Proof that mutual statistical independence implies pairwise independence

Question

This question about pairwise vs. mutual relations is related some extant questions: here and here.

Kobayashi, Mark & Turin's Probability, Random Processes and Statistical Analysis, 2012, states without proof:

three events, A, B, C are mutually independent when:

P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C], P[A,B,C]=P[A]P[B]P[C]

No three of these relations necessarily imply the fourth. [my italics]

However, Wikipedia and others generally agree that mutual independence implies pairwise independence, but also without a demonstration.

What is the simplest proof that mutual independence implies pairwise independence?

Note: GC Rota wrote that probability can be understood by focusing on random variables or focusing on distributions. However, the two views should be equivalent, correct?

The assertion is that no three of the relations imply the fourth. That is true. Of course the four relations imply the first three, but that is not what is being said. — André Nicolas, Jan 26 '13 at 04:47
Mutual independence is the fourth equality only, not all four. — alancalvitti, Jan 26 '13 at 04:52
@AndréNicolas, ok, I mistakenly assumed that mutual statistical independence behaved like "mutual relation". — alancalvitti, Jan 26 '13 at 04:56
The fourth relation does not imply the others. Let $A$ and $B$ be highly dependent events, and let $\Pr(C)=0$. (There are less trivial examples!) — André Nicolas, Jan 26 '13 at 04:57
Understood. I used the wrong definition of mutual statistical independence. — alancalvitti, Jan 26 '13 at 05:02
@AndréNicolas, what about in terms of factorization of the PDFs of 3 random variables, rather than in terms of events (since there are no zero PDFs)? — alancalvitti, Jan 27 '13 at 21:56

score 3 · Accepted Answer · answered Jan 26 '13 at 04:49

3

Mutual independence means the four identities you copied, pairwise independence means the first three of these identities. Ergo.

answered Jan 26 '13 at 04:49

Did

284,245

Mutual independence means the last of the identities, not all four. Please clarify? – alancalvitti Jan 26 '13 at 04:51
2

Mutual independence means all four of the identities, not only the last one. Please grab any textbook. – Did Jan 26 '13 at 04:52
I see, ok, +1. Then there is an ambiguity in mutual statistical dependence versus mutual relation as opposed to pairwise relation. – alancalvitti Jan 26 '13 at 04:55
Ok, my mistake is definitional. – alancalvitti Jan 26 '13 at 04:59
what I meant to ask - but did not phrase properly - is whether properly mutual independence (ie, the last equality as opposed to all four) implies pairwise independence. – alancalvitti Jan 27 '13 at 21:58

score 2 · Answer 2 · answered Feb 27 '19 at 23:33

For an example where $P[A,B,C]=P[A]P[B]P[C]$ does not imply $P[A,B]=P[A]P[B], P[B,C]=P[B]P[C], P[A,C]=P[A]P[C]$ consider the distribution with indicators:

A  B  C  Prob
1  1  1   1/8
1  1  0   3/8
0  0  1   3/8
0  0  0   1/8

So $P[A]=P[B]=P[C]=\frac12$ and $P[A]P[B]P[C]=\frac18$ but $P[A,B]=\frac12, P[B,C]=\frac18, P[A,C]=\frac18$

In such as case, $A,B,C$ are not mutually independent

Proof that mutual statistical independence implies pairwise independence

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