a problem of sample path

sean9991

This is a problem I face in my work.  I don't have a answer yet.  All discussions are welcome.

We are observing a time series C(t), the n th observation is realized at time t = tn, with value Cn.  C(t) is monotonic increasing in t.  Theoratical, we know C(t) / t converges (say, to R, an unknown number).

We would like to have a good rule for stopping time, T, so that C(T)/T is a reasonable approximation of R.

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