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](https://math.stackexchange.com/a/69417/15941) 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