Ranks of F-limits of filter sequences

We give an exact value of the rank of an F-Fubini sum of filters for the case where F is a Borel filter of rank 1. We also consider F-limits of filters Fi, which are of the form limFFi={A⊂X:{i∈I:A∈Fi}∈F}. We estimate the ranks of such filters; in particular, we prove that they can fall to 1 for F as well as for Fi of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.