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!