By Marinucci D.
The angular bispectrum of round random fields has lately received anenormous significance, in particular in reference to statistical inference on cosmologicaldata. during this paper, we learn its moments and cumulants of arbitrary order andwe use those effects to set up a multivariate imperative restrict theorem and better orderapproximations. the consequences depend on combinatorial tools from graph conception anda specified research for the asymptotic habit of coefficients coming up in matrixrepresentation thought for the crowd of rotations SO(3).
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Extra info for A central limit theorem and higher order results for the angular bispectrum
2k) is a rational multiple of π 2k and hence is transcendental by Lindemann’s theorem (1882) on the transcendence of π. For each k = 1, 2, 3, . . , ζ(2k + 1) is conjectured to be transcendental, but very little has been proved so far about the arithmetical nature of ζ(2k + 1). , for no explicitly given integer k ≥ 2 the irrationality of ζ(2k + 1) has been proved), and Zudilin (2004) proved that at least one of the four numbers ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. 2 The Bernoulli polynomials are defined by Bn (x) = ∂ n z ex z ∂z n e z − 1 (n = 0, 1, 2, .
2), 1 ≥ 1 1 S2n+1 2n S2n+1 . 2), (2n)!! (2n − 1)!! 2 1 = 2n + 1 n k=1 2k 2k 2k − 1 2k + 1 = 1 π . 3) For n → ∞ we get Wallis’ formula: π = 2 ∞ 2 2 4 4 6 6 4k 2 = ··· . 3), (2n)!! (2n − 1)!! 2 = 1 π (2n + 1) 1 + O 2 n = πn 1 + O 1 n . 4) By Taylor’s formula, (1 + x)1/2 = 1 + 21 x + 1/2 x 2 + . . = 1 + O(x) (x → 0). 2 Hence 1 1/2 1 1+O (n → ∞). = 1+O n n Also (2n − 1)!! (2n)!! = (2n)! and (2n)!! 2 (2n)!! = = . (2n − 1)!! (2n)! (2n)! 1 Stirling’s Formula for n! 2 = πn 1 + O (2n)! n (n → ∞). 1 (Stirling’s ‘elementary’ formula) For n → ∞ we have log n!
To see this, note exp(|z|n+ε ), so that that |P(z)| |z|n , whence e P(z) = eRe P(z) ≤ e|P(z)| P(z) n n−1 ≤ n. If P(z) = a0 z + a1 z + . . + an (a0 = 0), and if z → ∞ along ord e one of the n half-lines defined by arg z = − n1 arg a0 , whence arg(a0 z n ) = arg a0 + n arg z = 0 so that a0 z n > 0, then Re P(z) − |z|n−ε = Re(a0 z n ) + O(|z|n−ε ) = |a0 ||z|n + O(|z|n−ε ) → +∞. This shows that lim sup z→∞ e P(z) = lim sup exp(Re P(z) − |z|n−ε ) = +∞, exp(|z|n−ε ) z→∞ whence ord e P(z) ≥ n. We aim at relating the order of an entire function with the density of its zeros.
A central limit theorem and higher order results for the angular bispectrum by Marinucci D.