Lets say I simulate a immigration-death process:
$P(X(t + \delta t) = x+ 1 | X(t) = x) = \lambda \delta t$
$P(X(t + \delta t) = x-1 | X(t) =x) = v x \delta t$
using a Gillespie simulation - I pick a waiting time using
$\tau = \frac{1}{\lambda + v x(t)} \times ln(1/r_1)$ where $r_1 \sim U(0,1)$
then select a reaction using
$$x(t + \tau) = \begin{cases} x(t) + 1, \text{ if } r_2 < \lambda / (\lambda + v x(t)) \\ x(t) - 1, \text{ if } r_2 \geq \lambda / (\lambda + v x(t)) \end{cases} $$
where $r_2 \sim U(0,1) $.
If I run this for long enough it will eventually oscillate around a steady state. Lets say I want the variance around that steady state. In this simple case I could calculate it analytically, but that quickly becomes insoluble as the system becomes more complex.
I can calculate the mean by calculating a weighted mean of the value of $x$ multiplied by the amount of time spent at $x$ and dividing by the total length of time the system is simulated for (after a burn in to reach "steady state").
But what about the variance? I can do a biased estimate of the variance with
$$ \hat\sigma^2 = \frac{\Sigma_i \tau_i(x_i - \bar x)^2}{\Sigma_i \tau_i}$$
where $x_i$ is the progression of number of x's and $\tau_i$ is the length of time at each step.
Is there an unbiased estimate? The answer here suggsets it depends if the weights are probability weights or frequency weights. The weights, $\tau_i$ seem like they are fulfilling the role of a frequency weight. They are not normalized to 1 and they say how often $x_i$ was observed. But they are not unitless. So if we unbaised by making the denominator $\Sigma_i \tau - 1$, then the answer would be very different depending on whether $\tau$ was in hours, minutes or seconds.
Does it even matter?
