Difference between 'spatially' and 'instantaneously' independent
The components are instantaneously uncorrelated if 
$$
E[y_i(t_1) y_j(t_2)] = 0
$$
for all $i \not= j$ and for all $t_1,t_2$ and when $i = j$ :
$$
E[y_i(t_1) y_i(t_2)] = 0
$$
for all $t_1 \not= t_2$.

The components are spatially uncorrelated if
$$
E[y_i(t) y_j(t)] = 0
$$
for all $i \not= j$ (assuming a change in index corresponds to a spatial change).

Note that correlation is not quite the same thing as statistical independence.