Expansions for Nearly Gaussian Distributions
ASTRONOMY & ASTROPHYSICS MAY II 1998, PAGE 193 SUPPLEMENT SERIES Astron. Astrophys. Suppl. Ser. 130, 193–205 (1998)
Expansions for nearly Gaussian distributions
S. Blinnikov1,2 and R. Moessner2 1 Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia Sternberg Astronomical Institute, 119899 Moscow, Russia 2 Max-Planck-Institut f¨ur Astrophysik, D-85740 Garching, Germany
Received July 8; accepted November 10, 1997
Abstract. Various types of expansions in series of Cheby- Bertschinger 1991; Kofman et al. 1994; Moessner et al. shev-Hermite polynomials currently used in astrophysics 1994; Juszkiewicz et al. 1995; Bernardeau & Kofman 1995; for weakly non-normal distributions are compared, namely Amendola 1994; Colombi 1994; Moessner 1995; Ferreira the Gram-Charlier, Gauss-Hermite and Edgeworth expan- et al. 1997; Gazta˜naga et al. 1997), in analyzing the ve- sions. It is shown that the Gram-Charlier series is most locity distributions and fine structure in elliptical galaxies suspect because of its poor convergence properties. The (Rix & White 1992; van der Marel & Franx 1993; Gerhard Gauss-Hermite expansion is better but it has no intrinsic 1993; Heyl et al. 1994), and in studies of large Reynolds measure of accuracy. The best results are achieved with number turbulence (Tabeling et al. 1996). In this paper we the asymptotic Edgeworth expansion. We draw attention present a unified approach to the formalism used in those to the form of this expansion found by Petrov for arbitrary applications, illustrating it by examples of similar prob- order of the asymptotic parameter and present a simple al- lems arising in cosmology and in the theory of supernova gorithm realizing Petrov’s prescription for the Edgeworth line spectra. expansion. The results are illustrated by examples similar The first investigation of slightly non-Gaussian distri- to the problems arising when fitting spectral line profiles butions was undertaken by Chebyshev in the middle of the of galaxies, supernovae, or other stars, and for the case of 19th century. He studied in detail a family of orthogonal approximating the probability distribution of peculiar ve- polynomials which form a natural basis for the expansions locities in the cosmic string model of structure formation. of these distributions. A few years later the same polyno- mials were also investigated by Hermite and they are now Key words: methods: statistical; cosmic strings; line: called Chebyshev-Hermite or simply Hermite polynomi- profiles als (their definition was first given by Laplace, see e.g. Encyclopaedia of Mathematics 1988). There are several forms of expansions using Hermite polynomials, namely the Gram-Charlier, Gauss-Hermite 1. Introduction and Edgeworth expansions. We introduce the notation in 2 The normal, or Gaussian, distribution plays a promi- Sect. 2 and use the simple example of the χ distribution nent role in statistical problems in various fields of astro- for various degrees of freedom to illustrate the properties of Gram-Charlier expansions in Sect. 3. In subsequent sec- physics and general physics. This is quite natural, since 2 the sums of random variables tend to a normal distri- tions the χ distribution is used for testing the other two bution when the quite general conditions of the central expansions. We show in Sect. 4 that the Gram-Charlier limit theorem are satisfied. In many applications, to ex- series is just a Fourier expansion which diverges in many tract useful information on the underlying physical pro- situations of practical interest, whereas the Gauss-Hermite cesses, it is more interesting to measure the deviations of series has much better convergence properties. However, a probability density function (hereafter PDF) from the we point out that another series, the so called Edgeworth normal distribution than to prove that it is close to the expansion, is more useful in many applications even if it is Gaussian one. This has been done for example in the work divergent, since, first, it is directly connected to the mo- on peculiar velocities and cosmic microwave background ments and cumulants of a PDF (the property which is anisotropies in various cosmological models (Scherrer & lost in the Gauss-Hermite series) and, second, it is a true asymptotic expansion, so that the error of the approxima- Send offprint requests to:S.Blinnikov tion is controlled. 194 S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions
