CAN Estimators from Dependent Observations

Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)


In this chapter, we review some basic properties of stationary stochastic processes. Results on martingale limit theorems and laws of large numbers for mixing sequences, as well as central limit theorems for sums of dependent random variables have been discussed. We then discuss weak convergence of empirical processes obtained from stationary observations. These results have been applied to generate consistent and asymptotically normal estimators of parameters of a stationary stochastic process.


Weak Convergence Ergodic Theorem Empirical Process Empirical Distribution Function General Random Sequence 
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  1. Aitchison, J., Silvey, S.D.: Maximum-likelihood estimation of parameters subject to restraints. Ann. Math. Statist. 29, 813–828 (1958)Google Scholar
  2. Athreya, K.B., Lahiri. S.N.: Measure Theory and Probability. Springer, New York (2006)Google Scholar
  3. Athreya, K.B., Pantula, G.S.: Mixing properties of Harris chains and Auto-regressive Processes. J. Appl. Probab. 23, 880–892 (1986a)Google Scholar
  4. Athreya, K.B., Pantula, G.S.: A note on strong mixing ARMA processes. Statist. Probab. Lett. 4, 187–190 (1986b)Google Scholar
  5. Basawa, I.V., Prakasa Rao, B.L.S.: Statistical Inference for Stochastic Processes. Academic Press, London (1980)Google Scholar
  6. Bickel, P.J., Bühlmann, P.: A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli. 5, 413–446 (1999)Google Scholar
  7. Billingsley, P.: Statistical Inference for Markov Processes. The University of Chicago Press, Chicago (1961a)Google Scholar
  8. Billingsley, P.: The Lindeberg-Lévy theorem for martingales. Proc. Am. Math. Soc. 12, 788–792 (1961b)Google Scholar
  9. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)Google Scholar
  10. Boos, D.D.: A Differential for L Statistics. Ann. Statist. 7, 955–959 (1979)Google Scholar
  11. Bosq, D.: Nonparametric Statistics for Stochastic Processes Lecture Notes in Statistics 110. Springer, New York (1996)Google Scholar
  12. Bradley, R.A.: Basic properties of strong mixing conditions : A survey and some open questions. Probab. Surv. 2, 107–144 (2005)Google Scholar
  13. Bühlmann, P.: Blockwise bootstrapped empirical process for stationary sequences. Ann. Statist. 22, 995–1012 (1994)Google Scholar
  14. Deo, C.M.: A note on empirical processes of strong mixing sequences. Ann. Probab. 1, 870–875 (1973)Google Scholar
  15. Doukhan, P.: Mixing : Properties and examples lecture notes in statistics 85. Springer, New York (1994)Google Scholar
  16. Fernholz, L.T.: Von Mises Calculus for Statistical Functionals. Lecture Notes in Statistics, Vol. 19, Springer, New York (1983)Google Scholar
  17. Gorodetskii, V.V.: On strong mixing property for linear sequences theory. Probab. Appl. 22, 411–413 (1977)Google Scholar
  18. Götze, F., Hipp, C.: Asymptotic expansions for sums of weakly dependent random variables. Z. Wahr. verw. Geb. 64, 211–240 (1983)Google Scholar
  19. Hall, P., Heyde, C.C.:Martingale Limit Theory and its Applications Academic Press, London (1980)Google Scholar
  20. Ibragimov, I.A.: A central limit theorem for a class of dependent random variables theory. Probab. Appl. 8, 83–89 (1963)Google Scholar
  21. Lahiri, S.N.: Resampling Methods for Dependent Data. Springer, New York (2003)Google Scholar
  22. Lindsey, J.K.: Statistical Analysis of Stochastic Processes in Time. Cambridge University Press, New York (2004)Google Scholar
  23. Nze, P.A., Bühlmann, P., Doukhan, P.: Weak Dependence beyond mixing and asymptotics for nonparametric regression. Ann. Statist. 30, 397–430 (2002)Google Scholar
  24. Nze, P.A., Doukhan, P.: Weak dependence : models and applications. In: Dehling, H., Mikosch, T., Sørensen, M. (eds.) Empirical Process Techniques for Dependent Data, pp. 117–136. Birkhäuser, Boston (2002)Google Scholar
  25. Radulović, D.: On the Bootstrap and the Empirical Processes for Dependent Sequences. In: Dehling, H., Mikosch, T., Sørensen, M. (eds.) Empirical Process Techniques for Dependent Data, pp. 345–364, Birkhäuser, Boston (2002)Google Scholar
  26. Rajarshi, M.B. : Chi-squared type goodness of fit tests for stochastic models through conditional least squared estimation. J. Ind. Statist. Assoc. 24, 65–76 (1987)Google Scholar
  27. Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)Google Scholar
  28. Shao, J.: Mathematical Statistics. Springer, New York (1999)Google Scholar
  29. Tjøstheim, D.: Non-linear Time Series: A Selective Review Scand. J. Statist. 21, 97–130 (1994)Google Scholar
  30. Withers, C.S.: Convergence of empirical processes of mixing rv’s on [0,1]. Ann. Statist. 3, 1101–1108 (1975)Google Scholar
  31. Withers, C.S.: Conditions for linear processes to be strong mixing. Z. Wahr. Verw. Geb. 57, 477–480 (1981)Google Scholar
  32. Yoshihara, K.: Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors. Z. Wahr. Verw. Geb. 33, 133–137 (1975)Google Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsUniversity of PunePuneIndia

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