I have a random vector $Z = X_{1} + X_{2} + \ldots + X_{T}$, where each $X_{i}$ is a $n\times 1$ random vector with finite mean and covariance matrix. Each $X_i$ has a different mean and covariance matrix. I have $ L = E[Z'](E[ZZ']^{-1})$, where E is expectation, $E^{-1}(ZZ')$ is inverse of $E[ZZ']$.
Now, if I take $T$ to infinity, what can I say about L? I suspect (after doing some numerical examples) that L will be very small, but is there a way to show it analytically?