# Is “small” dependence enough for central limit theorem?

Writing down a paper about some estimation of some combinatorial quantities, i realized that i would have much more precise results if these two questions have positive answer:

1) Suppose you have a sequence of random variables $X_n$(boolean random variable with $Pr(X_i=1)=1/2$), such that $Cov(X_n,X_{n+1})=c$(in my case $c=-1/12$) and $X_i,X_j$ are independent whenever $|i-j|>1$. What can be said of $X_1+...+X_n$? Does the central limit theorem hold, altough there is no complete indipendence but "almost"?

2) Consider the well known relation ${2n} \choose {n}$$\sim\frac{4^n}{\sqrt \pi n}$. Is there a purely probabilistic way(using the central limit theorem or something near there) to prove this? The setting i've in mind is of course Bernoulli of parameter 1/2 $X_1,...,X_{2n}$ independent each other. And i can see that it would suffice to prove the convergence in 0 of the discrete densities to the gaussian in 0. Moreover if the answer in 1) is positive, does it allows, also there, the same asymptotics for the central term of $X_1+...+X_n$?

Thanks for any explanation!

For point 1, search for CLT for mixing sequences'' you will be drowned by the number of hits. Also, there are CLT's for stationary sequences in many textbooks - Hall and Heyde's book has a section on that.
For point 2, search for local CLT for lattice variables''