Operator stable laws: Series representations and domains of normal attraction

Series representation for operator semistable laws and domains of normal attraction

“…Note that in contrast to the operator stable case discussed in [6], tile random elements T and L in general are not independent and hence To and L, are not asymptotically independent.…”

“…This condition is, in contrast to the operator stable case discussed in [6], in general not necessary. However, if we consider a subclass of the operator semistable laws, where the function Ko(s) in (2.3) is independent of 0, i.e i, Ke(s) = k(s), it turns out that our condition is also necessary.…”

“…One of the features of LePage's series representation is that the centering terms are obtained from the characteristic functions rather than directly from the summands. Later Hahn et al [6] gave a different series representation with more classical centerings analogous to those in [13] for stable laws on the real line.…”

“…We will now prove a slightly different series representation with more classical centering analogous to those in [6,Theorem 2]. THEOREM 3.6.…”

“…Later, using a series representation, Hahn et al in [6] gave a different characterization for the operator stable case in terms of order statistics relative to the coordinate system defined by the equation x = r(x)B/(z) (see See. 2).…”