This is just a particular example of Weak Law of Large Number, which can be proved typically by the application of Chebyshev's inequality. In this case you can apply this inequality because all (4th) momemts of a normal random variable exists.
The first idea is alright - if you already have the result of convergence in mean.
i.e. Are you sure you can directly use the result
without proving it?
For the second idea, the step
is wrong. As a quick check: the RHS is independent of the limit , and as you see there is no random variable inside the probability and it is actually equal to zero.
Tweet |