I have the equation:
$x = \mathrm{V}^{\frac{1}{2}}y$
which takes a vector of random numbers $y$ and induces a correlation structure according to a matrix $\mathrm{V}$, which is a square symmetric toeplitz matrix constructed from the autocorrelation function required for $x$. $\mathrm{V}$ is positive-definite, and its square root can be easily found by the Cholesky decomposition.
Therefore, I can take a vector of random numbers and induce a required correlation structure. However, $\mathrm{V}$, can be expressed as $\mathrm{V}_1 + \mathrm{V}_2 + \mathrm{V}_3$, as there are three distinct structures in the autocorrelation function. Rather than inducing all three correlation structures at the same time, I want to do the three individually and then add them together, but:
$x = \mathrm{V}^{\frac{1}{2}}y \neq \left(\mathrm{V}_1^{\frac{1}{2}} + \mathrm{V}_2^{\frac{1}{2}} + \mathrm{V}_3^{\frac{1}{2}}\right) y$
How can this be done?
($\mathrm{V}_i$ are positive-definite, and their square roots can also be easily found by the Cholesky decomposition.)
Perhaps you could write $$(V_1 + V_2 + V_3)^{1/2} y = V_1^{1/2} y + \left((V_1 + V_2)^{1/2} y - V_1^{1/2} y\right) + \left((V_1 + V_2 + V_3)^{1/2} y - (V_1 + V_2)^{1/2} y\right)$$ so that $V_1^{1/2} y$ is the effect of the first component alone, $(V_1 + V_2)^{1/2} y - V_1^{1/2} y$ is the difference the second component makes once you have the first component, and $(V_1 + V_2 + V_3)^{1/2} y - (V_1 + V_2)^{1/2} y$ is the difference the third component makes once you have the first two.