This paper considers a sequence of Bernoulli random variables which are dependent in a way that the success probability of a trial conditional on the previous trials depends on the total number of successes achieved prior to the trial. The paper investigates almost sure behaviors for the sequence and proves the strong law of large numbers under weak conditions. For linear probability functions, the paper also obtains the strong law of large numbers, the central limit theorems and the law of the iterated logarithm, extending the results by. James etal. (2008).
Bibliographical noteFunding Information:
The authors would like to thank the reviewer for helpful comments and constructive suggestions that lead to significant improvement of the paper. Qi’s research was supported by NSF Grant DMS 0604176 , and Yang’s research was supported by the National Basic Research Program (973 Program) of China ( 2007CB814905 ) and the National Natural Science Foundation of China (Grant Nos. 10871008 , 11131002 ).
- Central limit theorem
- Dependent Bernoulli random variables
- Law of the iterated logarithm
- Strong law of large numbers