The asymptotics of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ for normal distribution using saddle point approximation In my probability and numerical analysis research I have come across the following predicament:

If we have a standard normal random variable X with CDF $ \Phi $, and PDF $ \phi $ I am interested in the asymptotic behavior of $\int_{-\infty}^{\infty} \phi(x) {\Phi(\frac{x}{a})}^{qa} dx $ where $ a,q \geq 1 $ are constants and when $ a \to \infty $?

I tried Laplace's method (quite trivially) but it did not yield good explicit results, so I thought the only possible answer to this might be the saddle point method which I barely know anything about using (in general or in probability theory) so I am writing here in the hopes of someone helping me do something with this here, I realize a similar question exists on this site  here
but that one got no attention (it is not mine either and I know not its owner), so my question is more concrete and I need help in applying the method of saddle point approximation in this case. I thank all helpers
 A: In my opinion, that other question has been answered by michael completely in the very first comment. However, since the question arose again, let me just spell the details of michael's answer out.
We have $\Phi(z)=\frac12e^{\sqrt{2/\pi}z+O(z^2)}$. Thus $\Phi(x/a)^{qa}=2^{-qa}e^{\sqrt{2/\pi}qx}e^{O(qx^2/a)}$. Now squeeze the desired integral between $2^{-qa}\frac 1{\sqrt{2\pi}}\int e^{\sqrt{2/\pi}qx}e^{-(\frac 12+C\frac qa)x^2}$ and $2^{-qa}\frac 1{\sqrt{2\pi}}\int e^{\sqrt{2/\pi}qx}e^{-(\frac 12-C\frac qa)x^2}$. If $q^3/a\to 0$, both bounds are just $2^{-qa}e^{q^2/\pi}(1+O(\frac {q^3}a))$.
Now, of course, this is just the main term. If you want to get more terms in the asymptotic series, you need to use more terms in the Taylor formula for $\log\Phi(z)$ near the origin and to do Laplace in honest. As to the saddle point method, I would not bother about it here, but, if you are really interested, de Bruijn's book "Asymptotic methods in analysis" is an excellent introduction for beginners (that was the second book I was assigned to read as a student and I would certainly recommend it as a compulsory reading for all students at master level and above; it is relatively short, concise, and pretty much to the point on every issue it touches).
