Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-25T12:41:53.525Z Has data issue: false hasContentIssue false
  • English
  • Français

Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables

Published online by Cambridge University Press:  18 October 2010

Donald W.K. Andrews
Donald W.K. Andrews*
Affiliation:
Cowles Foundation, Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides L1 and weak laws of large numbers for uniformly integrable L1-mixingales. The L1-mixingale condition is a condition of asymptotic weak temporal dependence that is weaker than most conditions considered in the literature. Processes covered by the laws of large numbers include martingale difference, ø(·), ρ(·), and α(·) mixing, autoregressive moving average, infinite-order moving average, near epoch dependent, L1-near epoch dependent, and mixingale sequences and triangular arrays. The random variables need not possess more than one finite moment and the L1-mixingale numbers need not decay to zero at any particular rate. The proof of the results is remarkably simple and completely self-contained.

Type
Research Article
Information
Econometric Theory , Volume 4 , Issue 3 , December 1988 , pp. 458 - 467
Copyright
Copyright © Cambridge University Press 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Andrews, D.W.K. Consistency in nonlinear econometric models: a generic uniform law of large numbers. Econometrica 55 (1987): 1465–1472.10.2307/1913568Google Scholar
2. Andrews, D.W.K. & Fair, R.C.. Inference in econometric models with structural change. Review of Economic Studies 60 (1988): forthcoming.Google Scholar
3. Billingsley, P. Convergence of probability measures. New York: Wiley, 1968.Google Scholar
4. Chow, Y.S. On the L p -convergence for n −1/p S n , 0 < p > 2. Annals of Mathematical Statistics 42 (1971): 393–394.10.1214/aoms/1177693530+2.+Annals+of+Mathematical+Statistics+42+(1971):+393–394.10.1214/aoms/1177693530>Google Scholar
5. Gallant, A.R. Nonlinear statistical models. New York: Wiley, 1987.10.1002/9780470316719Google Scholar
6. Gallant, A.R. & White, H.. A unified theory of estimation and inference for nonlinear dynamic models. New York: Basil Blackwell, 1988.Google Scholar
7. Hall, P. & Heyde, C.C.. Martingale limit theory and its applications. New York: Academic Press, 1980.Google Scholar
8. Herrndorf, N. A functional central-limit theorem for ρ−mixing sequences. Journal of Mul-tivariate Analysis 15 (1984): 141–146.10.1016/0047-259X(84)90073-3Google Scholar
9. Kolmogorov, A.N. & Rozonov, Yu. A.. On strong mixing conditions for stationary Gaussian processes. Theory of Probability and Its Applications 5 (1960): 204–207.10.1137/1105018Google Scholar
10. McLeish, D.L. A maximal inequality and dependent strong laws. Annals of Probability 3 (1975): 829–839.10.1214/aop/1176996269Google Scholar
11. McLeish, D.L. Invariance principles for dependent variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32 (1975): 165–178.Google Scholar
12. McLeish, D.L. On the invariance principle for nonstationary mixingales. Annals of Probability 5 (1977): 616–621.10.1214/aop/1176995772Google Scholar
13. Teicher, H. Almost certain convergence in double arrays. Z. Wahrscheinlichkeilstheorie verw. Gebiete 69 (1985): 331–345.10.1007/BF00532738Google Scholar
14. Wooldridge, J. & White, H.. Some invariance principles and central-limit theorems for dependent heterogeneous processes. Econometric Theory 4 (1988): forthcoming.10.1017/S0266466600012032Google Scholar