Some applications (e.g. Bernardeau 1992, 1994; of degrees n 6= m are orthogonal on the real axis with Moessner 1995) involve cumulants of higher order. respect to the weight function w(x)if These require the use of a correspondingly higher or- ∞ der Edgeworth series, but only the first few terms of w(x)Pn(x)Pm(x)=0. (6) Z−∞ the series are given in standard references (Cram´er 1957; We follow the notation of Abramowitz & Stegun (1972) Abramowitz & Stegun 1972; Juszkiewicz et al. 1995; where possible, so Hen (x) denotes the polynomial with Bernardeau & Kofman 1995). In Sect. 5 we popularize weight function w(x) = exp(−x2/2) ∝ Z(x). According a derivation of the Edgeworth expansion due to Petrov to Rodrigues’ formula, (1962, 1972, 1987) for an arbitrary order of the asymptotic n 2 d 2 parameter, and we present a slightly simpler and more He (x)=(−1)nex /2 e−x /2 . (7) n dxn straightforward way of obtaining his result. The formula We will refer to He (x) as Chebyshev-Hermite polynomi- found by Petrov (see Sect. 5 below) requires a summation n als following Kendall (1952). The ones with the weight over indices with non-trivial combinatorics which hindered w(x) = exp(−x2) ∝ Z2(x), its direct application. We find a simple algorithm realizing n 2 d 2 Petrov’s prescription for any order of the Edgeworth ex- n x −x Hn(x)=(−1) e n e , (8) pansion; this algorithm is easily coded, e.g. with standard dx will be called Hermite polynomials. Fortran, eliminating the need for symbolic packages. The expansions in Chebyshev-Hermite and Hermite We find that the Edgeworth-Petrov expansion is in- polynomials are used in many applications in astrophysics. deed very efficient and reliable. In Sect. 6 we apply the For a PDF p(x) which is nearly Gaussian, it seems natural formalism to the problem of peculiar velocities resulting to use the expansion from cosmic strings studied by Moessner (1995) and we ∞ dnZ(x) show how this technique allows one to reliably extract de- p(x) ∼ c , (9) n dxn viations from Gaussianity, even when they are tiny. n=0 X where the coefficients cn measure the deviations of p(x) from Z(x). From the definitions (5) and (7) it follows that 2. Background and notation dnZ(x) =(−1)nHe (x)Z (x) , (10) dxn n Let F (x) be the cumulative probability distribution of a so that random variable X. Then the mean value for the random ∞ n variable g(X) is the expectation value p(x) ∼ (−1) cnHen (x)Z (x) , (11) n=0 ∞ X Eg(X) ≡hg(X)i≡ g(x)dF (x) . (1) with (−1)n ∞ Z−∞ c = p(t)He (t)dt. (12) n n! n The PDF is p(x)=dF(x)/dx. The distribution F (x) Z−∞ is not necessarily smooth, so it can happen that p(x) This is the well-known Gram-Charlier series (of type A, is nonexistent at certain points. Nevertheless, the mean see e.g. Cram´er 1957; Kendall 1952 and references therein Eg(X) is defined as long as the integral in (1) exists. to the original work). Since Hen (x)isapolynomial,we Following the definition (1), the k-th order moment of X see that the coefficient cn in (12) is a linear combination is of the moments αk of the random variable X with PDF ∞ p(x). The combination is easily found by using the explicit k k αk ≡ EX = x dF (x) . (2) expression for Hen (x), which we derive in Appendix B:, Z−∞ namely Thus, the mean of X is m ≡ α1 = EX and its dispersion [n/2] k n−2k is (−1) x Hen (x)=n! . (13) ∞ k!