By Marinucci D.

The angular bispectrum of round random fields has lately won anenormous value, specifically in reference to statistical inference on cosmologicaldata. during this paper, we examine its moments and cumulants of arbitrary order andwe use those effects to set up a multivariate relevant restrict theorem and better orderapproximations. the implications depend upon combinatorial equipment from graph concept anda unique research for the asymptotic habit of coefficients bobbing up in matrixrepresentation thought for the crowd of rotations SO(3).

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28) f (x) dx = a 1 b−a b a+b 2 x− a f (x) dx. 28) is proved. The following theorem holds [34]: Theorem 24. 29) b f (x) dx ≥ 0. a Proof. 29) is proved. Remark 17. The above theorem contains a sufficient condition for the differentiable mapping f such that the second inequality in the H· − H· result remains true. In what follows, we need a lemma which is interesting in itself as it provides a refinement of the celebrated Chebychev’s integral inequality (see also [66]). Lemma 1. , (f (x) − f (y)) (g (x) − g (y)) ≥ 0 for all x, y ∈ [a, b].

Let p > 1 and [a, b] ⊂ [0, ∞) . 40) where q := p p−1 . p (b − a) 2 (p + 1) 1 p p [Lp (a, b)] q , 2. SOME RESULTS RELATED TO THE H· − H· INEQUALITY 36 Proof. By Theorem 26 applied to the convex mapping f (x) = xp we have: 1 b ap + bp 1 − 2 b−a x dx ≤ a 1 q b 1 (b − a) p 2 (p + 1) p1 p p−1 q px 1 1 q b 1 (b − a) p p 2 (p + 1) p1 = dx a x(p−1)q dx . 40) is proved. Another result which is connected with the logarithmic mean L (a, b) is the following one: Proposition 13. Let p > 1 and 0 < a < b. 41) 0 ≤ H −1 (a, b) − L−1 (a, b) ≤ (b − a) L 1 2 (p + 1) p 2p 1−p (a, b) p−1 p .

Let f, g : [a, b] → R be continuous on [a, b] and differentiable on (a, b) . 53) g (x) dx a b f (x) g (x) dx − b f (x) dx a a a b 2 b g 2 (x) dx − ≤ L (b − a) a b ≤ (b − a) 2 b g (x) dx g (x) dx . a a Proof. First of all, we will show that for all x, y ∈ [a, b] we have the inequality 2 2 l (g (x) − g (y)) ≤ (f (x) − f (y)) (g (x) − g (y)) ≤ L (g (x) − g (y)) . 54) If g (x) = g (y) , then the above inequality becomes an identity. If g (x) = g (y) , and (assume) x < y, then by Cauchy’s theorem , there exists an ξ ∈ (x, y) such that f (x) − f (y) f (ξ) = ∈ [l, L] , g (x) − g (y) g (ξ) and thus f (x) − f (y) l≤ ≤ L.

### A central limit theorem and higher order results for the angular bispectrum by Marinucci D.

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