# Central limit theorem for weak dependent bernoulli random variables

Suppose $\epsilon_1,\epsilon_2,...$ are i.i.d bounded random variables with compact support. Let $X_k=g_k(\epsilon_k,...,\epsilon_1)$ be Bernoulli random variables with the covariance between $X_i$ and $X_j$ decaying exponentially.

Does a CLT hold here for the $X_k$, i.e. does $\frac{X_1+...+X_n-\Sigma_{i=1}^n \mathbb{E}[X_i]}{\sqrt{n}}$ converge in distribution to $N(0,\sigma^2)$?

Is there any paper that adresses the subject?

## 2 Answers

I agree with michael, you should have more hypothesis for the CLT and as he says, one have to check that $\sigma^2 \neq 0$. Another problem is that small covariance doesn't imply that the system is close to iid variable, strong corellation are possible for $3$ or more variables.

But I suppose that your exponential decay come from a kind of Markov (or Ergodic) process, in that case some CLT are proved in the book "Markov Processes Characterization and Convergence" STEWART N. ETHIER THOMAS G. KURTZ. You can be also interested in the Martingale Functional CLT cited in "Recent progress on the Random Conductance Model" Marek Biskup

No, you need more than that, for example, if $X_i$ are iid rvs taking values $\pm 1$ with prob 1/2 each and $Y_{2i} = X_i, Y_{2i+1} = -X_i$. If you avoid next door neighbors these are independent, so they have covariance that declines exponentially, but the partial sums are $\pm 1$. You may want to score this as convergence to normal with st. dev. 0, but you can modify it so that the variance doesn't converge by doing as I've sketched for a time, then just take i.i.d's until you get close to normal, then let it let it get back to 0 etc.