Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE UNIFORM CLT FOR MARTINGALE DIFFERENCE ARRAYS UNDER THE UNIFORMLY INTEGRABLE ENTROPY

  • Bae, Jong-Sig (Department of Mathematics and Institute of Basic Science, Sungkyunkwan University) ;
  • Jun, Doo-Bae (Department of Mathematics, Sungkyunkwan University) ;
  • Levental, Shlomo (Department of Statistics and Probability, Michigan State University)
  • Published : 2010.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

Keywords

References

  1. R. J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes-Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990.
  2. J. Bae, An empirical CLT for stationary martingale differences, J. Korean Math. Soc. 32 (1995), no. 3, 427–446.
  3. J. Bae and M. J. Choi, The uniform CLT for martingale difference of function-indexed process under uniformly integrable entropy, Commun. Korean Math. Soc. 14 (1999), no. 3, 581–595.
  4. J. Bae and S. Levental, Uniform CLT for Markov chains and its invariance principle: a martingale approach, J. Theoret. Probab. 8 (1995), no. 3, 549–570. https://doi.org/10.1007/BF02218044
  5. R. F. Bass, Law of the iterated logarithm for set-indexed partial sum processes with finite variance, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 4, 591–608. https://doi.org/10.1007/BF00531869
  6. M. D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 1951 (1951), no. 6, 12 pp.
  7. P. Doukhan, P. Massart, and E. Rio, Invariance principles for absolutely regular empirical processes, Ann. Inst. H. Poincare Probab. Statist. 31 (1995), no. 2, 393–427.
  8. R. M. Dudley, Donsker classes of functions, Statistics and related topics (Ottawa, Ont., 1980), pp. 341–352, North-Holland, Amsterdam-New York, 1981.
  9. R. M. Dudley, Uniform Central Limit Theorems, Cambridge Studies in Advanced Mathematics, 63. Cambridge University Press, Cambridge, 1999. https://doi.org/10.2277/0521461022
  10. R. Durrett, Probability: Theory and Examples, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991.
  11. D. Freedman, On tail probabilities for martingales, Ann. Probability 3 (1975), 100–118. https://doi.org/10.1214/aop/1176996452
  12. M. I. Gordin and B. A. Lifsic, The central limit theorem for stationary Markov Processes, Soviet Math. Dokl. 19 (1978), no. 2, 392–394.
  13. J. Hoffmann-Jorgensen, Stochastic Processes on Polish Spaces, Various Publications Series (Aarhus), 39. Aarhus Universitet, Matematisk Institut, Aarhus, 1991.
  14. M. Ossiander, A central limit theorem under metric entropy with $L_2$ bracketing, Ann. Probab. 15 (1987), no. 3, 897–919. https://doi.org/10.1214/aop/1176992072
  15. D. Pollard, Convergence of Stochastic Processes, Springer-Verlag, New York, 1984.
  16. D. Pollard, Empirical Processes: Theory and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics, 2. Institute of Mathematical Statistics, Hayward, CA; American Statistical Association, Alexandria, VA, 1990.
  17. S. van der Geer, Empirical Processes in M-Estimation, Cambridge University Press, 2000.
  18. A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer Series in Statistics. Springer-Verlag, New York, 1996.
  19. K. Ziegler, Functional central limit theorems for triangular arrays of function-indexed processes under uniformly integrable entropy conditions, J. Multivariate Anal. 62 (1997), no. 2, 233–272. https://doi.org/10.1006/jmva.1997.1688

Cited by

  1. Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean vol.142, pp.11, 2012, https://doi.org/10.1016/j.jspi.2012.04.005
  2. THE SECOND CENTRAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE ARRAYS vol.51, pp.2, 2014, https://doi.org/10.4134/BKMS.2014.51.2.317
  3. Limiting law results for a class of conditional mode estimates for functional stationary ergodic data vol.25, pp.3, 2016, https://doi.org/10.3103/S1066530716030029
  4. Corrected portmanteau tests for VAR models with time-varying variance vol.116, 2013, https://doi.org/10.1016/j.jmva.2012.12.004
  5. The uniform central limit theorem for the tent map vol.82, pp.5, 2012, https://doi.org/10.1016/j.spl.2012.02.003
  6. On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications vol.30, pp.1, 2018, https://doi.org/10.1080/10485252.2017.1404059