Does the following "theorem" have a name?

Question

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.

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. — user27119, Apr 28 '20 at 21:38
I've edited my question to be clear I am talking about probability distributions. — user27119, Apr 28 '20 at 21:39
See https://stats.stackexchange.com/questions/331973 for much more about this. — whuber, Apr 28 '20 at 22:55
@mdewey This was already posted by whuber, but thanks anyway! — user27119, May 01 '20 at 15:25

score 1 · Accepted Answer · answered Apr 28 '20 at 21:13

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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.

answered Apr 28 '20 at 21:13

gunes

57,205

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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. – user27119 Apr 28 '20 at 21:40

Does the following "theorem" have a name?

1 Answers1