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I am trying to show that the limiting distribution (the limit of the distribution when [tex]n\rightarrow\infty[/tex]) of [tex]\sqrt{n} (B(\lfloor n/2\rfloor, n-\lfloor n/2\rfloor+1)-1/2)[/tex] is [tex]N(0,1/4)[/tex], where B is the Beta distribution and N is the normal distribution.

Wikipedia claims there is an expression of the Beta distribution as a quotient of sums of independent exponential distributions but I don't know how to prove this or how to go from there. I am thinking there must be an easy approach through CLT and such, but I don't seem to find it.

Any help on this problem would be greatly appreciated!