(n − 2k)! 2k k=0 σ2 ≡ E(X − EX)2 = (x − m)2dF (x) . (3) X −∞ The Gram-Charlier series was used in cosmological appli- Z cations in the paper by Scherrer & Bertschinger (1991), We denote the cumulative normal distribution by but note that it is incorrectly called Edgeworth expansion x 1 x P (x) ≡ Z(t)dt = √ exp(−t2/2)dt, (4) there. The Gram-Charlier series is merely a kind of Fourier 2π expansion in the set of polynomials He (x). This expan- Z−∞ Z−∞ n so its PDF is the Gaussian function sion has poor convergence properties (Cram´er 1957). Very often, for realistic cases, it diverges violently. We consider exp(−x2/2) Z(x)= √ . (5) such an example in the next section. The Edgeworth se- 2π ries, discussed in detail in Sect. 5, is a true asymptotic We also need to recall some definitions for sets of orthog- expansion of the PDF, which allows one to control its ac- onal polynomials. Any two polynomials Pn(x)andPm(x) curacy. S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions 195
3. An example based on the χ2 distribution
A good example for illustrating the fast divergence of the 2 Gram-Charlier series is given by its application to the χν distribution with ν degrees of freedom, since the moments of this distribution are known analytically and its PDF tends to the Gaussian one for large ν.IfX1,X2, ..., Xν , are independent, normally distributed random variables with zero expectation and unit dispersion, then the variable 2 2 ν 2 χ = χν ≡ i=1 Xi has the PDF (χ2)ν/2−1 exp(−χ2/2) ρ(χ2)= P ,χ2>0. (14) 2ν/2Γ(ν/2) 2 The expectation√ value of χ is ν and its dispersion is 2ν. If x =(χ2−ν)/ 2ν,thenxhas zero expectation and unit dispersion and its PDF p(x) asymptotically tends to the Gaussian distribution Z(x). Transforming ρ(χ2) from (14) to p(x), we obtain for x>− ν/2: √ √ √ ( 2νx + ν)ν/2−1 exp −( 2νx + ν)/2 p(x)= 2ν p . (15) 2ν/2Γ(ν/2) The χ2 distribution was employed by Matsubara & Yoko- yama (1996) for a representation of the cosmological den- sity field (see also Luo 1995 for an application of the χ2 dis- tribution in the context of cosmic microwave background temperature anisotropies). It can also serve as a crude model for approximating the profiles of spectral lines in moving media. If the lines are nearly Gaussian in the mat- ter at rest, then the motion with a high velocity gradi- ent, like that in supernova envelopes, leads to distortions 2 which can be approximated by the χν PDF (with ν =2 for highly non-Gaussian profiles, see e.g. Blinnikov 1996, and references therein). The comparison of the Gram-Charlier approximations of p(x) with the exact results is presented in Figs. 1 and 2 for an increasing number of terms in the expansion. It is clear that the series quickly becomes inaccurate with a larger number of terms included.
4. Fourier expansions
In order better to understand the poor convergence prop- erties of the Gram-Charlier series, let us first discuss how Fig. 1. The normalized χ2 PDF (15) for ν = 5 (dashed line), it is related to the Fourier expansion. A Fourier expansion and its Gram-Charlier approximations with 2 and 6 terms in (Szeg¨o 1978; Suetin 1979; Nikiforov & Uvarov 1988) for the expansion (solid line) any function f(x) in the set of orthogonal polynomials Pn is given by and ∞ √ f(x) ∼ a P (x) , (16) hn = 2πn!forHen (x) , (19) n n √ n=0 h = π2nn!forH(x). (20) X n n with Now we can see that the Gram-Charlier series (11) is just 1 ∞ an = w(t)f(t)Pn(t)dt. (17) the Fourier expansion (16) of f(x)=p(x)/Z(x)intheset h n n Z−∞ of Chebyshev-Hermite polynomials with cn =(−1) an. Here hn is the squared norm The properties of the Gram–Charlier approximations ∞ 2 of p(x) in Figs. 1 and 2 are to be considered in the gen- hn = w(t)Pn (t)dt, (18) eral context of the convergence of Fourier expansions. Z−∞ 196 S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions
