Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?

If $$X_1,X_2$$ are dependent but identically distributed, it seems obvious that $$P(X_1+X_2\geq2C) \leq P(X_1\geq C)=P(X_2\geq C)$$. At least if we additionally assume that the joint distribution is symmetric, i.e. $$P(X_1\leq a, X_2\leq b) =P(X_2\leq a, X_1\leq b)$$.

But how would I prove this?

• I assume the 'seems obvious' is because it would hold if $X_1$ and $X_2$ were perfectly correlated, and intuitively it seems that perfect correlation should maximise $X_1+X_2$. Commented May 16, 2022 at 3:40
• I'm pretty sure this exact question was asked and answered about a year ago--it's just hard to find with a search. Here is a generalization, though, that includes this question as a special case: stats.stackexchange.com/questions/525638.
– whuber
Commented May 16, 2022 at 12:42
• Yes that was my intuition @ThomasLumley ! Always humbling to learn that it was totally wrong!
– Jome
Commented May 16, 2022 at 18:14

Counterexample

$$P(X_1 = 3, X_2 = 7) = P(X_1 = 7, X_2 = 3) = 0.5$$

Then $$P(X_1 \geq 4) = 0.5$$ and $$P(X_1 + X_2 \geq 8) = 1$$

Thus in this case $$P(X_1 \geq c) < P(X_1 + X_2\geq c)$$

Below is a little sketch that describes which intuition I used to find these points.

If there are relatively many points in the triangular regions where $$X_1 + X_2 > 2c$$ while $$X_1 < c$$ or $$X_2 < c$$ then you get a counter example to the inequality.

• The distribution does not need to be negatively correlated or even dependent. For instance, imagine a pair of Bernoulli variables with with $$P(X_1,X_2 = 0,0) = P(X_1,X_2 = 1,0) = P(X_1,X_2 = 0,1) = P(X_1,X_2 = 1,1)$$ and consider $P(X_1 \geq 0.1) = 0.5$ with $P(X_1 + X_2 \geq 0.2) = 0.75$ Commented May 16, 2022 at 8:26
• The sketch reminds me an awful lot of stats.stackexchange.com/a/564911/919 ;-).
– whuber
Commented May 16, 2022 at 12:44
• @whuber I must say that this visual great and your answer over there (including the visual) might be better than mine (not including the visual). I actually don't even remember anymore that question and me answering it. I will have to review that other post again if I get the time. Commented May 16, 2022 at 13:01
• It is good to read back what I wrote a while ago. I sometimes don't get it anymore myself so how should the reader get it, and in this case the post is only three months old Commented May 16, 2022 at 13:07
• Forgetting stuff you posted is the curse of posting a lot of stuff :-).
– whuber
Commented May 16, 2022 at 13:11