We consider a class of correlated Bernoulli variables, which have the following form: for some 0 < p < 1, $$\begin{align}{P(X_{j+1}=1 \vert {\cal F}_{j})= (1-\theta_j)p+\theta_jS_j/j,}\end{align}$$where 0 ≤ θj ≤ 1, $S_n=\sum _{j=1}^nX_j$ and ${\cal F}_n=\sigma \{X_1,\ldots , X_n\}$. The aim of this paper is to establish the strong law of large numbers which extend some known results, and prove the moderate deviation principle for the correlated Bernoulli model.

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相关伯努利模型的渐近行为

我们考虑一类相关的伯努利变量，其形式如下：对于一些 0 <p< 1,$$\begin{align}{P(X_{j+1}=1 \vert {\cal F}_{j})= (1-\theta_j)p+\theta_jS_j/j,}\end{align}$ $其中 0 ≤ θj≤ 1,$S_n=\sum _{j=1}^nX_j$和${\cal F}_n=\sigma \{X_1,\ldots , X_n\}$. 本文的目的是建立扩展一些已知结果的强大数定律，并证明相关伯努利模型的适度偏差原则。