I want to define the covariance of a variable, say $X$, which is the integral over $Y(t)$:
$$ {\rm Cov}[X,X'] = \int\int {\rm Cov} [aY(t), a'Y(t')] dt dt'.$$
Just as an example, let the variable $Y$ be unitless, so that the total covariance has units of ${\rm sec}^2$.
However, in my case, samples over $t$ are independent, so the covariance when $t\neq t'$ is 0. What is the correct way to define my covariance within the integral?
My initial thought is to use ${\rm Cov}[Y(t),Y(t')] = K \delta(t-t').$, however, this has units $1/{\rm sec}$ (unless $K$ has units of ${\rm sec}$?), so that the final result has units of ${\rm sec}$, rather than ${\rm sec}^2$.
Is there a standard way of doing this?
