Limit theorems for dependent Bernoulli variables with statistical inference

ABSTRACT

We consider a class of dependent Bernoulli variables where the conditional success probability is a linear combination of the last few trials and the original success probability. We obtain its limit theorems including the strong law of large numbers, weak invariance principle, and law of the iterated logarithm. We also derive some statistical inference results which make the model applicable. Simulation results are exhibited as well to show that with small sample size the convergence rate is satisfying and the proposed estimators behave well.

KEYWORDS:

• Central limit theorem
• consistent estimator
• dependent Bernoulli random variables
• martingale

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F05
• 60F25
• 62F10

Acknowledgments

The authors are indebted to two anonymous referees whose valuable comments helped to improve the manuscript greatly.

Funding

This research is supported by the National Natural Science Foundation of China (No.11225104) and the Zhejiang Provincial Natural Science Foundation for Distinguished Young Scholars (No.R610019).