Hermite polynomials Hen (7), as for the Gram-Charlier series) is sometimes used: ∞ p(x) ∼ anHn(x)Z(x) , (21) n=0 X with √ π ∞ a = Z(t)p(t)H (t)dt. (22) n 2n−1n! n Z−∞ This series is often called the Gauss-Hermite expansion (see e.g. its application to spectral lines of galaxies in van der Marel & Franx 1993). Examples in Figs. 3 and 4 show its better convergence. Van der Marel & Franx (1993) use a theorem due to Myller-Lebedeff on the convergence of the Gauss-Hermite expansion: it converges when x3p(x) → 0 for any p(x) with finite and continuous second derivative. Actually, the conditions sufficient for the convergence are better: if p(t) obeys the Lipschitz condition |p(t) − p(x)|≤M|t−x|α,M=const, 0<α≤1,(23) in a vicinity of x and ∞ |p(t)|(1 + |t|3/2)dt<∞, (24) Z−∞ then the Gauss-Hermite series converges to p(x)atx(see e.g. Suetin 1979). These weaker conditions imply that the class of PDFs with convergent Gauss-Hermite expansions is much wider than suggested by the Myller-Lebedeff the- orem cited in van der Marel & Franx (1993). But the sim- ple relation between the coefficients in the expansion and the moments of the PDF typical for the Gram-Charlier series is now lost (cf. Eqs. (12), (13) and (22)). It might be not important in many practical applications when the moments cannot be accurately determined from observa- tions, but it is very important for a theoretical work based on the analysis of the moments. Our Figs. 3 and 4 show that the χ2 PDF is well-suited for approximation by a Gauss-Hermite series. However, one should be cautious about the accuracy of the compu- tation of the Fourier coefficients for higher order terms. We have not encountered the problem of numerical errors in our Gram-Charlier example, since there all coefficients can be calculated analytically. Yet in general care must Fig. 2. The same as in Fig. 1 but for 12 and 36 terms in the be taken of the computational accuracy to avoid spurious Gram-Charlier expansion numerical divergence of a series which converges theoret- ically.
The source of the divergence lies in the sensitivity of the Gram-Charlier series to the behavior of p(x) at infinity 5. The Edgeworth asymptotic expansion –thelattermustfalltozerofasterthanexp(−x2/4) for A random variable X can be normalized to unit variance the series to converge (Cram´er 1957; Kendall 1952). This by dividing by its standard deviation σ. The Edgeworth is often too restrictive for practical applications. Our ex- expansion is a true asymptotic expansion of the PDF of ample of the χ2 distribution in (15), with its exponential the normalized variable X/σ in powers of the parameter behavior at infinity, clearly demonstrates this. σ, whereas the Gram-Charlier series is not. This difference The Fourier expansion of p(x)/Z(x) in another set between the Gram-Charlier series and the Edgeworth ex- of Hermite polynomials Hn(x) (8) (not in Chebyshev- pansion was pointed out by Juszkiewicz et al. (1995), and S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions 197
Fig. 3. The normalized χ2 PDF (15) for ν = 5 (dashed line), Fig. 4. The same as in Fig. 3 but for 12 and 36 terms in the and its Gauss-Hermite approximations with 2 and 6 terms in Gauss-Hermite expansion the expansion (solid line)
In this section we give a simplified derivation of the in independent work by Bernardeau & Kofman (1995), Edgeworth series, following Petrov (1972). This deriva- followed by Amendola (1994) and Colombi (1994). tion, for an arbitrary order, is somewhat simpler than the Juszkiewicz et al. (1995) and Bernardeau & Kofman derivation given for example by Bernardeau & Kofman (1995) use 2 or 3 terms of the Edgeworth expansion de- (1995) for the third order only of the Edgeworth expan- rived e.g. in Cram´er (1957). We note that the full explicit sion. expansion for arbitrary order s was obtained already in The characteristic function Φ(t) of a random variable 1962 by Petrov (1962), see also Petrov (1972, 1987). X is the expectation E exp(itX) as a function of t, ∞ Petrov derived a powerful generalization of the itx Edgeworth expansion for a sum of random variables. Φ(t) ≡ e dF (x) , (25) Z−∞ 198 S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions
