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We continue investigating the set of limit points of averages of rearrangements of a given sequence. First we generalize results from the previous paper: if a sequence is a composition of two sequences, one of which is bounded and one of which tends to infinity, then we show necessary and sufficient condition for expecting non-trivial accessible points. A new case will be studied too: the sequences composed of two sequences, one tending to  the other tending to  Then we start to study accumulation points of the averages of rearranged sequences and prove that if a sequence has 4 accumulation points  then any closed set in  can be represented as the set of accumulation points of the averages of a certain rearranged sequence. 

rearrangement of sequence, arithmetic mean.