Journal of the
Korean Mathematical Society JKMS

For a linear operator $Q$ from $R^d$ into $R^d$ and $0$ $<$ $b$ $<1$, the $(Q,b)$-semi-stability and the strict $(Q, b)$-semi-stability of probability measures on $R^d$ are defined. The $(Q,b)$-semi-stability is an extension of operator stability with exponent $Q$ on one hand and of semi-stability with index $\alpha$ and parameter $b$ on the other. Characterization of strictly $(Q,b)$-semi-stable distributions among $(Q,b)$-semi-stable distributions is made. Existence of $(Q,b)$-semi-stable distributions which are not translation of strictly $(Q,b)$-semi-stable distribution is discussed.

Keywords: infinite divisibility, operator semi-stability, semi-stability, operator stabi1ity

MSC numbers: 6.00E+08