I have two random walks, which represent fishing mortality in season 1 and season 2 of year $t$ ($X_{t,\mathrm{summer}}$ and $X_{t,\mathrm{winter}}$). If I add up both series to obtain annual fishing mortality ($X_t$), what would be the expected variance?

I read that for independent time series, the total variance would be the sum of the individual variances. But it seems this might not be the case here, as fishing mortality depends on stock size and catches and the former does not change much between seasons.

  • $\begingroup$ I don't have series by year but by season, which I hope I made more clear by adapting my question. Fishing mortality was calculated with the Baranov catch equation and approximates the number of fish deaths divided by the number of fish in the water. $\endgroup$ – Wave Jan 3 '18 at 14:42
  • $\begingroup$ Do you have data for only 1 year (i.e. two observations: one for summer, one for winter) Or do you have data for multiple years? $\endgroup$ – Matthew Gunn Jan 3 '18 at 18:10
  • $\begingroup$ 50 years. Otherwise it would be point estimates and not a random walk? $\endgroup$ – Wave Jan 4 '18 at 18:27

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