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 Electronic Communications in Probability > Vol. 2 (1997) > Paper 3 open journal systems 


Strong Laws and Summability for Sequences of $phi$-Mixing Random Variables in Banach Spaces

Rüdiger Kiesel, University of London


Abstract
In this note the almost sure convergence of stationary, $varphi$-mixing sequences of random variables with values in real, separable Banach spaces according to summability methods is linked to the fulfillment of a certain integrability condition generalizing and extending the results for i.i.d. sequences. Furthermore we give via Baum-Katz type results an estimate for the rate of convergence in these laws.


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Pages: 27-41

Published on: May 14, 1997


Bibliography
  1. A. de Acosta, On large deviations of sums of independent random vectors, In: Beck A. et al: Probability in Banach Spaces 5, Lecture Notes in Mathematics 1153, (1984) 1--14, Springer Verlag. Math Review link
  2. A. de Acosta and J. Kuelbs, Limit Theorems for Moving Averages of independent Random Vectors, Z. W-theorie verw. Geb. 64 (1983) 67--123. Math Review link
  3. N. H. Bingham, Summability methods and dependent strong laws, In: Eberlein, E.; Taqqu, M.S.: Dependence in Probability and Statistics, (1986) 291--300, Birkhauser. Math Review link
  4. N. H. Bingham, Extensions of the strong law, in Analytic and Geometric Stochastics: Adv. Appl. Probab. Supplement (1986) 27--36. Math Review link
  5. N. H. Bingham, Moving averages, In: Edgar, G.A.; Sucheston, L.: Almost Everywhere Convergence Proc. Conf. Columbus OH, (1988) 131--145, New York: Academic Press. Math Review link
  6. N. H. Bingham and C. M. Goldie, Riesz Means and Self-Neglecting Functions, Math. Z. 199 (1988) 443--454. Math Review article not available.
  7. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, (1987) Cambridge University Press. Math Review link
  8. D. Borwein and W. Kratz, An O-Tauberian theorem and a High Indices theorem for power series methods of summability, Camb. Phil. Math. Soc. 115 (1994) 365--375. Math Review article not available.
  9. A. Hozorgnia and R. Bhaskara, On summability methods and limit theorems for Banach space random variables, Bull. Inst. Math. Acad. Sinica 7 (1979) 1--6. Math Review link
  10. R. Bradley, Basic properties of strong mixing conditions, In: Eberlein, E.; Taqqu, M.S.: Dependence in Probability and Statistics, (1986) 165--192, Birkhauser. Math Review article not available.
  11. L. Breiman, Probability, Classics in applied mathematics 7 (1992) SIAM. Math Review link
  12. Y. S. Chow, Delayed sums and Borel summability of independent, identically distributed random variables, Bull. Inst. Math. Acad. Sinica 1 (1973) 207--220. Math Review article not available.
  13. G. H. Hardy, Divergent Series, (1949) Oxford Press. Math Review article not available.
  14. B. Heinkel, On the almost sure summability of B-valued random variables, In: Dudley R. M. et al: Probability in Banach Spaces 8, (1991) 325--338, Birkhauser. Math Review article not available.
  15. B. Heinkel. On Valiron means of B-valued random variables, Lith. Math. Journal 32 (29) (1992) 162--173. Math Review link
  16. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups (198) AMS. Math Review article not available.
  17. M. Iosifescu and R. Theodorescu, Random Processes and Learning (1969) Springer Verlag. Math Review article not available.
  18. N. C. Jain, Tail probabilities for sums of independent Banach space random variables, Z. Wahrscheinlichkeitstheorie verw. Gebiete 33 (1975) 155--166. Math Review article not available.
  19. R. Kiesel, Power series methods and almost sure convergence Math. Proc. Camb. Phil. Soc. 113 (1993) 195--204. Math Review article not available.
  20. R. Kiesel, General Norlund transforms and power series methods, Math. Z. 214 (1993) 273--286. Math Review article not available.
  21. R. Kiesel, Starke Gesetze fur gewichtele Summen von Zufallsvariablen, Habilitationsschrift, Universitat Ulm (1995). Math Review article not available.
  22. R. Kiesel, Summability and strong laws for varphi-mixing random variables, J. Theo. Prob. (to appear). Math Review article not available.
  23. R. Kiesel and U. Stadtmuller, Erdos-Renyi-Shepp laws and weighted sums of independent identically distributed random variables, J. Theo. Prob., 9(4) (1996) 961--982. Math Review article not available.
  24. T. L. Lai, Summability methods for independent identically distributed random variables, Proc. Amer. Math. Soc. 45 (1974) 253--261. Math Review article not available.
  25. T. L. Lai, Convergence rates in the strong law of large numbers for random variables taking values in Banach spaces, Bull. Inst. Math. Acad. Sinica 2 (1) (1974) 67--85. Math Review article not available.
  26. N. Ledoux and M. Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 23 (1991) Springer Verlag. Math Review article not available.
  27. E. Liebscher, Strong convergence of sums of alpha-mixing random variables with applications to density estimation, Stoch. Proc. Appl. 65 (1996) 69--80. Math Review article not available.
  28. M. Peligrad, The r-quick Version of the Strong Law for Stationary varphi-mixing sequences, In: Edgar, G. A.; Sucheston, L., Almost everywhere Convergence, 335--348, Proc. Conf. Columbus OH, (1988), New York: Academic Press. Math Review link
  29. A. Peyerimhoff, Lectures on Summability, Lecture Notes in Mathematics, No. 107, (1969) Springer Verlag. Math Review article not available.
  30. U. Stadtmuller, On a family of summability methods and one-sided Tauberian conditions, J. Math. Analysis Appl. (to appear). Math Review link
  31. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, (1996) Springer Verlag. Math Review article not available.
  32. K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, (1970) Springer Verlag. Math Review article not available.
















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Electronic Communications in Probability. ISSN: 1083-589X