Fig. 5. The normalized χ2 PDF (15) for ν = 2 (dashed line), Fig. 6. The normalized χ2 PDF (15) for ν = 5 (dashed line), and its Edgeworth-Petrov approximations with 4 and 12 terms and its Edgeworth-Petrov approximations with 2 and 4 terms in the expansion (solid line) in the expansion (solid) that is the Fourier transform of p(x) if the probability den- A similar series for ln Φ(t), sity p(x)=dF(x)/dxexists. The definition (25) implies ∞ κn n that if the moment α (2) of X exists, ln Φ(t) ∼ (it) , (28) k n! n=1 Φ(k)(0) = ikα . (26) X k involves cumulants (semi-invariants) κn defined by Hence the Taylor series for Φ(t)isgivenby 1 dn κn ≡ ln Φ(t) . (29) n n t=0 ∞ i dt αk h i Φ(t) ∼ 1+ (it)k . (27) In Appendix A: we prove a fundamental lemma of calculus k! k=1 for the n−th derivative of a composite function f ◦g(x) ≡ X S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions 199 f(g(x)), which reads Suppose that the probability density p(x) of a ran- dn dom variable X exists. Then the PDF for X/σ is q(x) ≡ f(g(x)) = dxn σp(σx), and it is the inverse Fourier transform of the char- n acteristic function ϕ: 1 1 km (r) (m) ∞ n! f (y)|y=g(x) g (x) , (30) 1 km! m! q(x)= e−itxϕ(t)dt. (38) {km} m=1 2π X Y Z−∞ where r = k1 + k2 + ... + kn and the set {km} consists If Φ(t) is the Fourier transform of a function p(x), then of all non-negative integer solutions of the Diophantine (−it)nΦ(t) is the transform of the n-th derivative of p(x), equation n ∞ d 1 −itx n k1 +2k2 +... + nkn = n. (31) n p(x)= e (−it) Φ(t)dt. (39) dx 2π −∞ Using (30) we derive a useful relation (Petrov 1987) be- Z The Fourier transform of the Gaussian distribution Z(x) tween the cumulants κ and the moments α of a PDF in n k in (5) is exp(−t2/2) , see e.g. Bateman & Erd´elyi (1954). Appendix B:, Therefore each (it)n, multiplied by exp( t2/2) in the ex- n − km r−1 1 αm pansion of ϕ (see Eqs. (34) to (37)), generates (according κn = n! (−1) (r − 1)! . (32) km! m! to Eq. 38) the n-th derivative of Z(x), {km} m=1 X Y dn ∞ Here summation extends over all non-negative integers (−1)n Z(x)= e−itx(it)n exp(−t2/2)dt, (40) dxn {km} satisfying (31) and r = k1 +k2 +...+kn. We describe Z−∞ a simple algorithm for obtaining all solutions of Eq. (31) in the corresponding expansion for q(x), in Appendix C. ∞ Now we are ready to begin with the derivation of the q(x)=Z(x)+ σs Edgeworth expansion. Consider a random variable X with s=1 EX = 0 (this can always be achieved by an appropri- X s m+2 m+2 km ate choice of origin), and let X have dispersion σ2.If 1 Sm+2(−1) d × m+2 Z(x) . (41) Xhas the characteristic function Φ(t), then the normal- km! ( m +2)! dx {km} m=1 ized random variable X/σ has the characteristic function X Y Here the set k in the sum consists of all non-negative ϕ(t)=Φ(t/σ). Therefore we have from Eqs. (28) and (29) { m} that integer solutions of the equation ∞ κn k1 +2k2 +... + sks = s. (42) ln ϕ(t) = ln Φ(t/σ) ∼ (it)n . (33) σnn! n=2 Using (10) and r = k1 + k2 + ... + ks we can rewrite X Here the sum starts at n = 2 because EX = 0. Moreover, (41) in terms of the Chebyshev-Hermite polynomials: 2 ∞ since κ2 = σ (see Eq. (32)) we obtain q(x)=σp(σx)=Z(x) 1+ σs ∞ n−2 2 Snσ ( ϕ(t) ∼ e−t /2 exp (it)n , (34) s=1 n! X (n=3 ) s km 1 Sm+2 X × He (x) . (43) with s+2r k ! (m +2)! m=1 m 2n−2 {km} Sn ≡ κn/σ . (35) X Y This is the Edgeworth expansion for arbitrary order Let us write the exponential function in (34) as a formal s. See Petrov (1972, 1987) for a more general form of the series in powers of σ, expansion (for non-smooth cumulative distribution func- ∞ r ∞ Sr+2σ tions F (x) and for a sum of random variables) and for the exp (it)r+2 ∼ 1+ P (it)σs , (36) (r +2)! s proof that the series (43) is asymptotic (see also the clas- (r=1 ) s=1 X X sical references Cram´er 1957 and Feller 1966). This means where the coefficient of the power s is a function Ps(it). that if the first N terms are retained in the sum over s, ∞ r+2 r Now, using g(x) ≡ r=1{Sr+2(it) x /(r+2)!