Let $Y=(Y_1,Y_2,...,Y_n)$ be the vector of order statistics for a random sample from the Pareto distribution with pdf $f(x)=(1+x)^{-2}, x\ge 0.$ Compute the limiting distribution for rv's $nY_1$ and $Y_n/n$ as $n \to \infty$.

So I tried finding cdf first and I got $F(x)=\int_o^x(1+x)^{-2}= \frac {x}{x+1}.$ But I'm not sure if the integration is correct.

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    $\begingroup$ The integration is correct. $\endgroup$ – NCh May 4 '17 at 12:32
  • $\begingroup$ Thanks. So for $Y_n/n$ I got the limiting distribution $e^{-v^{-1}}$. But I'm not sure how to compute the limiting distribution for $nY_1$? $\endgroup$ – ViC May 4 '17 at 13:13
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    $\begingroup$ $P(nY_1>x)=P(Y_1>x/n)=(1-F(x/n))^n = \left(1-\frac{x/n}{x/n+1}\right)^n\to e^{-x}$. $\endgroup$ – NCh May 4 '17 at 14:09

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