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I'm currently reading the paper "Random matrices: The distribution of the smallest singular values" by '"Terence Tao and Van Vu" and have run into some terminology which I don't quite (rigorously) understand.

In Theorem 1.3, the authors state that $\mathbb{E}[|\xi|^{C_0}]<\infty$ for some sufficiently large absolute constant $C_0>0$. What does "sufficiently large absolute constant" mean? I googled it but I couldn't find a definition.

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  • $\begingroup$ So it holds for any $\xi$ random variables in $L^2$? $\endgroup$
    – ABIM
    Commented Apr 6, 2020 at 11:46
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    $\begingroup$ "Absolute constant" usually means it does not depend on any of the fixed data. $\endgroup$
    – YCor
    Commented Apr 6, 2020 at 12:35
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    $\begingroup$ Any random variable $\xi$ ? No. Any random variable $\xi$ as specified in Theorem 1.3. If $\xi \in L^2$, then it hods for $C_0=2$ by definition of $L^2$. Presumably the random variables in Theorem 1.3 need not be in $L^2$, so he has to use a larger constant $C_0$. Perhaps they state it that way because they do not actually compute the constant $C_0$ that works, they only prove that one exists. $\endgroup$
    – Gerald Edgar
    Commented Apr 6, 2020 at 13:07

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"Absolute constant" means that it does not depend on anything. For example, $3, 10^{12},\pi$ and Feigenbaum number are absolute constants. They are real numbers. "Sufficiently large" means that the authors did not care or could not compute or estimate it.

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