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Math Help - Is this random variable almost surely convergence?

  1. #1
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    Is this random variable almost surely convergence?

    Let  X_n be a sequence of random variable with probability density of  f_n(x) = \frac {n}{ \pi (1+n^2x^2) }  \ \ \ \ \forall n \geq 1

    Does  X_n converges almost surely?

    Proof so far:

    Claim:  X_n converges to 0 almost surely.

    Let  \epsilon > 0 and fixed  N \in \mathbb {N} , I want to show that  \lim _{n \rightarrow \infty } P(|X_n|< \epsilon \ \forall n \geq N) = 1 ,

    that meansing proving that  P(X_n< \epsilon \ \forall n \geq N) = 1 and  P(X_n > - \epsilon \ \forall n \geq N ) = 1

    First the first one, we have:

     P(X_n < \epsilon \ \forall n \geq N ) = P(X_N \leq \epsilon ) since X_n is decreasing monotonically. And that is the CDF of  X_n , which is:

     \frac {1}{\pi} [ \arctan (N \epsilon ) + \frac {\pi}{2} ] \rightarrow 1 as  n \rightarrow \infty

    Similar, the one will work as well. So is this it? Thanks.
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    Re: Is this random variable almost surely convergence?

    Hello,

    It makes no sense to say that X_n decreases monotically. There is no relationship between the behaviour of the sequence of pdf's and the behaviour of the sequence of rv's.

    But I have to think further for the problem itself
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    Re: Is this random variable almost surely convergence?

    Our professor said that you would have assume they are independent and use borel-contelli theorem for.this
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    Re: Is this random variable almost surely convergence?

    Then he gave you all the elements to solve this...

    Consider the events A_n=\{|X_n|>1/n\}, prove that \sum_n P(A_n)=\infty. By Borel-Cantelli's lemma, and because the events are independent, it lets us say that P(\limsup_n A_n)=0 \Leftrightarrow P(\liminf_n \{|X_n|<1/n\})=1.
    Then write the liminf with quantifiers and it should give you the almost sure convergence
    Thanks from tttcomrader
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