Let $\phi$ be the following function $$ \phi (\omega, n) \equiv \left( 4 e^{\gamma - 2} \right)^{\sqrt{n} \, i \omega / \Sigma} \left[ \frac{2}{\sqrt{\pi}} \Gamma \left( \frac{3}{2} + \frac{i \omega}{\sqrt{n} \, \Sigma} \right) \right]^n $$ where $\omega$ is a continuous variable and $n$ is a parameter. $e, \pi, \gamma$ are the Euler's number, Ludolph's constant and the Euler-Mascheroni constant, respectively. The capital $\Sigma$ is defined via $\Sigma^2 = \pi^2 / 2 - 4$, thus, also just a numerical parameter.
Now we define the following Fourier transform $$ f (x, n) \equiv \frac{1}{2 \pi} \int \limits_{- \infty}^\infty \mathrm{d} \omega \, e^{- i \omega x} \phi (\omega,n) $$
It's not super hard to prove, that as $n \to \infty$, $f (x, n) \to \mathcal{N} (x; \mu = 0, \sigma = 1)$ (normal distribution with zero mean and unit variance).
As a first order business, I would like to prove, that the integral defining $f$ results in a valid distribution only when $n$ is an integer. What I know: the integral of $f$ over the whole real line is always 1, regardless of $n$. What I suspect: $f$ is strictly positive only when $n$ is a positive integer. I don't know how to prove it.
Since computing the integral numerically (there's no other way for general $n$, as far as I'm aware) is expensive (especially in the tails, where greater numerical precision is needed), I'd like to find the asymptotic/limiting behavior of $f$ for $x \to \pm \infty$, for any integer $n$ (assuming we already proved the previous point). That way, the numerical integration gives the main shape, while the tails can be extrapolated with the asymptotic forms.
For example, we know immediately, that for $n \to \infty$, the function is the normal distribution $ \sim e^{-x^2/2}$ (for both $x \to \pm \infty$), which also gives its asymptotic behavior.
The integral is also computable for $n = 1$ and it's asymptotic behavior is much more complicated:
$$ f(x \to - \infty, 1) \sim \frac{\Sigma (e^{2 - \gamma} / 2)^{3/2}}{\sqrt{2 \pi}} \exp \left( \frac{3 \Sigma}{2} x \right) $$ $$ f(x \to + \infty, 1) \sim C \exp \left( - \frac{e^{2 - \gamma}}{4} e^{\Sigma \, x} \right) $$ with $C$ being a constant I have had trouble to determine.
Can an analogous analysis be done for any integer $n \geq 2$?