Are there any advanced results established regarding the behavior of the Covariance of two random variables other than the bounds on the correlation and independence when it is zero etc. which are usually summarized in introductory notes on this topic such as at this link? http://www.stat.yale.edu/~pollard/Courses/241.fall97/Variance.pdf
Some Possibilities for Advanced Properties
1)
For example, whether it is a convex or concave function and so on and under what restrictions on the density or joint density functions etc.?
---- It seems to be both convex or concave since it is linear in the variables.
2)
Where do the maximum and the minimum occur etc.?
3)
Also, to get the joint density (for example, the bivariate normal) we need the correlation coefficient which is based on the covariance. And to get the covariance we need the joint density? Seems like a cyclical; which came first - the chicken or the egg problem?
Please let me know if anything is not clear or if this question is too trivial or incorrect in some sense etc.
Steps Tried to get Maximum / Minimum
$$ Cov\left(X,Y\right)=\int\int\left(t-\mu_{X}\right)\left(u-\mu_{Y}\right)f_{XY}\left(t,u\right)\:dt\:du $$ $$ =\int\int\left(t\:u\right)f_{XY}\left(t,u\right)\:dt\:du-\mu_{X}\:\mu_{Y} $$ Taking derivatives and First Order Conditions, $$ \frac{\partial Cov\left(X,Y\right)}{\partial\mu_{X}}=\int\int ut\left[\frac{\partial\left\{ f_{XY}\left(t,u\right)\right\} }{\partial\mu_{X}}\right]\:dt\:du-\mu_{Y} $$ $$ \Rightarrow\mu_{Y}=\int\int ut\left[\frac{\partial\left\{ f_{XY}\left(t,u\right)\right\} }{\partial\mu_{X}}\right]\:dt\:du $$ $$ \frac{\partial Cov\left(X,Y\right)}{\partial\mu_{Y}}=\int\int ut\left[\frac{\partial\left\{ f_{XY}\left(t,u\right)\right\} }{\partial\mu_{Y}}\right]\:dt\:du-\mu_{X} $$ $$ \Rightarrow\mu_{X}=\int\int ut\left[\frac{\partial\left\{ f_{XY}\left(t,u\right)\right\} }{\partial\mu_{Y}}\right]\:dt\:du $$
Can we take derivatives as above, assuming the densities are differentiable? Is another approach advisable? Can we simplify this further?
The only material I could find was this. (related but not the same) http://www.math.tu-dresden.de/sto/schmidt/dsvm/dsvm2003-4.pdf
Related Questions Joint Density and Covariance between Two Random Variables with the same Mean and Variance
https://math.stackexchange.com/questions/74677/covariance-of-increasing-functions/74681