Hello,
i am trying to solve this problem, however, after reading the weak law large number theorem, slutzky theorem, deta method many tiimes, i still vaguely have a clue to solve this.

This is one question with 7 parts, and i guess if you can't do first you can't solve the others.

It would be great if you can provide a detail solution or at least a hint/guide so i can try and follow it.

Thanks in advance!

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