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I'm interested in how to deal with α-stabile probability density functions, for example one with the Laplace image like:

$ \phi(s) = s^{(\alpha-1)}*e^{-\nu s^{\alpha}} $

I can work with this image analytically or invert it numerically, but im worried about the normalization. Currently i'm specifically working with numerical inversions and the area under the PDF is not 1. Why is this?

The code i'm using to numerically invert this PDF is based on Talbot contours and the code is from Abate and Valko, (2003) IJNME A1032.C.

FT[F_, t_, M_:32]:=
Module[{np, r, S, theta, sigma},
np = Max[M, $MachinePrecision];
r = SetPrecision[2M/(5t), np];
S = r theta (Cot[theta] + I);
sigma = theta + (theta Cot[theta] - 1)Cot[theta];
(r/M)Plus @@ Append[Table[Re[Exp[t S](1 + I sigma)F[S]],
        {theta, Pi/M, (M - 1)Pi/M, Pi/M}],
            (1/2) Exp[r t] F[r]]
]

ny=1;
alpha=0.25;
g[s_]:=g[s]=s^(alpha-1) * Exp[-ny*s^alpha];
NIntegrate[FT[g, t], {t, 0, 10}]

This produces the result 3.86102, which already after integrating to t=10 is over 1. Since the pdf cannot take negative values, it will only increase. In fact it goes to orders of $10^{100}$..

I know that i'm dealing with a infinite-mean PDF, at least without truncation, but the area should still be normalized to 1, i believe. I need this specifically to then calculate the cumulative distribution and generate random samples based off my distribution (which is more complex, but of the same type as demonstrated in my question).

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