Thread: Limiting distribution - iid gamma random variables

here is another question i stuck. again if you help me and explain steps, i will be pleased. thanks again

** Suppose that
X1, X 2, , X n be i.i.d. random variables with Gamma (α,n) pdf with mean μ= nα and variance σ^2=αn^2. Let Yn →α in probability. Find the limiting distribution of

sqrt(Yn)* [sqrt( sampleXn / nα - 1 ) ]


**
Suppose that X1, X2,…, Xn are i.i.d. Uniform(0,1). Denote U n=sqrt(n)*(sample Xn − 1/2) , n ≥1.

Find the moment generating function of Un.
Find the limiting distribution Un using first part

I get....

If

then converges to a standard normal
and of course by slutsky I can replace that alpha with Y's