Formulas of Absolute Moments
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Formulas of Absolute Moments Gwo Dong Lin Hwa-Kang Xing-Ye Foundation, Taipei, Taiwan Academia Sinica, Taipei, Taiwan
Chin-Yuan Hu National Changhua University of Education, Changhua, Taiwan
Abstract The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat Hsu’s (J. Chinese Math. Soc. N.S. 1, 257–280, 1951) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace–Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly Hall’s (Z. Wahrsch. Verw. Gebiete 62, 355–359, 1983) result for distributions on the right half-line. AMS (2000) subject classification. Primary 60E10, 42A38, 42B10. Keywords and phrases. Absolute moments, Fractional moments, Translated moments, Characteristic function, Fourier–Stieltjes transform, Laplace–Stieltjes transform, Ramanujan’s master theorem
1 Introduction The absolute moments of probability distributions play important roles in both theoretical and applied fields (see, e.g., von Bahr 1965; Hall 1983; Nabeya 1951, 1952; Barndorff-Nielsen and Stelzer 2005 and Matsui and Pawlas 2016). Our main purpose in this paper is to investigate the presentation of (fractional) absolute moments of distributions in terms of their Fourier–Stieltjes or Laplace–Stieltjes transforms. We review first some important properties of the characteristic function (ch.f.). Consider a random variable X with distribution F (x) = Pr(X ≤ x)
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G. D. Lin and C.-Y. Hu
on R ≡ (−∞, ∞), denoted X ∼ F , and let φ be the ch.f. of F , namely, the Fourier–Stieltjes transform ∞ ∞ ∞ itx e dF (x) = cos(tx) dF (x) + i sin(tx) dF (x), t ∈ R, φ(t) = −∞
−∞
−∞
(1) or, φ(t) = E[exp(itX)] = E[cos(tX)] + i E[sin(tX)] ≡ Re(φ(t)) + i Im(φ(t)), t ∈ R. It is known that for any distribution F , its ch.f. φ in Eq. (1) always exists and uniquely determines F by the famous Weierstrass Approximation Theorem. Therefore, we can recover F and derive its properties via the ch.f. φ theoretically. In fact, we have the classical L`evy inverse formula: T 1 1 − e−ith −itx e φ(t) dt, F (x + h) − F (x) = lim T →∞ 2π −T it provided that both x and x + h (with h > 0) are continuity points of F (see, e.g., Lukacs 1970, Chapter 3). Another explicit expression is the following (improper) integral: for any continuity point x of F on R, T 1 1 ∞ dt 1 1 −itx dt F (x) = − Im(φ(t)e ) Im(φ(t)e−itx ) = − lim (2) 2 π 0 t 2 π T →∞ 0 t (see Zolotarev 1957, or Kawata 1972, p. 130, or Rossberg et al. 1985, p. 45). On the other hand, if X ≥ 0, then for any continuity point x of F on R+ ≡ [0, ∞), we have the inverse formulas via (2): dt 2 ∞ (sin xt)Re(φ(t)) , x ∈ R+ , (3) F (x) = π 0 t dt 2 ∞ (cos xt)Im(φ(t))
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