I Does the generalized gamma distribution have a mgf?

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The generalized gamma distribution has moments of all orders but lacks a moment generating function (mgf) due to the condition α<1, which leads to divergence in the integral for mgf. The density function is defined as f(x)=Cx^(β-1)e^(-x^α) for x>0, with normalization constant C. There is a debate over the conditions on α and β, with one contributor suggesting that the requirement β>-1 is necessary for all moments to exist, while questioning the relevance of α<1. The discussion highlights the need for clarity on the bounds for α and β, emphasizing that both could potentially be negative under certain conditions. Ultimately, the existence of the mgf remains unresolved, contingent on the correct interpretation of these parameters.
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I'm reading An Intermediate Course in Probability by Gut. I am confused about a statement made concerning the generalized gamma distribution and its existence of a moment generating function.
I quote from An Intermediate Course in Probability by Gut:

Another class of distributions that possesses moments of all orders but not a moment generating function is the class of generalized gamma distributions whose densities are $$f(x)=Cx^{\beta-1}e^{-x^{\alpha}},\quad x>0,\tag1$$where ##\beta>-1##, ##0<\alpha<1##, and ##C## is a normalization constant (that is chosen such that the total mass equals ##1##).

It is clear that all moments exist, but, since ##\alpha<1##, we have $$\int_{-\infty}^\infty e^{tx}x^{\beta-1}e^{-x^\alpha}\, dx=\infty\tag2$$ for all ##t>0##, so that the moment generating function does not exist.

First, I don't think it is clear that all moments exist. Integrating ##(1)## and making the substitution ##y=x^\alpha##, and rewriting the integral in terms of a gamma integral, I get that ##C=\Gamma(\beta/\alpha)##. The only condition I get is ##\beta/\alpha>0##, so I don't see that the restriction ##\beta>-1## makes sense. I also don't see why ##\alpha<1## would make sense. Does anyone know what the proper bounds should be on ##\alpha## and ##\beta##? In view of my finding that ##\beta/\alpha>0##, I don't see why they both could be negative too.

Second, with the observations made above, does the moment generating function exist?
 
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I think the correct ##C## should be ##\alpha/\Gamma(\beta/\alpha)##. And ##\beta>-1## actually makes sense if we want all moments to exist. Moreover, ##0<\alpha<1## is only specified I think to show this is when the integral diverges.
 

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