On some properties of Levy vectors and their variations

We investigate some properties of Levy vectors of vector series sigma(n is an element of omega) v(n). We consider a generalization of an example from [P. Klinga, Rearranging series of vectors on a small set, J. Math. Anal. Appl., 424(2):966-974, 2015] proving that for a special class of vector series, there exists a subseries that has no convergent subseries with the sum range equal to the plane. To prove this result, we introduce a useful notion of a Levy supervector and prove some of its properties.