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Given $z\geq 0$, denote $$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$ and $$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$ Does the following limit $$\lim\limits_{m\to\infty}\sqrt{m}(m-2)\int_0^1 (1-z)^{m-3}\frac{F_m(z)}{F_m(0)} dz$$ exist?

It is interesting if we know the value of this limit. If anyone know the references for this problem, please let me know.

Thanks in advances

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    $\begingroup$ Welcome to the site. For context could you please explain why is it interesting to know the limit? $\endgroup$
    – user9072
    Commented Oct 5, 2014 at 12:20
  • $\begingroup$ You have an $n-3$ in an exponent in the last integral. What is $n$? $\endgroup$
    – S. Carnahan ♦
    Commented Oct 11, 2014 at 13:56
  • $\begingroup$ @S.Carnahan: Almost surely that is an $m$. I have edited the post accordingly. $\endgroup$
    – Alex M.
    Commented Dec 8, 2021 at 21:25

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