The ordinal convergence and Glivenko–Cantelli type theorems in $L_p(-\infty ,\infty )$
Author:
I. K. Matsak
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 75 (2007), 83-92
MSC (2000):
Primary 60B12
DOI:
https://doi.org/10.1090/S0094-9000-08-00716-3
Published electronically:
January 24, 2008
MathSciNet review:
2321183
Full-text PDF Free Access
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Abstract: Let $F (t)$ be a distribution function and $F_n (t)$ the corresponding empirical distribution function. We find necessary and sufficient conditions for the ordinal convergence o-lim $F_n=F$ in the spaces $L_p (-\infty ,\infty )$.
References
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- Ī. K. Matsak, Ordinal law of large numbers in Banach lattices, Teor. Ĭmovīr. Mat. Stat. 62 (2000), 83–95 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 62 (2001), 89–102. MR 1871511
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References
- V. I. Glivenko, Sulla determinazione empirica delle leggi di probabilitá, Giorn. Ist. Ital. Attuari. 4 (1933), no. 1, 92–99.
- M. Csörgö and P. Révész, Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest, 1981. MR 666546 (84d:60050)
- P. Gänssler and W. Stout, Empirical processes: A survey of results for independent and identically distributed random variables, Ann. Probab. 7 (1979), no. 2, 193–243. MR 525051 (80d:60002)
- E. V. Khmaladze, Some applications of the theory of martingales in statistics, Uspekhi Mat. Nauk 37 (1982), no. 6, 194–212. (Russian) MR 683280 (84c:62066)
- M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991. MR 1102015 (93c:60001)
- I. K. Matsak, Ordinal law of large numbers in Banach lattices, Teor. Imovir. Mat. Stat. 62 (2000), 83–95; English transl. in Theory Probab. Math. Statist. 62 (2001), 89–102. MR 1871511 (2002k:60019)
- I. K. Matsak, A remark on the ordered law of large numbers, Teor. Imovir. Mat. Stat. 72 (2005), 84–92; English transl. in Theory Probab. Math. Statist. 72 (2006), 93–102. MR 2168139 (2006f:60011)
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, vol. 2, Springer-Verlag, Berlin, 1979. MR 0540367 (81c:46001)
- L. V. Kantorovich and G. P. Akilov, Functional Analysis, “Nauka”, Moscow, 1984; English transl., Pergamon Press, Oxford-Elmsford, New York, 1982. MR 664597 (83h:46002)
- V. V. Yurinskiĭ, Exponential bounds for large deviations, Teor. Veroyatnost. Primenen. 19 (1974), no. 1, 152–153; English transl. in Theory Probab. Appl. 19 (1974), 154–155. MR 0334298 (48:12617)
- W. Feller, An Introduction to Probability Theory and its Applications, vol. II, John Wiley & Sons, Inc., New York–London–Sydney, 1971. MR 0270403 (42:5292)
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Additional Information
I. K. Matsak
Affiliation:
Department of Operations Research, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
mik@unicyb.kiev.ua
Keywords:
Empirical distribution function,
ordinal convergence,
Glivenko–Cantelli theorem
Received by editor(s):
September 1, 2005
Published electronically:
January 24, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
