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Marcinkiewicz laws with infinite moments

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Abstract

Marcinkiewicz laws of large numbers for φ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Lévy-Ottaviani-Etemadi and Hoffmann-Jørgensen inequalities are revisited.

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References

  1. J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn., 1 (2001), 193–237.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, John Wiley & Sons (New York, 1980).

    MATH  Google Scholar 

  3. B. von Bahr and C.-G. Esseen, Inequalities for the r-th absolute moment of a sum of random variables, 1 ≦ r ≦ 2, Ann. Math. Stat., 36 (1965), 299–303.

    Article  MATH  Google Scholar 

  4. S. N. Bernstein, Sur l’extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes, Math. Ann., 97 (1927), 1–59.

    Article  MathSciNet  Google Scholar 

  5. P. Billingsley, Convergence of Probability Measures, Wiley (New York, 1968).

    MATH  Google Scholar 

  6. P. Billingsley, Probability and Measure, 3rd ed., Wiley (New York, 1995).

    MATH  Google Scholar 

  7. P. Billingsley, Convergence of Probability Measures, 2nd ed., Wiley (New York, 1999).

    MATH  Google Scholar 

  8. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge Univ. Press (Cambridge, UK, 1987).

    MATH  Google Scholar 

  9. R. C. Bradley, A central limit theorem for stationary ρ-mixing sequences with infinite variance, Ann. Probab., 16 (1988), 313–332.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions, Probab. Surv., 2 (2005), 107–144.

    Article  MathSciNet  Google Scholar 

  11. S. Csörgő, Rates of merge in generalized St. Petersburg games, Acta Sci. Math. (Szeged), 68 (2002), 815–847.

    MathSciNet  Google Scholar 

  12. S. Csörgő and G. Simons, On Steinhaus’ resolution of the St. Petersburg paradox, Probab. Math. Statist., 14 (1993), 157–172.

    MathSciNet  Google Scholar 

  13. S. Csörgő and G. Simons, Pooling strategies for St. Petersburg gamblers, Bernoulli, 12 (2006), 971–1002.

    Article  MathSciNet  Google Scholar 

  14. R. A. Davis, Stable limits for partial sums of dependent random variables, Ann. Probab., 11 (1983), 262–269.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Denker and A. Jakubowski, Stable limit distributions for strongly mixing sequences, Statist. Probab. Lett., 8 (1989), 477–483.

    Article  MATH  MathSciNet  Google Scholar 

  16. N. Etemadi, On some classical results in probability theory, Sankhyā, Ser. A, 47 (1985), 215–221.

    MATH  MathSciNet  Google Scholar 

  17. W. Feller, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung I, Math. Z., 40 (1936), 521–559; II, ibidem 42, 301–312.

    Article  MathSciNet  Google Scholar 

  18. W. Feller, Über das Gesetz der Grossen Zahlen, Acta Litt. Scient. Regiae Univ. Hungaricae Francisco-Iosephinae, VIII, (1936–37), 191–201.

  19. W. Feller, Note on the law of large numbers and “fair” games, Ann. Math. Stat., 16 (1945), 301–304.

    Article  MATH  MathSciNet  Google Scholar 

  20. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., (Wiley, New York, 1971).

    MATH  Google Scholar 

  21. A. G. Grin’, Domains of attraction for mixing sequences, Sib. Math. J., 31 (1990), 43–52.

    Article  MathSciNet  Google Scholar 

  22. A. Gut, An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game, J. Theoret. Probab., 17 (2004), 769–779.

    Article  MATH  MathSciNet  Google Scholar 

  23. L. Heinrich, Rates of convergence in stable limit theorems for sums of exponentially ϕ-mixing random variables with an application to metric theory of continued fractions, Math. Nachr., 131 (1987), 149–165.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. Hensley, The statistics of the continued fraction digit sum, Pacific J. Math., 192 (2000), 103–200.

    Article  MATH  MathSciNet  Google Scholar 

  25. P. Hitczenko, S. Kwapień, W. V. Li, G. Schechtman, T. Schlumprecht and J. Zinn, Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables, Electron. J. Probab., 3 (1998), 1–26.

    MathSciNet  Google Scholar 

  26. J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math., 52 (1974), 159–186.

    MathSciNet  Google Scholar 

  27. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff (Groningen, 1971).

    MATH  Google Scholar 

  28. M. Iosifescu and R. Teodorescu, Random Processes and Learning, Springer-Verlag, (Berlin — Heidelberg — New York, 1969).

    MATH  Google Scholar 

  29. M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications 547, Kluwer, (Dordrecht, 2002).

    MATH  Google Scholar 

  30. A. Jakubowski, Minimal conditions in p-stable limit theorems, Stochastic Process. Appl., 44 (1993), 291–327.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Jakubowski and M. Kobus, α-stable limit theorems for sums of dependent random vectors, J. Multivariate Anal., 29 (1989), 219–251.

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Jorland, The Saint Petersburg Paradox 1713–1937, in: The Probabilistic Revolution, (Eds. L. KrÄuger et al.), MIT Press, (Cambridge, US, 1987), pp. 157–190.