} and f ≡ then the difference between q(x) and the partial sum is exp in (30), we find that P of a lower order than the N-thterminthesum(Erd´elyi 1 ds 1956; Evgrafov 1961). Convergence plays no role in the P (it) ≡ f(g(x))| s s! dxs x=0 definition of the asymptotic series. s 1 S (it)m+2 km Strictly speaking, Petrov (1972) proves the asymptotic = m+2 , (37) k ! (m +2)! theorems for sums of ν independent random variables only m=1 m 1/2 {Xkm} Y when σ ∼ 1/ν , and not for any σ, which we used in our where the summation extends again over all non-negative derivation. But in all practical applications where nearly integers {km} satisfying (31). Thus the function Ps is just Gaussian PDFs occur (and in all applications that we con- a polynomial. sider in the present work), those PDFs basically are the 200 S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions
Fig. 7. The normalized χ2 PDF (15) for ν = 5 (dashed line) and its Edgeworth-Petrov approximations with 12 terms in the expansion (solid) sums of random variables, and the proofs of the asymp- totic theorems are relevant. In the next section we show how the theory works in practice. Figures 5–8 show some examples of the Edgeworth ex- pansion for the χ2 distribution. It is clear that for strongly 2 non-Gaussian cases, like χν for ν = 2, it has a very small domain of applicability in practical cases since it diverges like the Gram-Charlier series for a large number of terms (Fig. 5). But already in this case one can check that the order of the last term retained gives the order of the error correctly, and one can truncate the expan- sion when the last term becomes unacceptably large. For nearly Gaussian distributions the situation is much bet- ter: compare the cases for ν =5andν=20inFigs.6,7 and 8.
Fig. 8. The normalized χ2 PDF (15) for ν = 20 (dashed line), 6. Peculiar velocities from cosmic strings and its Edgeworth-Petrov approximations with 2 and 4 terms in the expansion (solid) As an example we consider the probability distribution of peculiar velocities within the cosmic string model of structure formation. Cosmic strings are one-dimensional up to a distance of a Hubble radius, one expects a non- topological defects possibly formed in a phase transition Gaussian result, in contrast to inflationary theories which in the early Universe (Brandenberger 1994; Hindmarsh & predict a Gaussian PDF. Therefore departures from a nor- Kibble 1995; Vilenkin & Shellard 1994). After the time mal distribution may be a way to distinguish between the of formation, the string network quickly evolves to a scal- two main classes of theories of structure formation, infla- ing solution with a constant number ns of strings passing tion and topological defects. However, since many strings through a Hubble volume in one expansion time. present between the time of equal matter and radiation Since individual topological defects give rise to velocity contribute to the perturbations, a nearly Gaussian PDF perturbations of the dark matter through which they move can result due to the central limit theorem. S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions 201
Fig. 11. Relative error tN /(1+ΣN ) in the Edgeworth expansion of q(x)/Z(x) for the PDF of peculiar velocities from cosmic strings, for two values of N and n Fig. 9. Edgeworth expansion up to 10th order for the PDF of s peculiar velocities from cosmic strings, within an analytical model for the string network, for two values of the number of In order to estimate the deviation of the PDF from the strings per Hubble volume (Moessner 1995) normal distribution we use Petrov’s formula (43) for the Edgeworth series in this section. We calculate the cumu- lants up to 12th order in the analytical model for the cos- mic string network presented in Moessner (1995), where the cumulants are given up to 8th order. For simplicity, let us write Eq. (43) schematically as
