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Dear All
I have two questions regarding Central Limit
Theorem:
1- We know that (classical) CLT may apply even when
the R.V.s neither are identical nor independent. For the i.i.d. case we also
know that rate of convergence is n^{0.5}. The first question is:
Is the most possible convergence rate to a proper
normal distribution equal to that of i.i.d.
scenario (=n^{0.5})?
2- suppose that Z_n = X_n + Y_n, where X_n =
sum_{i=1}^n(a_i) and Y_n = sum_{i=1}^n(b_i).
moreover, suppose that CLT applies for both of X_n
and Y_n with convergence rate of n^{0.5}.
Moreover, consider this case that a_i{i=1,...n} are
identically distributed but have local dependence with eachother,
b_i{i=1,...,n} are identically distributed but have local dependence with
eachother and finally some of a_i and b_i are dependent with
eachother. the question is: does the distribution of Z_n converge to a
proper normal distribution? and if yes, can we say that the convergence rate
is n^{0.5}?
Thanks in advance
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