# #StackBounty: #time-series #autocorrelation #covariance #stochastic-processes #brownian Time-series Auto-Covariance vs. Stochastic Proc…

### 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.

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