Limit behaviors for dependent Bernoulli variables

Abstract

In this paper, we consider a class of dependent Bernoulli variables, which has the following form: for k ≥ m,

$$P\left({{X_{k + 1}} = 1\left|{{{\cal F}_k}} \right.} \right) = \sum\limits_{i = 1}^m {{\theta _i}{X_{k + 1 - i}} + {\theta _0}p,} $$

where m is a positive integer, $\sum\nolimits_{i = 0}^m {{\theta _i} = 1,\,\,{{\cal F}_n} = \sigma \left\{{{X_1}, \cdots,{X_n}} \right\},\,\,0 < p < 1}$. The convergence rate of the strong law of large numbers and the moderate deviation principle for the model are established. Furthermore, we study some properties of parameter estimation for the model.