∞ q(x)=Z(x) 1+ ts , (44) ( s=1 ) X i.e. denote the sth term in the sum by ts. Let us further N denote the sum up to s = N by ΣN ≡ s=1 ts.InFig.9 we show the Edgeworth expansion of the PDF of peculiar P velocities up to 10th order, for the case of ns =1stringper Hubble volume. Expanding up to Nth order, the relative deviation of the PDF from a normal distribution is given by, q(x) − 1=Σ . (45) Z(x) N It is only significant if the error of the asymptotic expan- sion, which is of the order of the last term included, is smaller than this deviation. In Fig. 10 we show the rel- ative deviation ΣN and the error tN associated with it. The wiggles or cusps in the graphs appear at zeros of ΣN and tN which are both oscillating, changing their sign, and Fig. 10. Relative deviation ΣN from a normal distribution of the we have plotted the logarithms of the absolute values. So Edgeworth expansion up to Nth order of the PDF of peculiar only the maxima of the curves give a true indication of velocities from cosmic strings, for n = 1. Also shown is the s the deviations and errors. For N = 10 the error is clearly error t of this deviation associated with the expansion N below the deviation and thus the latter is significant. For 202 S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions
N = 4, the error is still below the deviation for most val- Supernovae” and by a grant from the Research Center for the ues of x, but it is not as clear, especially since the error is Early Universe, University of Tokyo, and he cordially thanks not exactly equal to tN but of that order only. In Fig. 11 K. Nomoto, as well as W.Hillebrandt, MPA, Garching, for their hospitality. we show the relative error tN /(1 + ΣN ) in the expansion of q(x)/Z(x). It is smaller for an expansion to higher or- This research has made use of NASA’s Astrophysics Data der for a given number of strings per Hubble volume. The System Abstract Service. error decreases more strongly with N for larger ns.The relative error is also smaller, at fixed N, for a larger ns, Appendices in which case the PDF is closer to a normal distribution. It is interesting to compare these results with the gen- Appendix A: Lemma eral theory. Petrov (1972) proves that under certain con- In Sect. 5, the relation (30) for the n−th derivative of ditions (which are fulfilled in most physically important a composite function f ◦ g(x) ≡ f(g(x)) is used for the cases) derivation of the Edgeworth asymptotic expansion. Here N a simplified derivation of Eq. (30) is given. Petrov (1972, N q(x)=Z(x) 1+ ts + o(σ ) (46) 1987) suggests a proof by induction. We note that this ( s=1 ) X derivation is more transparent if one simply considers the uniformly for −∞ Appendix B: Applications of the Lemma and we say that S1 Table C1. Table of solutions to Eq. (31) for n = 1 to 8. Zeros Amendola L., 1994, ApJ 430, L9 for higher orders are left blank Bartunov O.S., Blinnikov S.I., Popolitov V.A., 1997, Trefor: user’s guide and references. Sternberg Astron. Inst. report n1 2 3 4 Bateman H., Erd´elyi A., 1954, Tables of Integral Transforms. 12 3 4 McGraw-Hill, New York, Toronto, London 10 11 12 Bernardeau F., 1992, ApJ 392, 1 100 20 Bernardeau F., 1994, A&A 291, 697 101 Bernardeau F., Kofman L., 1995, ApJ 443, 479 1000 Blinnikov S.I., 1996, Pis’ma v AZh 22, 92; translation: Astron. n5 6 7 8 Lett. 22, 79 56 7 8 Bourbaki N., 1958, El´´ ements de math´ematique. Livre IV. 13 14 15 16 Fonctions d’une variable r´eelle (the´orie ´el´ementaire), 21 22 23 24 Hermann & Co., Paris, Ch. I, Sect. 3.2 102 30 31 32 Brandenberger R., 1994, Int. J. Mod. Phys. A9, 2117 110 103 104 40 Colombi S., 1994, ApJ 435, 536 1001 111 112 105 Cram´er H., 1957, Mathematical Methods of Statistics. 