*Bounty: 50*

*Bounty: 50*

My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance:

$$ lambda(u):=frac{1}{t}sum_{t}(Y_{t+u}-bar{Y})(Y_{t}-bar{Y}) $$

When I think of the covariance of Standard Brownian motion $W(t)$ with itself, i.e. $Cov(W_s,W_t)=min(s,t)$, the way I interpret the covariance is as follows: Since $mathbb{E}[W_s|W_0]=mathbb{E}[W_t|W_0]=0$, the Covariance is a measure of how "often" one would "expect" a specific Brownian motion **path** at time $s$ to be on the same side of the x-axis as as the same Brownian motion **path** at time **t**.

It’s perhaps easier to think of correlation rather than covariance, since $Corr(W_s,W_t)=frac{min(s,t)}{sqrt(s) sqrt(t)}$: with the correlation, one can see that the closer $s$ and $t$ are together, the closer the Corr should get to 1, as indeed one would expect intuitively.

The main point here is that at each time $s$ and $t$, the Brownian motion will have a distribution of paths: so if I were to "estimate" the covariance from sampling, I’d want to simulate many paths (or observe many paths), and then I would fix $t$ and $s=t-h$ ($h$ can be negative), and I would compute:

$$ lambda(s,t):=frac{1}{i}sum_{i}(W_{i,t}-bar{W_i})(W_{i,t-h}-bar{W_i}) $$

For each Brownian **path** $i$.

With the time-series approach, it seems to be the case that we "generate" just one path (or observe just one path) and then estimate the auto-covariance from just that one path by shifting throught time.

Hopefully I am making my point clear: my question is on the intuitive interpretation of the estimation methods.