论文信息 - Laws of Large Numbers for Dependent Heterogeneous Processes

Laws of Large Numbers for Dependent Heterogeneous Processes

Laws of large numbers (LLN's) for mixingale sequences are an essential tool in the proofs of consistency of estimators when dependent processes are considered. A mixingale sequence can be viewed as an asymptotic equivalent of a martingale difference sequence. In a recent paper, Andrews (1988) extended the mixingale concept and established weak LI -type LLN's for mixingales. Andrews' work extends the results of McLeish (1975), who introduced the mixingale condition. In Section 2 of this paper, Andrews' conditions for an LI -type LLN for Lp-mixingale sequences will be extended such as to cover the case of trended mixingale sequences. Recently, Hansen (1991, 1992) proved some new strong LLN's using Andrews' Lp-mixingale concept. Those results extend the results of McLeish. In Section 3 of this paper, the conditions of both McLeish and Hansen for a strong LLN to hold will be relaxed substantially for the important case of Lp-mixingales with uniformly bounded pth moments and bounded indices of magnitude. The results obtained in Section 3 are complementary to those obtained by Hansen and McLeish since our results will not be powerful in the case of increasing indices of magnitude or the case that no uniform moment bound exists. In Section 4, we apply our results to near epoch dependent sequences. Section 5 concludes this paper with the proofs of the theorems.

R. M. Jong | M. R. | DE Jong

[1] Bruce E. Hansen,et al. Strong Laws for Dependent Heterogeneous Processes , 1991, Econometric Theory.

[2] Robert Serfling,et al. Contributions to Central Limit Theory for Dependent Variables , 1968 .

[3] Sadayuki Sato,et al. On the Rising Prices in the Post-war Capitalist Economy , 1991 .

[4] Kai Lai Chung,et al. A Course in Probability Theory , 1949 .

[5] Donald W. K. Andrews,et al. An empirical process central limit theorem for dependent non-identically distributed random variables , 1989 .

[6] H. White,et al. A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , 1988 .

[7] A. Gut. The weak law of large numbers for arrays , 1992 .

[8] D. McLeish. A Maximal Inequality and Dependent Strong Laws , 1975 .

[9] Bruce E. Hansen,et al. Convergence to Stochastic Integrals for Dependent Heterogeneous Processes , 1992, Econometric Theory.

[10] D. Pollard. Convergence of stochastic processes , 1984 .

[11] Benedikt M. Pötscher,et al. Basic structure of the asymptotic theory in dynamic nonlinear econometric models , 1991 .

[12] Kazuoki Azuma. WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[13] H. Bierens. Model specification testing of time series regressions , 1984 .

[14] D. Andrews. Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables , 1988, Econometric Theory.