Log-normal as a special case of the generalized gamma distribution

Questions
1

A limiting special case of the generalized gamma distribution:

\underset{\kappa \to \infty }{\text{lim}} \frac{\beta}{\Gamma (\kappa )\, \theta} \left(\frac{t}{\theta}\right)^{\kappa\beta -1} e^{-\left(\frac{t}{\theta }\right)^{\beta }}

should result in something resembling the pdf of the log-normal distribution.

However, I get precisely zero if I run:

Limit[b/(Gamma[k] s) (t/s)^(k b - 1) Exp[-(t/s)^b], k -> Infinity]

Am I missing something here?

Reference:

  1. Generalized Gamma: log normal as limiting special case
2

I’m not sure if it makes sense to make k to go to infinity, as the mean of the distribution would also go to infinity:

Assuming[\[Theta]>0\[And]\[Beta]>0,Limit[Mean[GammaDistribution[k,\[Theta],\[Beta],\[Mu]]],k->\[Infinity]]]

So the conclusion might be that it is a LogNormalDistribution but with its first parameter also infinity:

Limit[PDF[LogNormalDistribution[\[Mu], \[Sigma]], x], \[Mu] -> \[Infinity]]

which also results in to ‘0’ as the PDF (of course it is a limit, so it is not really 0 but goes closer and closer to 0)

3

See Generalize gamma log normal as limiting special case.

4

The limit actually is zero. Not sure what else can be said here.

5

[Deleted]

6

So, is this a useless result?

7

Is what useless?

This entire thread is remarkably unclear to me. A certain limit was computed, and correctly so. What else was expected?