Convergence Laws in Probability

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Homework Statement


Hello, I'm trying to revise for my probability exam next week and am getting a bit hung up on how to show a sequence of random variables converges.
For example, how would I got about doing this question:
Let {X[tex]_{n}[/tex]:n[tex]\geq[/tex]1} be a sequence of random variables with distribution functions F[tex]_{n}[/tex](x) = P(X[tex]_{n}[/tex][tex]\leq[/tex]x) defined by:

F[tex]_{n}[/tex](x) = [tex]\left\{\stackrel{0 if x<0}{\left(\frac{x}{2\theta}\right)^{n} if 0\leq x \leq 2\theta}[/tex]
and 1 if 2 theta is less than or equal to x

and let X be a degenerate random variable at the point x=2[tex]\theta[/tex]. Analyze the convergence in probability of X[tex]_{n}[/tex] to X when n tends to infinity.

Homework Equations





The Attempt at a Solution


I know that convergence in probability implies convergence in districbution, but I need to prove the reverse implication, which I know to be untrue...I think.
Also, the actual method of analysing the convergence is slightly confusing me as well.

Any pointers would be greatly appreciated!
Thanks.
PS Apologies for the Latex fail!
 
Last edited:

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