    Google Scholar 

  33. A. Ya. Khinchine, Metrische Kettenbruchprobleme, Compos. Math., 1 (1935), 361–382.

    Google Scholar 

  34. A. Ya. Khinchine, Sul dominio di attrazione della legge di Gauss, Giorn. Ist. Ital. Attuari, 6 (1935), 371–393.

    Google Scholar 

  35. A. Ya. Khinchine, Su una legge dei grandi numeri generalizzata, Giorn. Ist. Ital. Attuari, 7 (1936), 365–377.

    Google Scholar 

  36. A. N. Kolmogorov, Über die Summen durch den Zufall bestimmter unabhängiger Größen, Math. Ann., 99 (1928), 309–319.

    Article  MathSciNet  Google Scholar 

  37. A. N. Kolmogorov, Über das Gesetz des iterierten Logarithmus, Math. Ann., 101 (1929), 126–135.

    Article  MathSciNet  Google Scholar 

  38. S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser (Boston, 1992).

    MATH  Google Scholar 

  39. M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag (Berlin, 2002).

    Google Scholar 

  40. P. Lévy, Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchainées, J. Math. Pures Appl., 14 (1935), 347–402.

    MATH  Google Scholar 

  41. P. Lévy, Théorie de l’Addition des Variables Aléatoires, Gauthier-Villars, (Paris, 1937).

    Google Scholar 

  42. P. Lévy, Fractions continues aleatoires, Rend. Circ. Mat. Palermo, 2 (1952), 170–208.

    Article  Google Scholar 

  43. Z. Y. Lin and C. R. Lu, Limit theory for mixing dependent random variables, Mathematics and its Application v. 378, Science Press (Beijing, 1996).

    MATH  Google Scholar 

  44. M. Loève, Probability Theory, 2nd ed., Van Nostrand (New York, 1960).

    MATH  Google Scholar 

  45. R. A. Maller, Relative stability and the strong law of large numbers, Z. Wahr. verw. Gebiete, 43 (1978), 141–148.

    Article  MATH  MathSciNet  Google Scholar 

  46. J. Marcinkiewicz, Quelques théorèmes de la théorie des probabilités, Bull. Sem. Math. Univ. Wilno, 2 (1939), 22–34 (in: J. Marcinkiewicz, Collected Papers, PWN (Warszawa, 1964)).

    MathSciNet  Google Scholar 

  47. J. Marcinkiewicz and A. Zygmund, Sur les fonctions indépendantes, Fund. Math., 29 (1937), 60–90.

    Google Scholar 

  48. A. Martin-Löf, A limit theorem which clariffies the ‘Petersburg paradox’, J. Appl. Probab., 22 (1985), 634–643.

    Article  MATH  MathSciNet  Google Scholar 

  49. S. V. Nagaev, Probabilistic and moment inequalities for dependent random variables, Theory Probab. Appl., 45 (2000), 152–160.

    Article  MATH  MathSciNet  Google Scholar 

  50. G. Ottaviani, Sulla teoria astratta del calcolo delle probabilitá proposita dal Cantelli, Giorn. Ist. Ital. Attuari, 10 (1939), 10–40.

    MathSciNet  Google Scholar 

  51. M. Peligrad, An invariance principle for φ-mixing sequences, Ann. Probab., 13 (1985), 1304–1313.

    Article  MATH  MathSciNet  Google Scholar 

  52. M. Peligrad, On Ibragimov-Iosifescu conjecture for φ-mixing sequences, Stochastic Process. Appl., 35 (1990), 293–308.

    Article  MATH  MathSciNet  Google Scholar 

  53. W. Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math., 105 (1988), 195–206.

    Article  MATH  MathSciNet  Google Scholar 

  54. B. A. Rogozin, Relatively stable walks, Theory Probab. Appl., 21 (1976), 375–379.

    Article  MATH  Google Scholar 

  55. J. D. Samur, Convergence of sums of mixing triangular arrays of random vectors with stationary rows, Ann. Probab., 12 (1984), 390–426.

    Article  MATH  MathSciNet  Google Scholar 

  56. J. D. Samur, On some limit theorems for continued fractions, Trans. Amer. Math. Soc., 316 (1989), 53–79.

    Article  MATH  MathSciNet  Google Scholar 

  57. E. Seneta, Regularly Varying Functions, Lectures Notes in Mathematics 508, Springer-Verlag (New York, 1976).

    MATH  Google Scholar 

  58. A. V. Skorokhod, Limit theorems for stochastic processes with independent increments, Theory Probab. Appl., 2 (1957), 138–171.

    Article  Google Scholar 

  59. A. V. Skorokhod, Random Processes with Independent Increments, Nauka (Moscow, 1964) (in Russian).

    Google Scholar 

  60. H. D. Steinhaus, The so-called Petersburg paradox, Colloq. Math., 2 (1949), 56–58.

    MATH  MathSciNet  Google Scholar 

  61. Z. S. Szewczak, Relative stability for strictly stationary sequences, J. Multivariate Anal., 78 (2001), 235–251.

    Article  MATH  MathSciNet  Google Scholar 

  62. Z. S. Szewczak, On relative stability for strictly stationary strongly mixing sequences, preprint.

  63. Z. S. Szewczak, On limit theorems for continued fractions, J. Theoret. Probab., 22 (2009), 239–255.

    Article  MATH  MathSciNet  Google Scholar 

  64. I. Vardi, The limiting distribution of the St. Petersburg game, Proc. Amer. Math. Soc., 123 (1995), 2875–2882.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Z. S. Szewczak

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Szewczak, Z.S. Marcinkiewicz laws with infinite moments. Acta Math Hung 127, 64–84 (2010). https://doi.org/10.1007/s10474-010-9091-0

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