tl;dr: I have limited information on the characteristic function of a PDF $W(x)$ I seek. I try to deduce a fitting qualitative picture of $W(x)$ from this information. I ask if you would agree with qualitative picture of my results. Much text incoming.
First of, the distribution follows from a physical model that has a parameter $s$. I will call the random quantity of interest $x$ here. Thus, $$ W(x;s) $$ is the PDF of x with a parameter $s$. The parameter is non-negative. For $s=0$, we have $W(x,0) = \delta(x)$. The problem is that I don't know the PDF. The model I use allows me to calculate the characteristic function first and thus, in principle, the PDF. The characteristic function is $$ \varphi(t;s) $$ in this notation. Unfortunately, I don't even have a closed expression for $\varphi$. What I do have, however, are asymptotic solution for $s \to 0$ and $s \to \infty$. I will explain how this has to understood:
From the model, I can show that the characteristic function has the following form: $$ \varphi(t;s) = e^{-C(t;s)} $$ where $C(t;s)$ is a symmetric function, it is strictly monotically for t>0 and we have C(0;s) = 0 and C(t;s) > 0 elsewhere. It is real valued.
There is another property regarding the parameter $s$: $$ C(t;s) = |t|^{3/2} C(1;s t^{-1/2})$$ If we define $D(y) := C(1;y)$, I can show that $D(y)$ is an analytic function. The asymptotic solution for the characteristic function are found by expansions of $D(y)$ for $y \to 0$ and $y \to \infty$. Again, I don't want to go into detail, but even though I don't have closed expression for $D(y)$ either, the model can be solved in the respective limits, giving some taylor and laurent series. In the end, I have the following asymptotics: $$ D(y) = y - C_3 y^3 + \dots \quad (|y| < 1) \\ D(y) = 1 - C_1 y^{-1} + \dots \quad (|y| > 1) $$ Putting this into the above equation, I instead get $$ C(t;s) = |t| s - C_3 s^3 + \dots \quad (t > s^2) \\ C(t;s) = |t|^{3/2} - \dots \quad (t < s^2) $$ So, if we let $s \to 0$, the characteristic function look like that of a Cauchy distribution in a large domain. On the other hand, if $s$ becomes large enough, the characteristic function looks like that of a stable distribution with $\alpha = 3/2$. It is effectively an expansion in $s t^{-1/2}$, thus, the larger or smaller $s$ is, the better is the expression we get by truncating after the first order.
So, in principle, I know that the PDF looks like a Cauchy distribution if $s$ is small and look like a stable distribution with $\alpha = 3/2$ if $s$ tends to infinity. This result is verified by Monte Carlo simulation I ran on my model.
However, I try to get a qualitative pricture as to how the convergence works in the PDF for a fixed $s$. For example, lets look at the limit $s \to 0$: A Cauchy distribution has no moments, even the mean is not defined. However, this is clearly an error that results from the limit. Because, for any fixed, finite $s$, the characteristic function looks like $e^{-|t|^{3/2}}$ near $t=0$, whose first derivative exists. Whether the moments of distribution exists is determined by its tails. Therefore, it is clear that, for a finite, small $s$, the tails will not follow a Cauchy distribution, even though the center might.
At first, I had the following fallacy: Since, for a fixed $s$, the characteristic function looks like that of a Cauchy distribution for $t>>s^2$ (although there is this troublesome term $C_3 s^3$) and it looks like that of a different stable distribution for $t<<s^2$, I thought that, in "real space", the PDF $W(x)$ would look like Cauchy for $x<<1/s^2$ and look like the different stable distribution for $x>>1/s^2$.
But this clearly cannot be the case. $W(0)$ is equal to the total integral of the characteristic function, which means that all limits will contribute. For $s\to 0$, this is probably not much of a problem. The interval where the characteristic function does not look like that of a Cauchy distribution is given by $t<<s^2$, which goes to zero.
However, I still expect there to be some value $x_0$ where the PDF would transition between both limits for a fixed $s$. As I already explained, even for small $s$, the tails of the PDF should look like that of a stable distribution with $\alpha = 3/2$, since the first moment does exist. Thus, my qualitative picture is as follows: There is a $x_0$ that scales as $x_0 \propto s^{-2}$. For $x >> x_0$, the PDF looks like a stable distribution with $\alpha = 3/2$. For very large $s$, basically the whole PDF looks like that. Furthermore, for $s<<1$, the PDF looks like a Cauchy distribution for $x << x_0$. The transition between both limits is certainly highly non-trivial.
I know this was much text, my question is simply if you think if my qualitative picture (last paragraph) of the PDF is correct, given the information on the characteristic function I have.