Abstract
We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.
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References
Akhiezer, N.I.: The Classical Moment Problem. Oliver and Boyd, Edinburgh (1965) (English transl.)
Belinschi, S.T., Bożejko, M., Lehner, F., Speicher, R.: The normal law is ⊞-infinitely divisible, arXiv:0910.4263v2
Bercovici, H., Pata, V.: Stable laws and domains of attraction in free probability theory (with an appendix by Philippe Biane). Ann. Math. (2) 149(3), 1023–1060 (1999)
Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)
Chung, K.L.: A Course in Probability Theory. Harcourt, Brace & World, New York (1968)
Franz, U.: Unification of boolean, monotone, anti-monotone, and tensor independence and Lévy Processes. Math. Z. 243, 779–816 (2003)
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading (1954)
Hasebe, T.: Monotone convolution semigroups. Studia Math. 200, 175–199 (2010)
Hasebe, T.: Monotone convolution and monotone infinite divisibility from complex analytic viewpoint. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13(1), 111–131 (2010)
Hasebe, T., Saigo, H.: The monotone cumulants. Ann. Inst. Henri Poincaré Probab. Stat. arXiv:0907.4896v3 (to appear)
Hasebe, T., Saigo, H.: Joint cumulants for natural independence. arXiv:1005.3900v1
Levin, B.Ya.: Lectures on Entire Functions, in collaboration with Yu. Lyubarskii, M. Sodin and V. Tkachenko. Transl. Math. Monographs, vol. 150. Amer. Math. Soc., Providence (1996)
Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106, 409–438 (1992)
Muraki, N.: Monotonic convolution and monotonic Lévy–Hinčin formula. Preprint (2000)
Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Math. Soc. Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006)
Pérez-Abreu, V., Sakuma, N.: Free generalized Gamma convolutions. Electron. Commun. Probab. 13, 526–539 (2008)
Shiryayev, A.N.: Probability. Springer, New York (1984)
Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298, 611–628 (1994)
Speicher, R., Woroudi, R.: Boolean convolution. In: Voiculescu, D. (ed.) Free Probability Theory, Toronto, Canada, 1995, Fields Inst. Commun., vol. 12, pp. 267–280. Am. Math. Soc., Providence (1997). Papers from a Workshop on Random Matrices and Operator Algebra Free Products
Voiculescu, D.: Addition of certain non-commutative random variables. J. Funct. Anal. 66, 323–346 (1986)
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Hasebe, T. Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of Probability Measures. J Theor Probab 25, 756–770 (2012). https://doi.org/10.1007/s10959-011-0344-9
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DOI: https://doi.org/10.1007/s10959-011-0344-9
Keywords
- Fourier transform
- Stieltjes transform
- Cauchy distribution
- Non-commutative probability theory
- Paley–Wiener theorem