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!



Is "small" dependence enough for central limit theorem?
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'' 


