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I just learned the theory of convolution between functions, for a probability density function, is there always existing another function g such that f = g * g (convolve with itself?)

daily dreamers
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    What did you try? Two-element sets? – markvs Nov 04 '21 at 01:14
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    You are probably talking about probability density on $\mathbb{R}$. Have you learned about characteristic functions associated to a probability density function? See https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory) – orangeskid Nov 04 '21 at 01:26
  • Related: https://math.stackexchange.com/questions/58525/when-can-a-random-variable-be-written-as-a-sum-of-two-iid-random-variables – Clement C. Nov 04 '21 at 07:09
  • (In particular, the link given there, https://en.wikipedia.org/wiki/Indecomposable_distribution, answers you question in the negative) – Clement C. Nov 04 '21 at 07:11

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