Are there bounds on the Spearman correlation of a sum of two variables? Given $n$-vectors $x, y_1, y_2$ such that the Spearman correlation coefficient of $x$ and $y_i$ is $\rho_i = \rho(x,y_i)$, are there known bounds on the Spearman coefficient of $x$ with $y_1 + y_2$, in terms of the $\rho_i$ (and $n$, presumably)? That is, can one find (non-trivial) functions $l(\rho_1,\rho_2,n), u(\rho_1,\rho_2,n)$ such that 
$$l(\rho_1,\rho_2,n) \le \rho(x,y_1+y_2) \le u(\rho_1,\rho_2,n)$$
edit: per @whuber's example in the comment, it appears that in the general case, only the trivial bounds $l = -1, u = 1$ can be made. Thus, I would like to further impose the constraint:


*

*$y_1, y_2$ are permutations of the integers $1 \ldots n$.

 A: Spearman's rank correlation is just the Pearson product-moment correlation between the ranks of the variables. Shabbychef's extra constraint means that $y_1$ and $y_2$ are the same as their ranks and that there are no ties, so they have equal standard deviation $\sigma_y$ (say). If we also replace x by its ranks, the problem becomes the equivalent problem for the Pearson product-moment correlation.
By definition of the Pearson product-moment correlation,
$$\begin{align}
\rho(x,y_1+y_2) 
  &= \frac{\operatorname{Cov}(x,y_1+y_2)}
          {\sigma_x \sqrt{\operatorname{Var}(y_1+y_2)}} \\
  &= \frac{\operatorname{Cov}(x,y_1) + \operatorname{Cov}(x,y_2)}
          {\sigma_x \sqrt{\operatorname{Var}(y_1)+\operatorname{Var}(y_2)
                          + 2\operatorname{Cov}(y_1,y_2)}} \\
  &= \frac{\rho_1\sigma_x\sigma_y + \rho_2\sigma_x\sigma_y}
          {\sigma_x \sqrt{2\sigma_y^2 + 2\sigma_y^2\rho(y_1,y_2)}} \\
  &= \frac{\rho_1 + \rho_2}
          {\sqrt{2}\left(1+\rho(y_1,y_2)\right)^{1/2}}. \\
\end{align}$$
For any set of three variables, if we know two of their three correlations we can put bounds on the third correlation (see e.g. Vos 2009, or from the formula for partial correlation):
$$\rho_1\rho_2 - \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2} \leq \rho(y_1,y_2) \leq 
  \rho_1\rho_2 + \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2} $$
Therefore
$$\frac{\rho_1 + \rho_2}
       {\sqrt{2}\left(1+\rho_1\rho_2 + \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}\right)^{1/2}}
  \leq \rho(x,y_1+y_2) \leq
 \frac{\rho_1 + \rho_2}
       {\sqrt{2}\left(1+\rho_1\rho_2 - \sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}\right)^{1/2}}
$$
if  $\rho_1 + \rho_2 \geq 0$; if $\rho_1 + \rho_2 \le 0$ you need to switch the bounds around.
