Q: What is the tail behaviour of $\log \Phi(t)$ as $t \to \infty$?
Since $\Phi(t) \to 1$ as $t \to \infty$, we know that $\log \Phi(t)\to 0$, but I would like to know at what rate this function vanishes.
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1The results described at https://stats.stackexchange.com/a/7206/919 make short work of this, because they relate the tail behavior of $1-\Phi(t)$ to that of $\phi(t)$ whose logarithm is $-t^2/2.$ – whuber Mar 21 '19 at 13:41
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2The tail behavior of $Q(t) = 1-\Phi(t)$ as $t \to \infty$ has been well-studied and I am sure that there are many questions and answers on this site regarding this. Perhaps looking for Mill's ratio will help. But if not, here is my answer on math.SE where you can find upper and lower bounds on $Q(t)$ from which you can get some results on the asymptotic behavior of $\log \Phi(t)$ as $t\to\infty$. – Dilip Sarwate Mar 21 '19 at 13:47
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@DilipSarwate Thanks. Indeed, there are plenty of questions regarding $Q$ but $1-Q$ seems to have received less attention. Nonetheless, I should be able to get the answer eventually from the links you and whuber point out. – Logan Mar 21 '19 at 13:49