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I want to find the moment generating function of the product of two independent random variables: $X \sim \mathcal{N}\left(0, \sigma^2 \right)$ and $Y \sim \mathcal{N}\left(0, \sigma^2 \right)$.

I think I have found the characteristic function:

$$ \varphi(t) = \sqrt{\frac{1}{1+ \sigma^4 t^2}} $$

Does this mean that the moment generating function is:

$$ M(t) = \varphi(-jt) = \sqrt{\frac{1}{1- \sigma^4 t^2}}$$

Or does it actually not exist?

Questions:

  1. Am I correct about the characteristic function?
  2. Is that, therefore, the moment generating function?
  3. If you find a characteristic function, how can you tell the moment generating function exists or does not exist?

Sorry for such a basic question but I am learning all of this on my own.

Thanks!

StubbornAtom
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The Dude
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  • @mrsamy Check here: https://math.stackexchange.com/questions/74013/characteristic-function-of-product-of-normal-random-variables Or just use the Mellin transform. – The Dude Jan 27 '21 at 15:05

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