance of the field components in Fourier space is determined by the power spectrum, namely, for one realization, (A.4)To prove the equivalence of the two methods, we insert the Fourier transforms as given in Eq. (A.2) into the estimator, Eq. (A.1): (A.5)The sum over n simply gives a Kronecker δ, since (A.6)Thus, we obtain (A.7)where in the last step, we used the fact that the g_i are real, implying . In order to show that this is equivalent to Eq. (A.3), we now split the sum into two parts, omitting the zero term (since , as we explained before): (A.8)Inserting Eq. (A.4) we end up with (A.9)This is exactly the way we calculate ξ_m in our new method – thus, we proved analytically that the two methods are indeed equivalent. As mentioned before, we also confirmed this fact numerically.

Appendix B: Analytical calculation of the ξ₀-dependence of mean and covariance matrix

As mentioned in , our analytical calculation of the ξ₀-dependence of the mean and the covariance matrix does not produce practically usable results – nonetheless, it is interesting from a theoretical point of view and is thus presented in this appendix.

We are ultimately interested in the mean y and the covariance matrix C_y, however, we will first show calculations in ξ-space before addressing the problem of how to transform the results to y-space.

	Fig. B.1 Mean of ξn for different n as function of ξ0, determined from simulations (black points with error bars) and analytically to zeroth (red crosses), first (blue circles), second (green filled triangles; left panel only), third (purple empty triangles; left panel only), and tenth (brown squares) order.
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Appendix B.1: Calculation in ξ-space

The ξ₀-dependence of the mean ⟨ ξ₁ ⟩ (where the index is purely a numbering and does not denote the lag) can be computed as (B.1)with the conditional probability (B.2)We define the corresponding characteristic function as Fourier transform of the probability distribution (Φ ↔ p in short-hand notation; for details on characteristic functions see Keitel & Schneider 2011, hereafter KS2011, and references therein): Making use of the characteristic function Ψ(s₀,s₁) (where Ψ(s₀,s₁) ↔ p(ξ₀,ξ₁)) computed in KS2011, we can also write (B.5)Comparison with Eq. (B.3) yields (B.6)Now, we can calculate the mean (i.e. the first moment) from the characteristic function (equivalent to Eq. (B.1) – again, see KS2011 and references there): Using the result from KS2011 for the bivariate characteristic function, (B.9)where , we can calculate the derivative as (B.10)where we inserted the univariate characteristic function computed in KS2011, (B.11)To calculate the integral in Eq. (B.7), we use a Taylor expansion of Y_n(s₀) from Eq. (B.10): (B.12)We insert the derivative into Eq. (B.7) and thus obtain (B.13)According to the definition of Ψ(s₀) ↔ p(ξ₀), (B.14)and thus, after changing the order of summation and integration, Eq. (B.13) can finally be written as (B.15)Inserting the known result for p(ξ₀) and calculating its derivatives allows us to compare the analytical result to simulations. The results can be seen in ; here, the black points with error bars show the mean of ξ_n for different lags n as determined from simulations (100  000 realizations, Gaussian power spectrum with Lk₀ = 100), and the colored symbols show the analytical results to different order (see figure caption). It seems that, although the Taylor series in Eq. (B.15) does not converge, a truncation at order 10 yields sufficient accuracy, barring some numerical issues for very low ξ₀-values.

	Fig. B.2 Different elements of the covariance matrix C({ ξn }), determined from simulations (black points with error bars) and analytically to zeroth (red crosses), first (blue circles), fifth (purple triangles), and tenth (brown squares) order.
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The ξ₀-dependence of the covariance matrix C_ξ can be calculated in a similar way. We start from the general definition of covariance, The integral A can again be expressed in terms of the characteristic function Φ(s₁,s₂;ξ₀) ↔ p(ξ₁,ξ₂ | ξ₀): (B.16)Similar to the previous calculations, (B.17)with the trivariate characteristic function (B.18)Calculating the second derivative of B.18 yields (B.19)The Taylor expansions of Z_n(s₀) and Z_k,n(s₀) read Using it as well as the expansion (B.12), we finally obtain (B.22)We show a comparison of the results (for different elements of the covariance matrix) from simulations and the analytical formula in . Again, the black dots are obtained from simulations and the colored symbols represent the results from Eq. (B.22), where the last term (i.e. the one containing the mean values ⟨ ξ_n ⟩) was calculated up to tenth order, thus providing sufficient accuracy, as previously shown. As before, there are some numerical problems for very small values of ξ₀. Additionally, the analytical results do not agree with the simulations for small lags, as can be seen from the left-most panel (the same holds for other covariance matrix elements involving small lags). However, for the the higher-lag examples (i.e. the right two panels), a truncation of the Taylor series at tenth order seems to be accurate enough.

Appendix B.2: Transformation of mean and covariance matrix to y-space

In the previous section, we showed how to calculate the (ξ₀-dependent) mean and covariance matrix in ξ-space. The computation of the quasi-Gaussian approximation, however, requires the mean y and the covariance matrix C_y in y-space, which cannot be obtained from those in ξ-space in a trivial way due to the highly non-linear nature of the transformation ξ → y.

Thus, instead of settling for a linear approximation, we have to choose a more computationally expensive approach. Namely, we calculate the first and second moments (in ξ) of the quasi-Gaussian distribution as functions of the mean and (inverse) covariance matrix in y-space and equate the result to the analytical results, i.e. we solve a set of equation of the form where we did not write down the ξ₀-dependence explicitly for the sake of readability.

Note that this is a complicated procedure, since the integration on the equations’ left-hand sides can only be performed

numerically (we make use of a Monte-Carlo code from Press et al. 2007). In order to solve the equation set (consisting of equation for an N-variate distribution) we use a multi-dimensional root-finding algorithm (as provided within the GSL, Galassi et al. 2009). However, due to the high dimensionality of the problem, this procedure does not seem practical, since it is computationally very expensive – in addition to that, any possible gain in accuracy is averted by the required heavy use of purely numerical methods.

Thus, as described in , we refrain from using our analytical results for the mean and covariance matrix and simply determine them (as well as their ξ₀-dependence) from simulations, which we have shown to be sufficiently accurate.

© ESO, 2013