Convergence in distribution example

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Homework Statement


prob.png


Homework Equations


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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.

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?
 
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Answers and Replies

  • #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)##) .
 
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  • #3
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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)##) .
That makes sense, thanks a lot! I thought I was going crazy xD
 

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