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I am aware that if one has random variables, and sums them, then the result belongs to a distribution which is the convolution of the parent probability distributions of the initial random variables.

For example, let $X$ and $Y$ be random variables drawn from the probability distributions $F(t)$ and $G(t)$ respectively. If $Z = X + Y$, then $Z$ belongs to the probability distribution $$Z\sim (F*G)(t) = \int_{-\infty}^{+\infty} F(\tau)G(t - \tau) \ d\tau \text{.}$$

Does this principle or theorem have a name? Is it part of central limit theorem? I want to be able to refer to it quickly rather than explain the above every time.

Q.P.
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    The result is _not_ part of the central limit theorem. Also. note that if by _distribution_ you mean the Cumulative Probability Distribution Function (CDF), then the result you state is false: The CDF of the sum is _never_ the convolution of the CDFs. – Dilip Sarwate Apr 28 '20 at 21:13
  • No it doesn't answer my question as it doesn't give a name of a theorem so far as I can see. – Q.P. Apr 28 '20 at 21:38
  • I've edited my question to be clear I am talking about probability distributions. – Q.P. Apr 28 '20 at 21:39
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    See https://stats.stackexchange.com/questions/331973 for much more about this. – whuber Apr 28 '20 at 22:55
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    @whuber This was a really useful one, thank you! – Q.P. Apr 28 '20 at 23:17
  • @mdewey This was already posted by whuber, but thanks anyway! – Q.P. May 01 '20 at 15:25

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The RVs that are to be summed need to be independent for that formula to hold. And, the formula is for PDFs. I have seen it referred as convolution formula/rule (of course this makes sense if there is context), but didn't encountered with a formal name so far such as Central Limit Theorem. So, I don't think there is one that is commonly known. See here for the wikipedia entry.

gunes
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    That's great. Thanks. I knew it was part of convolution theory but was curious if it was a named component thereof. – Q.P. Apr 28 '20 at 21:40