10000 200 120 113 Princeton Univ. Press, Princeton 1002 201 121 Encyclopaedia of Mathematics, 1988, Translated from the 1010 1003 202 Russian. Kluwer Academic Publishers, Dordrecht, Holland 10001 1011 210 Erd´elyi A., 1956, Asymptotic Expansions, Dover, U.S.A. 100000 1100 1004 Evgrafov M.A., 1961, Asymptotic estimates and entire func- 10002 1012 tions, Translated from the Russian. Gordon and Breach, 10010 1020 New York 100001 1101 Feller W., 1966, An Introduction to Probability Theory and its 1000000 2000 Applications, Vol. II, Ch. XVI. Wiley & Sons, New York, 10003 London, Sydney, Toronto 10011 Ferreira P.G., Magueijo J., Silk J., 1997, Phys. Rev. D56, 4592 10100 Gazta˜naga E., Fosalba P., Elizalde E., 1997, Non-Gaussian 100002 traces in the variance from small scale CMB observations, 100010 astro-ph/9705116 1000001 Gerhard O.E., 1993, MNRAS 265, 213 10000000 Hauschildt P.H., Baron E., Allard F., 1997, ApJ 483, 390 Heyl J.S., Hernquist L., Spergel D.N., 1994, ApJ 427, 165 Hindmarsh M., Kibble T., 1995, Rep. Prog. Phys. 58, 477 Hubeny I., Lanz T., 1995, ApJ 439, 875 sumcur=sumcur-k(m)*m+m+1; Juszkiewicz R., Weinberg D.H., Amsterdamski P., k(m)=0; Chodorowski M., Bouchet F., 1995, ApJ 442, 39 k(m+1)=k(m+1)+1; Kendall M. C., 1952, The advanced theory of statistics, Vol. 1, m=m+1; Distribution theory. Fifth Edition Charles Griffin & Co., _until sumcur <= n _or m>mold; London _if m>mold _then mold=m _fi; Kofman L., Bertschinger E., Gelb J.M., Nussser A., Dekel A., k(1)=n-sumcur; 1994, ApJ 420, 44 nsol=nsol+1; Luo X., 1995, ApJ 439, 517 Matsubara T., Yokoyama J., 1996, ApJ 463, 409 Here the node nextset ends, and the whole algorithm Moessner R., 1995, MNRAS 277, 927 is finished. We have written it here in a pseudocode which Moessner R., Perivolaropoulos L., Brandenberger R., 1994, can be translated mechanically into any machine language. ApJ 425, 365 We actually use a special preprocessor Trefor which au- Nikiforov A.F., Uvarov V.B., 1988, Special Functions of tomatically transforms the text above to standard Fortran Mathematical Physics, translated from the Russian. Birkhauser, Basel, Boston (see Weinstein & Blinnikov 1984, and Bartunov et al. Petrov V.V., 1962, Vestnik Leningrad. Univ. No. 19, 150-153 1997). Petrov V.V., 1972, Summy nezavisimykh sluchainykh velichin, For reference, we present the first 8 sets of solutions Moskva,Nauka;translation:PetrovV.V.Sumsofin- found by this algorithm. In practice it is easier not to use dependent random variables. Series title: Ergebnisse der the Table C1 even for low n, but generate all coefficients Mathematik und ihrer Grenzgebiete; Vol. 82., 1975. in the code. Springer-Verlag, Berlin, New York Petrov V.V., 1987, Predel’nye teoremy dlia summ nezav- isimykh sluchainykh velichin. Moskva, Nauka; translation: References Petrov V.V., Limit theorems of probability theory: se- quences of independent random variables. Series title: Abramowitz M., Stegun I., 1972, Handbook of Mathematical Oxford studies in probability; 4, 1995. Clarendon Press, Functions. New York: Dover Oxford, Oxford University Press, New York S. Blinnikov and R. Moessner: Expansions for nearly Gaussian distributions 205 Rix H., White S.D.M., 1992, MNRAS 254, 389 Tabeling P., Zocchi G., Belin F., Maurer J., Willaime H., 1996, Scherrer R., Bertschinger E., 1991, ApJ, 381, 349 Phys. Rev. E. 53, 1613 Suetin P.K., 1979, Klassicheskie Ortogonal’nye Mnogochleny, van der Marel R.P., Franx M., 1993, ApJ 407, 525 Moscow: Nauka Vilenkin A., Shellard E.P.S., 1994, Cosmic strings and other Szeg¨o G., 1978, Orthogonal Polynomials. Providence: topological defects. Cambridge Univ. Press, Cambridge American Mathematical Society colloquium publications Weinstein S.I., Blinnikov S.I., 1984, Trefor – a preprocessor for XXIII structured program design in Fortran. Report ITEP 1984– 79