Suppose I have two finite sets of data
B, with equal length
What's the best upper and lower bound I can make on
var(A+B), in terms of
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The lower bound will occur when $B = -A$ so that $A + B = 0$ and the variance is $0$. This also occurs when $B$ is shifted from $A$, but in any case you cannot have a variance less than $0$ and this shows a case where it can equal $0$.
The upper bound comes when $B = A$ or $B = c1 + c2 \times A$ which gives a correlation of $1$. In this case you get the variance is the comments above.