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Convergence in distribution example

  1. Apr 25, 2017 #1
    1. The problem statement, all variables and given/known data
    prob.png

    2. Relevant equations

    Definition: A sequence [itex] X_1,X_2,\dots [/itex] of real-valued random variables is said to converge in distribution to a random variable [itex]X[/itex] if [itex]\lim_{n\rightarrow \infty}F_{n}(x)=F(x)[/itex] for all [itex]x\in\mathbb{R}[/itex] at which [itex]F[/itex] is continuous. Here [itex]F_n, F[/itex] are the cumulative distributions functions of the random variables [itex]X_n[/itex] and [itex]X[/itex] respectively.

    3. The attempt at a solution

    I'm trying to understand/recreate the following solution to the problem.

    prob1.png

    My working so far is that
    $$F_{X}(x)=P(X\leq x)=\begin{cases} 0, &x<-1 \\ 1/2, &x\in[-1,1) \\ 1, &x\geq 1\end{cases}$$ and since [itex]X[/itex] only takes values 1 and -1 then [itex]X_n = (-1)^{n+X}+\frac1n=(-1)^{n+1}+\frac1n[/itex] and so $$F_{X_n}(x)=P(X_n\leq x)=\begin{cases} 0, &x<(-1)^{n+1}+\frac{1}{n} \\ 1, &x\geq (-1)^{n+1}+\frac{1}{n}\end{cases}$$ I can't understand how the limits to this have been achieved in the solution. Why does [itex]F_{X_n}(x)\rightarrow 1/2[/itex] for [itex]t\in(-1,1)[/itex], say?
     
    Last edited: Apr 25, 2017
  2. jcsd
  3. Apr 25, 2017 #2

    andrewkirk

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    I agree with your analysis. It looks like the problem has been incorrectly stated. The ##X_n## are not even random, since ##X_n=(-1)^{n+1}## for all integer ##n##.

    The ##X_n## do not converge in distribution to ##X## because ##F_X## is continuous at 0 and equal to ##1/2##, but ##F_{X_n}(0)## is alternately ##0## and ##1## as ##n## increments, hence ##F_n(0)## does not converge to ##1/2##.

    The specific error in the text's attempted proof is the statement that 'for large enough ##n##, ##F_{X_n}(t)=1/2##' (for ##t\in(-1,1)##) .
     
  4. Apr 26, 2017 #3
    That makes sense, thanks a lot! I thought I was going crazy xD
     
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