Say we have $n$ observation $\{X_i, i=1,...,n\}$ from a realisation of a Wiener process. We don't know when the process began. We want to estimate the autocovariance of this process. If we form two samples, say $\mathbf{X_1}=\{X_i, i=1,...,n-10\}$ and $\mathbf{X_2}=\{X_i, i=11,...,n\}$ and calculate sample auto-covariance then what do we get? The covariance of $W_s$ and $W_t$ is $\min\{s,t\}$ but this is not what the sample autocovarince seems to give. If I take the expecation of $\mathbf{X_1'X_2}$ then we get (I think) a sum of times $S=\Sigma_{i=1}^{n-10}t_i$ (which we don't know since we don't know when the process began). How do we get to the $\min\{s,t\}$ results from our one sample?


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