# 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

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).